THE METAPLECTIC CASE OF THE WEIL-SIEGEL FORMULA by William Jay Sweet, Jr . .. ~ Dissertation submitted to the Faculty of the Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1990 Advisory Committee: Professor Stephen Kudla, Chairman/ Adviser Professor Alex Dragt Professor Rebecca Herb Professor Ronald Lipsman Associate Professor Jeffrey Adams ABSTRACT Title of Dissertation: THE METAPLECTIC CASE OF THE WEIL-SIEGEL FORMULA William Jay Sweet, Jr., Doctor of Philosophy, 1990 Dissertation directed by: Dr. Stephen S. Kudla, Professor, Department of Mathematics The Weil-Siegel formula, in the form developed by Weil, asserts the equality of a special value of an Eisenstein series with the integral of a related theta series. Recently, Kudla and Rallis have extended the formula into the range in which the Eisenstein series fails to converge at the required special value, so that Langlands' meromorphic analytic continuation must be used. In the case addressed by Kudla and Rallis, both the Eisenstein series and the integral of the theta series are automorphic forms on the adelic symplectic group. This thesis concentrates on extending the Weil-Siegel formula in the case in which both functions are automorphic forms on the two-fold metaplectic cover of the adelic symplectic group. First of all, a concrete model of the global metaplectic cover mentioned above is constructed by modifying the local formulas of Rao. Next, the meromorphic analytic continuation of the Eisenstein series is shown to be holomorphic at the special value in question. In the course of this work, we develop the functional equation and find all poles of an interesting family of local zeta-integrals similar to those studied in a paper of Igusa. Finally, the Weil-Siegel formula is proven in many cases by the methods of Kudla and Rallis. ACKNOWLEDGEMENT I would like to express my sincerest gratitude to my advisor, Dr. Stephen Kudla, for all the time and effort he has contributed to my education over the years. His help has been invaluable. Thanks also to my wife, Cathy, for her tolerance during the writing of this thesis, and for her help at all times. Finally, I thank my family for their support in the completion of the un- dergraduate and graduate study which made this thesis possible. 11 TABLE OF CONTENTS Section Page 0. Introduction 1 1. The Contruction of the Metaplectic Group 10 §1.1 The symplectic group 10 §1.2 The Heisenberg group 13 §1.3 Rao's Construction of the local Weil representation, I 16 §1.4 Characters of second degree and the cocycle 19 §1.5 Rao's Construction of the local Weil representation, II 23 §1.6 A renormalization 26 § 1. 7 The global Weil representation 36 §1.8 The local and global metaplectic groups 37 2. Dual Pairs and the Metaplectic group 45 §2.1 Construction of an (O,Sp) dual pair 45 §2.2 The covering of Sp(W) 48 §2.3 The covering of O(V) 55 §2.4 Global coverings 56 3. The Weil-Siegel Formula 61 §3.1 Introduction 61 §3.2 Past work 64 §3.3 Statement of thesis results 64 4. Reduction to the Constant Term 68 §4.1 Notation 68 §4.2 Automorphic forms on metaplectic covers 70 §4.3 The reduction 74 §4.4 The constant term of E and the intertwining operators 80 5. The Local Intertwining Operator: Spherical Sections 86 §5.1 The method of Gindiken and Karpelevich 86 §5.2 Simplification 108 6. The Local Intertwining Operator: Non-Archimedean Places 113 §6.1 Global setup 113 §6.2 Reduction 115 §6.3 Zeta integrals 119 §6.4 Computation of the functional equation 122 lll §6.5 §6.6 Poles of the zeta integral Final results 7. The Local Intertwining Operator: Archimedean Places §7.1 Coverings of the unitary group §7.2 Gaussians §7.3 Bounding the order of poles §7.4 The complex places 8. The Middle Terms §8.1 The intertwining operators §8.2 Degenerate Eisenstein series 9. Bookkeeping: Holomorphy of the Constant Term §9.1 The global intertwining operator §9.2 The middle terms §9.3 Diagrams 10. The Induction: the Diamond Argument §10.1 The unitary case §10.2 The cases m = n + 2 and m = n + 3 §10.3 The cases 3 < m < n + l References IV 135 142 146 146 149 156 166 169 169 173 179 179 182 186 191 191 192 199 207 0. INTRODUCTION The Weil-Siegel formula, in the form developed by Weil in [Wl] and [W2], asserts the equality of a special value of an Eisenstein series with the integral of a related theta series. Recently, Kudla and Rallis [K-Rl], [K-R2] have extended the formula into the range in which the Eisenstein series fails to converge at the required special value, so that Langlands' meromorphic analytic continuation must be used. In the case addressed by Kudla and Rallis, both the Eisenstein series and the integral of the theta series are automorphic forms on the adelic symplectic group. The goal of this thesis is to extend the Weil-Siegel formula in the case in which both functions are automorphic forms on the two-fold metaplectic cover of the adelic symplectic group. In the course of this work, the following problems anse: (1) A concrete model of the metaplectic cover mentioned above must be con- structed. Although Weil develops the abstract existence and uniqueness of such a cover in [Wl], there seem to be no concrete formulas available in the literature for the global 2-cocycle which defines the cover, with the exception of the Sp(l) = SL(2) case (see [G]). (2) The meromorphic analytic continuation of the Eisenstein series must be shown to be holomorphic at the special value in question. (3) The Weil-Siegel formula itself must be shown to hold. 1 This last goal has been only partially fulfilled, in that there are some remaining cases for which the formula should hold, but for which the proof is incomplete. Now we describe the setting of the Weil-Siegel formula in more detail. To begin with, fix a number field k , and write A for the ring of adeles of k . We then consider a symplectic vector space W = k2n with a standard symplectic form <, > , and an m -dimensional k vector space V with a non-degenerate symmetric form ( , ) . Let the automorphism groups of these two spaces be denoted by G = Sp(W) and H = O(V). Tensoring the spaces V and W yields a new symplectic space W with form ~, ~ = ( , ) 0 <, > . We then have an embedding G x H <-4 Sp(W) ~ Sp( mn, k) so that the images of G and H form a dual reductive pair in Sp(W) . Considering the adelic points of these groups, we obtain dual pairs G(A) X H(A) ~ Sp(W)A Given a fixed character 'ljJ of k\A , the oscillator or Weil representation w = W¢ of Sp(W)A is a certain projective representation acting on the space S(V(Ar) of Schwartz-Bruhat functions on V(Ar . This defines a two-fold metaplectic cover - 7r Sp(W)A -+ Sp(W)A ' with w lifting to an honest group representation of Sp(W)A . There is also a unique splitting Sp(W)k ---, Sp(W)A of the k-rational points, and so we 2 identify Sp(W)k with its image. This is developed in a paper of Weil [Wl]. Then 1r-1 (G(A)) forms a two-fold cover of G(A) which is non-trivial if and only if m = dim(V) is odd. We will write w for the oscillator representation restricted to 1r-1 (G(A)). For any subgroup L ~ G(A), we let L denote 1r-1 ( L) in the case where m is odd. Now given a function c.p E S(V(At) , we may consider the theta function O(g, h, c.p) = L w(g) c.p(h- 1 x) for g E 1r-1 ( G(A)), h E H(A). xEV(k)" By [Wl ], this function is left G( k) -invariant. If we integrate out the orthogonal variable, we obtain a function I(g,c.p)= J O(g, h, c.p) dh H(k)\H(A) which converges if either (1) {V,( ,)} is anisotropic, so that H(k)\H(A) is compact, or (2) m - a > n + 1 , where a ~ 1 is the dimension of a maximal isotropic subspace of V . Given these restrictions, equation ( *) above defines an automorphic form on G(k)\G(A), G(k)\G(A), if m is even, or if mis odd. In preparation for defining the Eisenstein series, we take a maximal para- bolic subgroup 3 and write P = M · N , where M = { m(a) = ( a ta-I) E G I a E GL(n)} and N = { n(b) = ( 10n /n) E G I b = tb} · If K is a standard maximal compact subgroup of G(A) , then we have the Iwasawa decompositions G(A) = N(A)·M(A)·K and G(A) = N(A)·M(A)·J( (there is a natural splitting N(A) ~ G(A), so this last is well-defined). Finally, we need the functions on G(A) and G(A) defined by { n m(a) k} g = _ _ f----+ ja(g) j d I det(a)I n m(a) k (adelic abs. value). We may then define an Eisenstein series for a K -finite function <.p E E(g,s,1.p) = (,g,s,<.p)' ,EP(k)\G(k) where g E 1r-1 (G(A)), s EC, and (g, s, <.p) = ja(g )ls-sow(g) 1.p(O) with m n+ l So=----. 2 2 This series converges absolutely for Re( s) > nil , and has a meromorphic continuation to the s -plane, and a functional equation relating values at s to those at -s (see Arthur [A) for the G(A) case, and Morris [M) for the G(A) case). 4 The original Weil-Siegel formula [W2] asserts that as long as m > 2n + 2 (so that s 0 > nil ), we have E(g, S 0 , cp) = J(g, cp) . In other words, this identity holds in all cases in which the Eisenstein series is absolutely convergent at s = s 0 • In two recent papers, [K-Rl] and [K- R2], Kudla and Rallis extend the identity in the case where both sides are automorphic forms on G(A) (i.e., when m = dim(V) is even): THEOREM [K-R2]. Let m be even, and let a be the dimension of a maximal isotropic subspace of V(k). Assume that a= 0 or that m - a> n + I . Then for all K -finite cp E S(V(At) (i) E(g, s, cp) is holomorphic at s = s0 , and (ii) E(g,s 0 ,cp) = K·l(g,cp) for all g E G(k)\G(A), where K = { 1, if m > n + I 2, if m ~ n + I This thesis concerns the case in which m is odd and a = 0 . The main result follows: THEOREM. Let {V, (,)} be an anisotropic symmetric k -vector space of odd dimension m, and let cp E S(V(At) be a K -finite function. Define the constant K by "' = { 1, 2, if m > n + I or m = I if ln+3. If we accept Conjecture 10.2.3 (see chapter 10), then all cases with m = n + 2 and m = n + 3 also hold, with the exception of (m, n) = (7, 4) . The proof begins with a quick reduction to the problem of proving the equality of the constant terms: (the last equality being immediate by interchanging the order of integration). Next, the constant term of the Eisenstein series is easily written as the sum of n + I terms: n-1 Ep(g, s, 'P) = (g, s, 'P) + L E~(g, s, 'P) + M( s )(g, s) ' r=l where the first term will match I p(g, ) is a symplectic vector space of dimension 2n over k , so that <, > is a non-degenerate, skew-symmetric, k -bilinear form on W . The symplectic group Sp(W) is the group of linear automorphisms of W which preserve the form <, > . We will frequently identify Sp(W) with the subgroup Sp( n, k) of G L(2n, k) defined as follows: setting J ( - ~ .. 10n) E GL(2n, k), we get 10 a symplectic form on k 2 n ( viewed as a space of row vectors) via writing tv for the transpose of a vector or matrix. We then define Sp(n, k) to be {g E GL(2n, k) / gJ tg = J} . Letting matrices act on row vectors in k2 n by right multiplication, it is clear that Sp( n, k) is the symplectic group of linear transformations of k2n preserving the form defined by J above. The isomorphism between Sp(W) and Sp(n, k) comes by the choice of a sym- plectic basis for (W, < , >) : that is, a basis { e1 , .•• , en, e;, . .. , e~} chosen such that < ei, e; >= 6ij , and < ei, ej >=< e;, e; >= 0 for all i and j with 1 ~ i, j ~ n . Given our choice of basis above, we will write X = spank { e1, ... , en} and Y = spank{e;, ... , e~}, so that X and Y are maximal isotropic subspaces of W, placed in duality via <, >. We then have W = X EBY. It is necessary to say something here about the notational convention used to represent elements of the group Sp(W). An element g E Sp(W) may be naturally written in block form as g = (; ! ) , where we are thinking of these entries as being on the one hand linear transformations ( acting on the right) a : X ~ X, b : X ~ Y, c : Y ~ X, and d : Y ~ Y. In this guise, we write (x, y) (; !) = (xa + ye, xb + yd) E X EBY for x E X and y E Y . This is thoroughly explained in the beginning of Weil [WI J. On the other hand, in certain contexts it will be very convenient to recall that we have 11 fixed a symplectic basis, and so we may consider a, b, c , and d as n by n matrices. Hopefully this will not cause much confusion; the sliding back and forth between viewpoints should make the statements and proofs to come more concise and easier to follow. Now, tensoring all of the vector spaces above with kv over k , we write Wv = W(kv) = W(k) ®k kv, etc., for the local objects attached to a place v E Ek , where Ek = { all places of k}. Consider a finite place v , and let w: C Wv be the Ov -lattice Then letting Kv C Sp(Wv) be the subgroup of transformations preserving w:, we see that Kv is a maximal compact (open) subgroup of Sp(Wv). As in Tate's thesis [T], the adelized symplectic group Sp(W)A .!!.. Sp(W(A)) is the restricted direct product of the groups {Sp(Wv)}vEEk with respect to the collection {Kv}v ). 13 It is easily seen that the center of 1iv is the set {(O,t) E 1fv It E kv} '.:::: k"/;, so that we may consider 1/)v to be a character of the center in a natural way. Abusing notation a bit, we write 1i(Yv) = {(y, t) E 1iv I y E Yv} , and note that since Yv is isotropic, 1i(Yv) '.:::'. Yv X kv as a group, and 1/)v extends to a character of 1i(Yv) via 1/)v(Y, t) d 'lj;v(t) . N t • d .!, f Ji(Y,) 1{ d fi I d1{(Wv)(•1• ) b h ex , we 1n uce 'f'v rom v to v : e ne n 1i(Yv) 'f'v to et e space of measurable functions f : 1iv ~ C such that (1) f(hg) = 1/)v(h)f(g) for all h E 1i(Yv), g E 1iv, and (2) J Xv IJ(x, 0)1 2 dx < CX) • We let p = P1/Jv denote the representation of 1iv acting on the space above by and one checks that the center kv of 1iv acts by the character 1/)v . In fact, p is an irreducible unitary representation of 1iv , and since it has central character 1/)v , the Stone-Von Neumann theorem guarantees us that it is unique up to unitary equivalence. While the above is a natural development of this representation, a more com- monly used model is the Schrodinger model. Since functions in Ind~~~)) ( 1/)v) are determined by their values on 1i(Yv)\1iv '.:::: {[x, OJ E 1iv Ix E Xv} '.:::: Xv , 14 and noting (2) above in the definition of IndZ[~}(¢v), it is easily seen that the map f f---t {x r-+ f([x,01)} is an isomorphism of Hilbert spaces. Carrying over the 1iv -action, and denot- ing the new representation by U = U t/J., , we obtain the formulas (1.2.1) U([O, tJ))cp(x) U([x 0 ,0J)cp(x) = cp(x + Xo) for

) on X s x Ys , and using the analo- gous notion of Fourier transform. The projective representation r is defined as follows: DEFINITION 1.3.2. (1) For p = ( ~ !) E Pv, and cp E L 2(Xv), we define r(p )cp(x) = I det al½ ¢v( ½ < xa, xb > )cp(xa). (2) For any set S as above, and cp E S(X) (the Schwartz-Bruhat functions on X ), define r(rs)cp(x) = J ¢v(< z,xsr >)cp(xs1 + z)dsz, Xs where we write x E Xv = Xs EB Xs1 as x = xs + xs, , and let This is just a partial Fourier transform of J witl1 17 respect to the vector subspace Xs of Xv . Since S(Xv) is dense in L 2(Xv), this defines r(rs) on all of L2(Xv), as is usual in Fourier analysis. (3) Given an arbitrary g E Gv , choose a decomposition g = p 1 rsp2 as in Lemma 1.3.1, and define Rao proves that this last definition gives a well-defined operator for every element g E Gv , and that these operators r(g) satisfy (1.2.2). Actually, he uses a slightly different Heisenberg group and a different representation U , but his conclusions translate to give the statement above. Rao also gives uniqueness conditions satisfied by r (see Theorem 3.6 [R]) and proves the following useful fact: PROPOSITION 1.3.3. For any arbitrary subset S of {1, 2, ... , n} , let rs(-) denote the standard (projective) Weil representation of Sp(Ws) cor- responding to the data W s = X s EB Ys , the symplectic basis being { e j, e; I j ES} . Let S1, ... Sm be a partition of {1, 2, ... , n} and let g1 E Sp(Ws;) be given. If g = diag(g1, ... , 9m) E Sp(Wv) m then on the space @ S(Xs;), we have i=l 18 m Note that for the finite places, S(Xv) = ® S(Xs;). i=l §1.4 Characters of second degree and the cocycle. The system of operators defined above by r gives only a projective representation because it fails to respect the group law: for any 91 , 92 E Gv , we know only that r(91) o r(92) = c(g1,92)r(9192) for some constant c(91 , 92 ) depending on the 9i . This comes from the fact that the operators on both sides of the equation above satisfy (1.2.2) for 9 = 9192 , which shows that this constant c(g 1 , g2) , called the cocycle defined by r , must in fact take values in the unit circle T . In order to investigate the properties of this cocycle, we must first discuss characters of second degree. This material comes mainly from Weil [Wl]. DEFINITION 1.4.1. Let A be any locally compact abelian group. A con- tinuous function f : A --+ T is called a character of second degree on A if the mapping AxA-T (x,y) 1--+ f(x + y)f(x)- 1 f(y)- 1 is a bicharacter (i.e. a group homomorphism in each variable). Then writing A for the analytic dual group of A and denoting the pairing of x E A with x* E A by [[x, x*]] = [[x*, x]] E T , we see that we have f(x + y)f(x)- 1 J(y)- 1 = [[x,ypl] = [[y,xpl] 19 for some continuous homomorphism p = p(f) : A -+ A . Call p(f) the symmetric morphism associated to f . f is said to be non-degenerate if p(f) is an isomorphism. Given two locally compact abelian groups A and B with fixed Haar mea- sures da and db, respectively, we define the modulus ja j of an isomorphism a: A-+ B by j F(b) db= lal j F(aa) da B A for FE Cc(B) . Now fix a locally compact abelian group A, and choose Haar measures da and da* on A and A , respectively, such that the measures are dual with respect to the Fourier transforms defined by F(F)(a*) = j F(a) [[a,a* ]J da and F(F*)(a) = j F(a*)[[a, a* ]] da* for FE Cc(A) and F* E Cc(.A.) . Then Weil proves that for a non-degenerate character of second degree f on A , there exists a constant ,(!) of absolute value 1, which we will call the Weil index of f, such that I F(cl> * f) = ,(f)jp(f)l-~F(cl>) · g for all cl> E Cc(A) , where g is the character of second degree on A defined by g(a*) = f(a*p - 1 )-1 . Here cl>* f stands for the usual convolution of cl> and J . See section 14 of [Wl] for details. 20 The appendix of Rao's paper (R] introduces a notation for the above which is extremely useful. Let kv be a fixed local field, and let TJv be any non- trivial additive character of kv . Then for any a E k; , we will let aT/v denote the character x 1---+ TJv( ax) . Now define ,v( T/v) to be the Weil index of the character of second degree given by x 1---+ TJv(x 2 ) , and let ( ) de/ ,v(aTJv) 1v a, T/v = ,v(TJv) for any a E k; . The Weil index of such characters is closely related to the Hilbert symbol of kv , which is defined to be (a, b)v de/ { +1 -1 if ax2 + by2 = z2 has a solution (x,y,z) =J (0,0,0) ink~, otherwise. Its most important properties are given in PROPOSITION 1.4.2. The Hilbert symbol (, )v of kv is a non-degenerate bi character of the group k; / ( k; )2 if kv 1- C . In other words, for any a, b, c Ek; , (1) (ab,c)v = (a,c)v(b,c)v, (2) (ac2 ,b) = (a,b), and (3) if a is not a square, then there exists d E k; such that ( a, d)v = -1 . Furthermore, if k is a number field with completion kv at v E ~k , then we have the product formula IT (a, b)v = l vEEk 21 for all a, b E k . See Serre [Se] for details. Now, from the appendix of [R], we have the following properties of the Weil index of a non-trivial character T/v . PROPOSITION 1.4.3 [R]. For all a, b, c E kv, 1v(ac2 ,r,v) = ,v(a,r,v) and the function a 1--t ,v( a, T/v) is a character of second degree on k'; / ( k'; )2 . Moreover, The following properties follow immediately: (1) 1 v(a,cr,v) = (a,c)v1v(a,r,v). (2) 1v(-l, T/v) = 1v(TJv)-2 • (3) 1'v(a,r,v)2 = (-1,a)v = (a,a)v. (4) 1'v(a,r,v)4 = 1 and 1 v(TJv)8 = 1. We will have frequent need for the following facts describing the behavior of both the Hilbert symbol and the Weil index when the local field is non- archimedean and has odd residual characteristic: LEMMA 1.4.4. Let kv be a non-archimedean local field whose residue field has odd characteristic. Then (1) the Hilbert symbol (, )v of kv is trivial on the set Uv x Uv . (2) If the character T/v of k; is trivial on the set P;a for some a E Z , but non-trivial on P;a-l , then the Weil index 11,(x, T/v) will equal 1 22 for all x E Uv . The first fact above is standard: see [Se], for example. The second is con- tained in the appendix of [R]. Given this background, we have the following theorem, which tells us how to compute c( ·, ·) . THEOREM 1.4.5 (RAO). arbitrary. (3) If S1 , S2 , .•. , Sm is a partition of {1, 2, ... , n} and cs(·,·) denotes the cocycle of rs for any S (see Proposition 1.3.3), then m c(g, g') = IT cs; (gj, gj) j=l where g = diag(g1 , ••• , gm) and g' = diag(g~, ... , g~) . ( 4) c( r ( ~ : ) , T) = the Weil index of X ~ 1/'( ½ < X, X · p >) where x EX. §1.5 Rao's construction of the local Weil representation, II. The purpose behind this work with the cocycle is to eventually construct a new group on which the projective representation will be a true group represen- tation. Ignoring for the moment the fact that our cocycle c takes values 23 m µs = { all 8th roots of unity in C} , and supposing it were µ2 -valued, we could define a group with underlying set Sp(Wv) x µ 2 and with group law [g,t] · [g',t'] = [gg',u'c(g,g')]. Then a new representation given by r(g, t )c.p = t · r(g )c.p would satisfy r([g, t][g', t'])c.p = r(gg', u' c(g, g'))c.p = u' c(g, g')r(gg')c.p = u'r(g)r(g')c.p = r(g, t)r(g', t')c.p. Our new group would then be a two fold covering group of Sp(Wv) , which may be either a trivial or a non-trivial covering (may split or not). Now we will write r = rv and c = Cv to indicate the place v . As it turns out, we may renormalize rv so that the cocycle does have values in µ 2 • Weil proved this indirectly in (Wl]. It was done explicitly by Kubota (see (G]) for Sp(l) = SL(2) , and by Rao for the general case. We state Rao's results. LEMMA 1.5.1 (RAO). There exists a unique map gt-+ x(g) of Sp(Wv) into k; / ( k; )2 such that the following properties hold: (1) x(p1gp2) = x(p1)x(g)x(p2) for all P1,P2 E Pv. (2) x( rs) = l for all subsets SC {1, 2, ... , n} . (3) x(p)=det(ply)·(k;)2 for all pEPv. Moreover, such a function is uniquely defined by x(p1 rsp2) = det ( (P1P2 )IY) · ( k: )2. 24 We then define the normalizing constants for 9 E n1 = PvrsPv , with j = ISi . This gives us a new system of operators for 9 E Gv with cocycle £\(·, ·) defined by (1.5.1) It is elementary that cv and Cv have the relationship (1.5.2) where, for any function d : Gv --+ C , we will write 6d( ) - d(91 )d(92) 91, 92 - d( ) · 9192 Equation (1.5.2) above states that the 2-cocycles Cv and Cv are cohomologous ( they differ by a co boundary 6mv ). For more on the relationships between group extensions and cohomology, see Jacobson ([J], pp. 363-369) and Moore [Me]. Rao derives the following explicit formula for Cv : THEOREM 1.5.2 (RAO). For any g1,g2 E Gv, ((-ll, )t.p(x) = t.p(x) t.p(x + x 0 ) = t.p(x) for all t E Ov, for all Yo E Yv0 , and for all Xo E xi. From the definition of "Pv in Tate, we see that the first equation holds auto- matically. Similarly, the second equation tells us that whenever x r/:. xi , we 28 must have cp(x) = 0. The last equation then shows that y,(x) = y,(O) for all x E xi . The conclusion follows. D Now, we see from equation (1.2.2) that taking any fixed g E Kv = stab(W:) C Gv , we must have for all h E 1{~ . In other words, fv(g)cp~ is an 1{~ -fixed vector, and so by the lemma must be some multiple of cp~ . Hence, for v ))dsz x~ 31 = mv(Ts)cp~ 1 s,(xs1 ) j 1Pv(~ < Ziei,Xiei >) II dvzi 0" iES iES tJ = mv(Ts)cp~ 1 s,(xs1 ) II j 1Pv(ZiXi)dvzi. iESo., But now 1Pv is a character and Ov is a compact group, so we may use the basic identity J 1Pv(ab) dvb = { O, meas(Ov) 0., Since we have chosen our character 1Pv nicely (as in Tate) for v 1 , ... , m} vary over all finite subsets of elements in L2 (Xv) · There are two natural maps to be considered here: projection on the first factor gives a two-fold covering homomorphism that we will usually denote by 7r: Mp(Wv) -t Sp(Wv), and projection on the second factor gives a unitary group representation of M p(Wv) . As in [Wl], one may show that 1r is a continuous, open surjection. The map 1r also gives a non-trivial cover if and only if kv -=I= C . Non-trivial here means that there is no continuous group homomorphism s : Sp(Wv) --+ Mp(Wv) satisfying 1r o s = id (such a map would be called a splitting of the covering), or equivalently, that M p(Wv) is not isomorphic as a topological group to Sp(Wv) x Z/ 2Z in a manner compatible with 1r • This is proven 38 abstractly in Weil, and again in Rao using the concrete realization developed there. It is important to note, however, that we do have splittings over certain subgroups of Sp(Wv) . Specifically, if V rt-. sk ' then the map Kv _. Mp(Wv) g i-----t (g,wv(g)) is an isomorphism onto its image, and gives a compact open neighborhood of the identity in M p(Wv) . We may use this to realize the topology on M p(Wv) if we like. Similarly, for the remaining finite places v < oo, v E Sk , there is a compact open subgroup Ki C Kv on which the cocycle f3v is trivial, and for which one may make the analogous statements. Another important splitting, which holds for any v E ~k ( or later globally), concerns the subgroups defined by Nv = {n(b) I b = bt E M(n, kv)} and is again given by n(b) 1-+ (n(b),wv(n(b))). The global metaplectic group Mp(W)A is defined in virtually the same way as the local group: Mp(W)A = { (g, 0 E Sp(W)A x Un(L2 (X(A))) I l = ±w(g)}, 39 with the topology also defined similarly. Projection on the first and second factors defines, respectively, a covering map, which we shall still call 1r : Mp(W)A - Sp(W)A , and a unitary representation of Mp(W),q . Rather than using these rather cumbersome definitions for routine calcula- tions, we will usually identify Mp(Wv) and Mp(W),q with the groups formed from the sets Sp(Wv) x µ 2 and Sp(W),q x µ 2 , respectively, with multipli- cation defined as in § 1.5 via the co cycles f3v and /3 . The isomorphisms are given by Sp(Wv) X µ2 ~ Mp(Wv) (g, t ] t---+ (g, t · Wv(g )) and similarly for the adelic case. These "new" groups we will denote by Sp(Wv) and Sp(W)A , and they will take the topologies pulled back from Mp(Wv) and Mp(W),q . While the Sp definitions are more convenient for algebraic computations, we really need to refer back to the "true" Mp definitions for considerations of topology. The covering maps 1r : Sp(·) - Sp(·) are now given by 1r([g , t]) = g , and the Weil representations by wv([g, t]) = t · wv(g ), and w([g, t]) = t · w(g). If kv ,...., C , then we still define Sp(Wv) as above for pedagogical reasons, but note that the cocycle is trivial, and so Sp(Wv) ~ Sp(Wv) x µ 2 in this case. Although the global metaplectic group Sp(W),q is not a restricted direct product of local groups, there is a local-global relationship which will be im- 40 portant later. Regarding the maximal compact subgroups Kv , v r/. Sk , as actual subgroups of the Sp(Wv) as above (via g 1-+ [g , 1] ), we may form the restricted direct product fl~ Sp(Wv) with respect to the Kv . This then gives us the following commutative diagram: fl~ Sp(Wv) ~ Sp(W)A (1.8.1) \,, l 7r fl~ Sp(Wv) = Sp(W)A The vertical and diagonal maps are the obvious ones, while the horizontal map is given by n~[gv, Ev] ~ [Ilv 9v, nv Ev] . This may be used to relate the representations Wv on Sp(Wv) to the global representation w on Sp(W)A . ,- The group nv Sp(Wv) acts on S(X(A)) by the representation @~Wv . But as we see in the following diagram, fl~ Sp(Wv) _Pl Sp(W)A Un(S(X(A))) /w this map factors through to give w . This allows us, for example, to represent a function such as Sp(W)A - C g f--t w(g)c.p(O) for c.p = ®~'Pv E S(X(A)) , as a tensor product of other functions: w(g)c.p(O) = ®~wv(9v)'Pv(O) for g = p(ITgv)· V Next, we must consider the embedding of the k -rational points in the global metaplectic group. The usual embedding of k '---+ A as a discrete subgroup 41 given by x 1-+ ( x, x, ... ) induces an embedding Sp(W) = Sp(W)k <--+ Sp(W)A ( the subscript k is added for emphasis). One of the most important facts proven in [Wl] (abstractly) is that there is an analogous splitting of Sp(W)A over Sp(W)k . Given the concrete formulas developed above, it is easy to see what this should be. PROPOSITION 1.8.1. There is a. unique homomorphism l Sp(W)k -+ Sp(W)A such that the diagram Sp(W)A l / l 1r Sp(TV)k -t Sp(W)A commutes. It is defined by l(g) = [g, .X(g)J, where .X(g) = flv Av(g). Proof Writing G(k), G(A), and G(A) for the three groups, suppose there are two such homomorphisms: /1 , /2 : G(k)-+ G(A). Then since they are lifts of the bottom mapping in the diagram, they must be of the form li(g) [g,ai(g)J for some functions ai: G(k)-+ µ 2 , i = 1,2. The fact that the li are homomorphisms amounts to saying that ( ') a1 (gg') f3 g, g = ( ) ( a1 g a1 g a2(gg') a2 (g )a2 (g') for all g,g' E G(k). This means that the map G(k) -+ µ 2 given by g 1-+ a1(g)a2(g) is a homo- morphism. But G( k) equals its own commutator subgroup, so this last map 42 must be trivial, showing that a1 = a 2 , and hence that 11 = 12 . This proves umqueness. To prove that the map l given above is in fact a homomorphism, we first need to check that >.(g) = IL >.v(g) is well-defined. Writing g = P1 rsp2 E P(k)rsP(k) , choose a finite set E (with Sk C E C Ek) large enough so that if v (/:. E then p 1 , p 2 E P( k) n K v • By Corollary 1.5.3, we see that V (/:. E then implies that Cv(P1,rsp2) = 1 = Cv(rs,P2)' so that rv(g) = r v(P1 )rv( rs )rv(P2) . By the definition of E , then, each of these three operators fixes .v(g) = 1 for v (/:. E, and so >.(g) is well-defined. We must now note that the cocycles cv are "well-behaved" globally with re- spect to the k -rational points G( k) of G . The claim is that IL cv(91, g2) = 1 for 91,92 E G(k). To prove this, we use Rao's explicit formula for cv(·,·) given in Theorem 1.5.2. First of all, all we need to know about the Leray invari- ant mentioned in the theorem is that it is the isometry class of a certain inner product space, or quadratic module, attached to 91 and 92 • We can realize this concretely as a non-degenerate quadratic form p with coefficients in k (in this case) which is then thought of as acting on various finite-dimensional vector spaces Vv = V 0k kv where v varies over Ek , and V is a fixed k -vector space. Now the product formula for the Hilbert symbol tells us that for any a, b Ek, we have Ilv(a, b)v = 1 . Since the Hasse invariant hv(P) of p may be defined by means of the Hilbert symbol ( as in Serre [Se]), it is also a basic fact that Ilv hv(P) = 1 ( p is defined over k ). This proves our claim 43 about the<\(·,·) . Finally, we show that l(9) = [9,,\(9)] is a homomorphism. For 91 ,92 E G( k) arbitrary, recalling that Av d~ 1 if v E Sk . But this is exactly what we needed to show (perhaps it would look more natural if we had written l(9) = [9, ,\(9)-1] ). D Since this mapping Sp(W)k --+ Sp(W)A is unique, and an injection, we will often ignore it and identify Sp(W)k with its image in Sp(W)A . One of the principal applications of this splitting is in the following theorem, proven by Weil in [Wl) , and stated in this form in [HJ. THEOREM 1.8.2. Let G = Sp(W) as before. There exists a unique (up to a scalar multiple) linear functional 0 on S(X(A)) satisfying 0(U(h)<.p) = 0( <.p) for all h E H( k) . This functional is also G( k) -invariant under the action of the Weil representation of G(A). It is given by the following formula: 0( <.p) = I: <.p( X) for <.p E S(X(A) ). xEX(k) This functional is called the Theta distribution. It will be used to con- struct one of the automorphic forms occurring in the Weil-Siegel formula. 44 2. DUAL PAIRS AND THE METAPLECTIC GROUP §2.1 Construction of an (O,Sp) dual pair. It turns out, as discussed in Howe's article [HJ in the Corvallis proceedings, that we wish to consider not the Weil representation of the metaplectic group Sp , but rather the restriction of the Weil representation to subgroups of this last, and especially to the inverse image under 1r : Sp--+ Sp of "dual reductive pairs" G and H in Sp. At this point, the level of exposition will go up somewhat, and we will assume, for example, such things as the definition of a linear algebraic group. DEFINITION 2.1.1 [HJ. Let L be a linear algebraic group defined over k . A pair ( G, H) of algebraic subgroups of L is called a dual reductive pair if (1) G and H are reductive groups (they are algebraically connected and have trivial unipotent radical), and (2) G is the centralizer of H in L , and vice versa. We will focus on constructing a particular dual reductive pair of type (0, Sp) in a larger symplectic group. Begin by fixing a symplectic vector space (W, <, >) over k as before, with a fixed symplectic basis { e1, . .. , en, e;, . .. , e~} (i.e. W has dimension 2n ). Since we have fixed this basis, we will write G = Sp( n) for the symplectic group, so that G( k) = Sp( n, k) is the group 45 of transformations of W which preserve the form <, > . Now consider a k -vector space V of finite dimension m , furnished with a non-degenerate symmetric bilinear form (,) . Let H = O(V) denote the orthogonal group of transformations of the space (V, (,)) , also writing H( k) to emphasize that we are looking at the k -points of the group. While it is not completely necessary, it will ease the exposition to come if we choose a basis { v1 , .•. , Vm} for V . Identifying V with km (viewed as a space of column vectors), the form (,) gives rise to a matrix Q = ((vi, Vj)), so that (x, y) = txQy and H = O(Q) = {h E GL(m, k) I thQh = Q}. It is worth m entioning that we are following the convention ofletting H act on V on the left, while G acts on W on the right. This seems to be standard practice in the literature, and it would probably be too confusing to attempt to change. As before, vector spaces and groups over the various completions of k and over the adeles will be denoted by Wv, Gv = G(kv), H(A) , etc. Taking the tensor product of V and W gives a new symplectic space W via W = V 0 k W with form <{:'. , » = ( , ) 0 < , >, by which we mean that <{:'. v 0 w, v' 0 w' »= ( v, v') < w, w' > . It is easily checked that this defines a non-degenerate symplectic form on W . We choose a symplectic basis for W by setting m eij = Vi 0 ej and e;j = I)Q-1 )1i(v1 0 e; ) J= l 46 for any i,j with 1 ~ i ~ m and 1 ~ j ~ n . The complete polarization W = X EB Y of W introduced in chapter 1 defines another complete polarization on the new space W via X = V @X and Y = V@ Y. If it were necessary, we could choose an ordering here- say the dictionary order on (i,j). But one can actually show that Rao's formulas for the Weil representation of Sp(Wv) on S(Xv) depend only on the choice of a complete polarization W = X EBY of W , and not on the choice of bases for X and Y . So aside from taking the e;j 's before the eij 's, we leave the ordering unspecified. Remembering that Sp acts on the right, we then have a monomorphism A: H(k) x G(k) - Sp(W) (h,g) 1-t h@g, defined by where the action of h@g on W is given by (v@w)·(h@g) = (h- 1v)@(wg). may check that leaving the ordering of the ( i, j) unspecified, the matrix of A( l v, (: ! ) ) with respect to the { e;j, eij} basis is given in block form by Q®b) l @d E Sp(W). This means, for example, that ! ) ) = L 8;kaj1ek1 + Q;kbj1ek1 ( k,l) = L)l @a)(i,j)(k,l)ekl + (Q@ b)(i,j)(k,l)ekl ( k,l) 47 where (k, l) runs over 1 ~ k ~ m, l ~ l::; n; and (10a)(i,j)(k,/) is meant to indicate the (i,j)th row, (k,l) th column entry of 10a. Similarly, A(h,lw) for h E O(V) will yield a matrix The corresponding local maps on Hv x Gv will be denoted by Av . In any case, this embedding defines a dual reductive pair (H, G) in Sp(W) . §2.2 The covering of Sp(W). For the rest of this section, fix a place v E :Bk , and suppose that our local character ¢v has been chosen as a component of a global character 'lj; of A/ k as in §1.6. We will also suppose that ¢u has been used to construct projective Weil representations on Gu and Sp(W v) , and hence metaplectic covers Gv and Sp(W u) , respectively. First, we focus on n-1 (Av(Gu)) C Sp(W v) (see the diagram below). This 1s a two-fold cover of (a copy of) Gv , and so by a result of Moore [Me], if ku '1- C then it must be either trivial or isomorphic to Gv , the local metaplectic group. In a temporary notation, let Gv be the set Gv x µ2 , and consider the following diagram: Gv Av x l Sp(Wv) l l IT Gu Sp(Wv) Av where the vertical maps are the canonical projections, and the map Au x 1 is given by (g, t:) t-t (Avg, t:) . To distinguish between the various cocyles 48 and other objects attached to Gv and Sp(W v) m chapter 1, we will use (temporarily) for objects belonging to Gv, and for those belonging to Sp(W v ). If we pullback the cocycle Bv to define a group structure on Gv via - d e f the cocycle f3v(91,g2) = Bv(Avg1,Avg2) for 9i E Gv, then Av X 1 clearly defines an injective homomorphism giving Gv ~ rr - 1 (Av(Gv)). The question of whether rr - 1 (Av(Gv)) is a trivial cover may then be answered by examining whether f3v is cohomologically trivial or homologous to f3v . Using Rao's explicit formula for cv (Theorem 1.5.2) Kudla has calculated the following: LEMMA 2.2.l[K2]. With notation as above, where dv(g) denotes the function on Gu given by . m(m-1) ili.=..U . dv(g)=(x(g),detV):;11+1(x(g),-l)v 2 (detV,-l)v 2 hv(V)1 for g E D1 C Gv . Since the proof involves a rather long and technical computation, it will be omitted. From this lemma, we immediately obtain 49 LEMMA 2.2.2. For any 91, g2 E Gv , Hence if kv ';/. C , then the cover Gv -+ Gv defined above is trivial if and only if m is even. If kv "' C , then the cover is trivial in any case. Relating this to the two standard covers of Gv gives the following: PROPOSITION 2.2.3. Suppose that kv 1 C . Then there exists a unique lifting Av of Av to an embedding of either Gv X µ2 (if m is even) or Gv (if m is odd) in Sp(W v) such that the following diagrams commute: Gv X µ2 A., Sp(Wv) Gv A., Sp(Wv) 1 1 1 1 Gv Sp(Wv) Gv Sp(Wv) A., A., This is defined in either case by Av(g, €) = (Av(g), € · bv(g)) where bv(g) = Av(Av(g)) · ..\v(g)m · dv(g). If kv ~ C, then the same definitions work, but Proof First we show uniqueness, treating the two cases together. Suppose At and A; are two liftings as above. Then taking (l~ )-lo (A;) , we obtain an automorphism of the group Gv x µ 2 , where multiplication is defined either using /Jv or not depending on the parity of m . In either case, one shows that such an automorphism must be of the form (g, €) 1-+ (g, € · J(g)) for some group homomorphism f : Gv -+ µ 2 • But Gv is simple, and so J must be identically 1. 50 The proof that the Av defined above is in fact a homomorphism is an easy consequence of the preceding lemma. D The final task to be accomplished in this section is to write down formulas for the pullback from Sp(W v) to Gv of the Weil representation of the first group. We will only do this for the odd m case, as this is the situation which will concern us for the rest of the paper. The formulas for even m involve only a slight modification, and may be easily derived from the work in this section. Using Wv to denote the Weil representation of the group Sp(W v) , we make a small change in the way the vector space W v , and hence the space L 2 (Xv) , are realized, as is customary in the literature of dual reductive pairs. By definition, Xv = Vv@ Xv and Yv = Vv ® Yv . Since we have fixed bases {e1, ... ,en} and {e;, ... ,e~} of X and Y respectively, we consider the isomorphisms n x = (x1, ... ,xn) f--+ LXi ® ei i=l n Y = (YI,···,Yn) t---t LYi ®e( i=l This realization gives W v ""' vvnEB vvn . One may then check that the symplectic form ~ , » is given now by ~ [x,y],[x',y'J »= tr((x,y')-(x',y)) where 51 and (x, y') stands for the n x n matrix of inner products (xi, Yj) . When dealing with dual pairs, we will consider the Weil representation Wv of Sp(W v) to act on the space L 2(Vt) . A word on Haar measures: given any symplectic space (U, <, >) over kv and additive character 1Pv of kv , there is a natural isomorphism U ~ U defined by u 1--4 1Pv( < ·, u > ). As m §1.2, this isomorphism determines a unique self-dual Haar measure du on U . If we choose a symplectic basis { c: 1 , ... , er, c:;, ... , c:;} for U , then for u = I:~=l aiC:i + bic:; , it is easy to see that this measure is given by du = da1 ... dar db1 ... dbr , where dai and dbi are the Haar measures on kv chosen in §1.2 . On the other hand, our particular symplectic space W v ~ Vvn EB Vvn inherits another measure from Vv : dz is uniquely determined by Vv ~ 11v , z 1--4 1Pv(( ·, z )) , and this yields a measure d[x, y] on W v via d[x, y] = dx dy = dx1 ... dxn dy1 ... dyn, where X = (xl,··· ,xn), y = (YI,·· ·,Yn) E vvn· Since we have changed our representation space to L2 (Vvn) , it is certainly more natural to use the measure dx = dx 1 ••• dxn in our formulas. Changing notation from that defined in Chapter 1, let 52 ·•,, ' ::·:: p ~ • ' ., • Also write ..=--:-·---·.A. -:--..:. -."":~":..-----::- "'.::"':"- ~ :,~--~ _--?:;- --~-- -~--==-=::- -===--~~-~ Pv = { ( a !) E Gv} = MvNv, where Mv = { m(a) = ( a a) I a E GL(n, kv)}, Nv= {n(b)= ( 1 ~) lb= tbEM(n, kv)}. In the following lemma, and throughout, we will frequently change a function c : k ; - C into a function on GL(n, kv) (without mentioning this explicitly) by composition with determinant, writing c(a) def c(det a) for a E GL(n, kv) · LEMM A 2.2.4. Let 1r : Gv - Gv be the usual covering map, and for any s ubgroup LC Gv, write L = 1r-1 (L). Then (1) multiplication in Mv is given by (2) the formula ([ () l) _t(a, (-l)~det(V))v x v m a , E - ( 1 .,. ) 'Yv a, 2 'f' v defines a character Xv : Mv - µ4 . Proof. These facts are immediate from Corollary 1.5.3, and §1.4 . 0 PROPOSITION 2.2.5. Let m = dimk(V) be odd, and fix notation as above. Then the pullback wv to Gv of the W eil representation associated to Sp(W v) , 53 acting in the space L 2 (Vvn) , is determined by the following formulas. (1) H g E Mv and 1r(g) = m(a), then for c.p E L 2(V:), wv(g)c.p(x) = Xv(g) lalj c.p(xa), (2) H n(b) E Nv ~ Gv (see §1.8 regarding the splitting over Nv ), then wv(n(b))c.p(x) = 1Pv(½tr(b(x,x))) · c.p(x), where (x, x) stands for the n xn matrix of V -inner products (xi, x1 ). (3) For S C {1, 2, ... , n} with ISi = j , write vvn = vvs EB v/' , so that X E vvn decomposes as X = Xs + X5 1 • Then for c.p E S(vvn) ' wv([rs, l])c.p(x) = ,v(V)-j j 7Pv(-tr(xs, z))c.p(xs1 + z) dsz. V.,5 Here, tr(xs, z) stands for the trace of the j x j matrix with entries (xi,z1) for i,l ES; dsz = ITiESdzi isaproductofself-dualmeasures on Vv as above; and ,v(V) is the Weil index of the map z ~ 7Pv( ½(z, z )). As before, this definition of wv([rs, 1]) extends to define an operator 54 §2.3 The covering of O(V). Although we will be less concerned with the action of rr- 1(Av(Hv)) on L 2(Vvn), it is still necessary to compute it explicitly once and for all. As in the previous section, let H v = Hv x µ2 , and consider the diagram A., Xl Sp(Wv) l l A., From §2.1, recall that Av(Hv) C Pv , the maximal parabolic of Sp(W v) sta- bilizing Y v . This makes the computations much easier. The cocycle Bv of Sp(W v) , when pulled back to H v , gives just = ( det( h1 ), det( h2) )~ Hence rr - 1 (Av(Hv)) is always a trivial cover when n is even, and is trivial in any case for finite places v not dividing 2 (recall that h E Hv =} det h = ±l , and v f 2 :::} (, )v = 1 on Uv x Uv ), or for real places. The action of (h, €) E H v on a function ip E L2 (Vvn) is computed to be Kudla's paper [Kl], rather than dealing with this non-standard 2-fold covering of Hv , it is traditional to twist the representation above by the character x~(h, €) = €1 v(det h, ½¢v) in the case where n is odd. So we will take the 55 action of Hv on L 2 (Vvn) to be just the left action given by: Note that this action still commutes with the action of Gv on L 2(V:). §2.4 Global coverings. Let m = dim V remain odd throughout this section. We wish to reproduce the local results of the last two sections for the global groups G(A) and H(A) . In other words, lift A: G(A)-+ Sp(W)A A: G(A) - Sp(W)A, to a map realize similarly a model for rr- 1(Av(H(A))), and write down formulas for the pullback of w (defined on Sp(W)A) to G(A) and H(A). For the first of these tasks, consider the diagram 0 (2.4.1) G(A) A Sp(W)A where a is the product of the maps Av , and the vertical maps are given by diagram (1.8.1). If we show that the map a is in fact well-defined (maps into the restricted product), then it is trivial to verify that there exists a unique monomorphism A causing the diagram to commute. As the entire diagram lies over A=ll Av G(A) - -- Sp(W)A, 56 this will yield the desired lifting of A . The condition on a amounts to asking that Av : Gv _. Sp(W v) map one standard maximal compact Kv C Gv into the other Sp(mn, Ov) c Sp(W v) for almost all v E 'Ek . The following lemma shows that this is in fact the case. LEMMA 2.4.1. Let v: be the Ov-latticegivenby spanov{v1,···,vn}. Writing ( Vv0 ) * = { v E Vv I ( v, w) E O v for all w E Vv0 } , define S k, V = sk u {v E 'Ek I (Vu")*=/:- Vv0 }. Then for all places V (/:. sk,V' (1) Av(Kv) C Sp(mn, Ov), and (2) the function Dv : Gv _. µ 2 is identically 1 on Kv . With the identifications Kv C Gv and Sp(mn, Ov) C Sp(Wv) devel- oped in §1.8, this proves that Av(Kv) C Sp(mn, Ov) for v (/:. Sk,V , since Av([g, El)= [Av(g), EOv(g)]. Thus, we have proven most of PROPOSITION 2.4.2. Let m = dim V be odd. Then there exists a unique monomorphism A which makes the following diagram commute: G(A) A Sp(W)A ~1 1 n A G(A) -- Sp(W)A· It is defined by A([g,E]) = [A(g),E8(g)], where 8(g) = Tiv8v(g). A also preserves the k -rational points: in other words A( G( k)) C Sp(W)k . Proof Existence and uniqueness have already been proven. That the for- mula for A is as given is an easy consequence of the commutativity of diagram 57 (2.4.1). To show that A(G(k)) c Sp(W)k, note that g E G(k) embeds as [g, >.(g)] , and we have A([g, >.(g)]) = [A(g), >.(g)8(g)] = [A(g) , A(A(g))]. This follows from the fact that Tiv dv(g) = 1 (see Lemma 2.2.1). D As preparation for giving the formulas for the pullback of the Weil repre- sentation, we need to describe the two-fold cover of M(A) ~ GL(n, A) and a certain character of the cover. LEMMA 2.4.3. As in Lemma 2.2.4, write 7r : G(A) - G(A) for the cover- ing homomorphism, and L = 7r- 1 (L) for a subgroup LC G(A). (1) Multiplication in M(A) is given by where we write (a,f3)k = TI)av,f3v)v for the product of all the local Hilbert symbols, given a= (av), /3 = (f3v) EA. (2) The unique embedding G(k) <----.t G(A) restricts to M(k)--+ M(A) m(a) t-----t [m(a), 1]. (3) For all finite places v not dividing 2, the subgroup Mi = { m( a) E Mv I a E GL(n, Ov)} of Mv also embeds in Mv via g 1--+ [g , 1] . 58 .=------ ·- ""'"': ";.-<:"~:".- ,.:;;"::- .~~~~ ~ ..,;r - ~ J .. -:;;,;;.--:-~-~-...,_,. -·_:~~~ ( 4) Tbe tensor product of all the local characters ®vXv defines a character of the restricted direct product ft Mv with respect to the Mi . This character factors through the product map It Mv --+ M(A) to give a character X: M(A)-T satisfying 1:(a,(-1)¥ det(V))k x([m(a), t:]) = ,(a, ½7P) Here ,(btp) stands for the Weil index of the map A --+ T given by x f-t 7P(bx2 ): it satisfies ,(btp) = ITv ,v(bv1Pv). Global properties of , a.re explained thoroughly in section 30 of {Wl }. A consequence of the discussion there is that x(g) = 1 for all g E M ( k) , so that X is actually a character of M(A)/M(k). Now, since the representation @~Wv: rr: Sp(Wv)--+ Un(L2 (V(A)n)) fac- tors through to w: Sp(W)A-+ Un(S(V(At)) (see §1.8), we see from diagram (2.4.1) above that the formulas for w def w o A: G(A) - Un(S(V(At)) result immediately from the local formulas given in Proposition 2.2.5. PROPOSITION 2.4.4. Let m be odd, and consider a. function c.p E S(V(At) . Then the representation w of G(A) defined above is determined by the following formulas. Notation follows closely that of Proposition 2.2.5. (1) If g E M(A) and 1r(g) = m(a), a E GL(n, A) , w(g)c.p(x) = x(g) laJ.rq.c.p(xa). 59 (2) If n(b) E N(A) ,_ G(A), then w(n(b))cp(x) = v,(½tr(b(x,x))) · cp(x). (3) For SC {1,2, ... ,n} with j = ISi, Ts E G(k) C G(A) acts via w( Ts)cp(x) = ,(v)-i j v,(-tr(xs, z) )c.p(xs, + z) dsz V(A) 5 Here, we note that Ts E G(k) embeds as [Ts, l] E G(A), and the measure dzi on V(A) is the product (as defined in Tate) of the local measures on the Vv . In addition, it is the unique self-dual measure on V(A) with respect to the identification V(A) ~ V(A),,__ given by z rt v,((·, z )). As in §2.3, one checks that the pullback of the cocycle B = flv Bv on Sp(W)A to the group H(A) is given by and H(A) acts on L2 (V(A?) via (w o A)(h, t)cp(x) = t,((det h)n, ½1f )-1c.p(h-1x). Note that H(k) ,_ H(A) via h rt [h, 1], and if n is odd, we may twist the representation above by the character x': H(A)/H(k) - T defined via x'([h,t]) = t,(deth,½v,). Hence, as in the local case, we will use the H(A) action defined simply by w(h)cp(x) def cp(h-1 x). 60 3. THE WEIL-SIEGEL FORMULA §3.1 Introduction. The Weil-Siegel formula asserts that a special value of an Eisenstein series equals the integral of a related theta series, where both of these are automorphic forms on a two-fold cover of Sp(n,A). In this chapter, we describe the general problem, state the cases in which the formula is known, and give the main result of this thesis. Let all notation be as in the preceding chapter. Then suppressing the map A , we have a dual pair H(A) x G(A) Sp(W)A which induces a representation of H(A) x rr-1 (G(A)) in the space S(V(A?) . For the purposes of this introduction, we make no assumptions about the parity of m = dim(V). If m is odd, then rr- 1 (G(A)) ~ G(A), and as before, we write L for the inverse image under 1r : G(A) -+ G(A) of any subgroup LC G(A). In the even case, rr- 1 (G(A)) is a trivial extension of G(A), and all of our formulas reduce to formulas on G(A) . First, recall the theta distribution 0 E Homsp(Wh(S(V(At), 1) from Theorem 1.8.2. Given a function cp E S(V(At) , we may consider the theta 61 function B(g,h,c.p) de/ 0(w(g,h)c.p) = L w(g)c.p(h-1 x), xEV(k)" for h E H(A), { G(A), if 2 Im, g E - G(A), if 2 f m. Since the k -points of G(A) and H(A) are mapped into Sp(W)k (see Propo- sition 2.4.2), this function is left G( k) and H( k) invariant. If we integrate out the orthogonal variable, we obtain a function (3.1.1) I(g, c.p) = J B(g, h, c.p) dh H(k)\H(A) which converges if either (1) {V( k ), (,)} is anisotropic, or (2) m - a > n + 1 , where a 2'. 1 is the dimension of a maximal isotropic subspace of V . The first of these is a consequence of reduction theory, which implies that if V(k) is anisotropic, then H(k)\H(A) is compact. The second is proven in Weil's paper [W2). In either case, the Haar measure dh is normalized so that H(k)\H(A) has volume one. Given these restrictions, equation (3.1.1) above defines an automorphic form on G(k)\G(A), G(k)\G(A), if m is even, or if mis odd. 62 -- - ~~- -~- · -= ...&_~-:--;---... - ...;:;.._._,.,,. · ... ~ - ~-;'"::P.::?:--~:;;r - ~..,- -:;:._ --- · :;-:-- _ -- - - -- :.-.=.-..!.:- - Next, m order to define the Eisenstein series, we consider the maximal parabolic P = M · N as before. Choose standard maximal compact subgroups of the various Gv by taking Kv to be Sp(n, Ov) for v < oo, Sp(n, R) n 0(2n) ,..._, U(n) for kv ~ R, and Sp(n, C) n U(2n) for kv ~ C. Writing K = Ilv Kv C G(A) , the local Iwasawa decompositions give rise to global decompositions G(A) = N(A) · M(A). K, and G(A) = N(A) · M(A) · K. Here, as always, we identify N(A) c.......+ G(A) as a subgroup. Now define func- tions on G(A) and G(A) (respectively) by g = { n :) : } 1----t la(g)I !.. I det(a)I (adelic abs. value). nm(a) k Note that while the element a(g) E G L( n, A) is defined only as a coset of a maximal compact of GL(n, A), the adelic absolute value of this i3 well-defined. Finally, for s E C , s0 = s0 (m, n) = '; - ntl , and a J{ -finite (respectively, K -finite) function <..p E S(V(Ar), we define the section (g,s,c.p) = la(g)ls-sow(g)c.p(O) and hence an Eisenstein series E(g,s,c.p)= ~ ('yg,s,c.p) for gE{g~:~}· ,EP(k)\G(k) By either the standard theory of Eisenstein series (see Arthur [Al) or a meta- plectic extension of this theory (see Morris [Ml), this series converges absolutely for Re( s) > nil , and has a meromorphic continuation to the complex s -plane and a functional equation relating values at s to those at -s . 63 §3.2 Past work. The original Weil-Siegel formula [W2] asserts that as long as m > 2n + 2 (so that s - .!!! - .!!.±l > .!!.±l ) we have o - 2 2 2 ' E(g,s 0 ,c.p) = I(g,c.p). In other words, this identity holds in all cases in which the Eisenstein series is absolutely convergent at s = 80 • In two recent papers, [K-Rl] and [K- R2], Kudla and Rallis extend the identity in the case where both sides are automorphic forms on G(A) (i.e., when m = dim(V) is even): THEOREM 3.2.1. [K-R2} Let m be even, and let a be the dimension of a maximal isotropic subspace of V . Assume that a = 0 or that m- a > n + 1 . Then for all K -finite c.p E S(V(At) (1) E(g,s,c.p) is holomorphic at s = sa(m,n), and (2) E(g,s 0 ,c.p) = K • I(g,c.p) for all g E G(k)\G(A), where K = { 1, 2, if m > n + 1, if m::;n+l. §3.3 Statement of thesis results. This paper will be concerned with the case where m is odd and a= O. The main result is as follows: THEOREM 3.3.1. Let {V, (,)} be an anisotropic symmetric k -vector space of odd dimension m , and let c.p E S(V(A)n) be a K -finite function. Define the constant K by K = { 1, 2, if m > n + 1 or m = 1 if 1 < m ::; n + 1. 64 Then the Eisenstein series E(g, s, c.p) is holomorphic at s = So for all pairs ( m, n) , and the equality E(g, s0 ,c.p) = K • l(g,c.p) , g E G(k)\G(A) holds in the following cases: (i) m= 1, (ii) m = 3, n = l or 2 ' (iii) 3n+3. If we accept Conjecture 10.2.3 (see chapter 10), then all cases with m = n + 2 and m = n + 3 also hold, with the exception of (m, n) = (7, 4) . We give a brief sketch of the proof, which follows along the lines of that in [K-Rl] . For the remainder of the paper, let the hypotheses be as given in Theorem 3.3.1 above. Noting that (N(k)\N(A)r ~ N(k) via 1/Jr(n(b)) = tf;(tr(bT)) for n(b) E N(A), n(T) E N(k) , an automorphic form f on G(k)\G(A) has Fourier coefficients (with respect to P = NM ) given by fr(g) = j J(ng)t/J-r(n) dn. N(k)\N(A) 65 ,, : ~ ,, .,· ' Here dn is normalized to give vol(N(k)\N(A)) = 1 . With this definition . . - th m mmd, define the constant term of f with respect to P to be the 0 Fourier coefficient of f , and denote this by l-j; . The proof begins with a quick reduction to the problem of proving ( 1) that Ep(g, s, c.p) is holomorphic at s = s0 , and (2) that the constant terms satisfy But the constant term of I is easily computed: writing c.p' = w(g)c.p , Ip(g,c.p) = J J L w(ng)c.p(h- 1x)dhdn N(k)\N(A.) H(k)\H(A.) xEV(k)n - J J L ¢(½tr(b(h- 1 x,h- 1 x)))c.p'(h-1 x)dhdn(b) N(k)\N(A) H(k)\H(A) xEV(k)n - j L ( j 1/;(½tr(b((x, x)))dn(b)) c.p'(h-1x) dh H(k)\H(A.) xEV(k)n N(k)\N(A) = c.p'(O) = w(g)c.p(O) = cI>(g,s 0 ,c.p) smce n(b) 1---+ 1/;( ½tr(bT)) , TE N(k) , is a non-trivial character of the compact group N(k)\N(A) if and only if T = 0. Next , the constant term of E is easily written as the sum of n + l terms: n - 1 Ep(g, s, c.p) = cI>(g, s, c.p) + L E;;(g, s, 'P) + M( s )cI>(g, s) ' r=l where the first term will match I p(g, c.p) at s = s0 , the middle n-1 terms re- strict todegenerateEisensteinserieson M(A) ~ atwo-foldcoverof GL(n,A), 66 and where M(s)(g,s) = j (( ~l ~) xg,s)dx N(A) is a G(A) -intertwining operator from a certain induced representation space to another. One then shows that the meromorphic continuations of all of these terms are finite at s 0 , and that all but the first term either vanish or cancel each other at s 0 , or in the cases 1 < m ~ n + l , that exactly one of them survives to match the first term (g, s0 , c.p) , giving the constant "' = 2 of the theorem. 67 4. REDUCTION TO THE CONSTANT TERM §4.1 Notation. For the remainder of this paper, let the hypotheses be as given in Theorem 3.3.1. As the title indicates, this chapter will be concerned with the reduction of our problem to proving the holomorphicity at so of Ep(s), and the equality of the constant terms of E(s 0 ) and I. Although the proof basically parallels that given in sections 2 and 3 of [K-Rl] , there are differences which arise due to the fact that we are working with automorphic forms on a metaplectic cover of G(A) = Sp(n, A) . First of all, we establish notation, repeating some definitions from [K-Rl]. Fix the Borel subgroup B of G given by B = { ( a ! ) E GI a is upper triangular in GL(n) } . Then a Levi decomposition of B is given by B = T t>< NB , where is a maximal torus of G , and 1 * * 0 1 1 0 EG * 1 is the unipotent radical of B . The standard parabolics of G are defined to be those algebraic subgroups of G containing B . They are parameterized 68 by sequences s = (s 1 , . .. ,sr) of positive integers with Isl= :Esi ~ n (see [Kl]). For 1 ~ s ~ n , define Ys = span{ ei', ... , e:} . Then the parabolic Ps associated to s is defined to be the stabilizer of the flag of isotropic subspaces of W given by The maximal (proper) parabolics among these are determined by a sequence of length 1: for any integer r , 1 ~ r ~ n , let Pr be the maximal parabolic stabilizing Yr . Pr then has unipotent radical given by N,={ C and we may take as Levi factor X ln- r z ) } E G , ln-r M,={ C: h :) EG}~GL(r)xSp(n-r) Note that the parabolic used to define the Eisenstein senes E , previously denoted by P , is now P n . Defining w j setting N:: r = Nn n Nr and ' we also observe that Tj-l (see Lemma 1.6.2) and -i P N N" Wn-r n Wn-r n n = n r ' 69 Finally, denote by a subgroup such that N = N' . N" n r n,r · §4.2 Automorphic forms on metaplectic covers. In this section we briefly describe the elements of the theory of automorphic forms which will be needed in this chapter, and the manner in which the theory for ordinary adele groups generalizes to the metaplectic situation. This last follows Morris [M]. The essential features possessed by the finite central covering 1r : G(A) - G(A) which allow the theory to carry over are the following: (1) The extension must split over the k -rational points G(k) of G(A) . For any standard parabolic subgroup P = M · N C G , (2) there exists a splitting over N(A) which is natural in the sense that if P => Q are parabolics with NQ => Np , then the splittings are compatible. (3) P(A) = Norm- (N(A)) G(A) . (4) There is a positive integer l such that (Z(M(A)) 1)"' C Z(A1(A)) Property (I) was developed fully in Proposition 1.8.1. The next two properties hinge on the following: any standard parabolic P = M · N => B = T · NB 70 satisfies NB =:) N . But the map u ~ [u, 1] defines a splitting, since N 8 c Pn and we have f3(u1,u2) = (x(u1),x(u2))k = ( l , l)k = 1 for all u 1 ,u2 E N 8 . Hence the splitting over NB(A) is compa tible with (and defines by restriction) splittings over Np for any P, showing (2) . Property (3) follows from an easy cocycle computation, given that P (A) = NormG(A)(N(A)), and that x(g) = 1 for any g E N(A). Finally, (4) is a r esult of the fact that z E Z(M(A)) =} f3( z, g) = (x(z),x(g))k = /3(g, z) for all 9 E G(A) (hence l = 1 suffices). From the discussion above we see that the definition of the constant term ' of an automorphic form made in §3.3 for the parabolic i\ remains perfectly va lid for an arbitrary standard parabolic f, = M · N , although the interpre- tation as a Fourier coefficient only holds for Pn (since Nn is abelian). Note that G itself may b e considered a parabolic wit h trivial unipotent radical, so that fc = f. DEFINITION 4.2.1. L et f : G(A) -+ C be an automorphic form as de- scribed in Borel and Jacquet [B-J}. Then f is called a cusp form if lj; = 0 for all proper parabolics P(A) C G(A) . Now, if f is an automorphic form on G(A) =:) P (A) = N(A)M(A) , then m 1--+ ! p( mg) defines an automorphic form on M( A) . Note that if f is 71 right K -finite, then for finitely many g E G(A) , the functions mi-+ fp(mg) determine fp (by the Iwasawa decomposition). Since any automorphic form on M(A) is Z(M(A)) -finite, we may write J-(mg) = ~ f- (mg) p L- P,( ( where ( ranges over a finite set of characters of Z(M(k))\Z(M(A)), and f P,( is the component of fp transforming with central character ( . DEFINITION 4.2.2. We say that f is negligible along the parabolic P = N · M if, for all g E G(A), f P,/·g) is perpendicular to all cusp forms a on M(A) with central character ( . For a given a, this means that J fp (mg)a(m)dm=O. ,( Z( M(A))·M(k)\M(A) LEMMA 4.2.3. If an automorphic form f on G(A) is negligible along P for all parabolic subgroups j5 of G(A), including G(A) itself, then f 1s identically zero. Note. This lemma is mentioned (in the real group situation) in Harish- Chandra [HC]. It derives from the decomposition of L 2 (G(k)\G(A)) given m section 2 of Morris [M], which parallels that in Langlands [L2). Similarly to the non-metaplectic case, we say that parabolics P and Q are associate if their Levi factors Mp(A) and MQ(A) are conjugate by an element of G( k) . Let { P} denote the equivalence class of associates of P . 72 DEFINITION 4.2.4. A form f is said to be concentrated on {P} (or more briefly, on P ) if f is negligible along all parabolics Q such that Q ik+z > · · · > in . Then the element w E fl corresponding to S is given by Xk I-+ Xik Xk+I I-+ -Xik+l which may be realized as a matrix in G( k) with entries equal to ±1 or O . Proof To see that the w above satisfy ( 4.3.1 ), consider the following. An element w E We acts on a root a E e to give w · a E c via t w·a d,J (w- 1 tw)a .ror t E T. . d b th t N 1, NB 1s generate y e roo groups a , where a ranges over ~ . But wNaw-1 = Nw,a , and so it is an easy check that each of the w ED above does satisfy Nw·a n Pn C Ns for all a E ~. On the other hand, multiplying one of the w E fl on the left by a non-trivial element of WM" '.:::: Sn amounts to introducing a permutation of the Xi 's. The check mentioned above will show that any such permutation would destroy property (4.3.1 ). O We now have enough information to prove the following basic fact. 75 PROPOSITION 4.3.2. E(g, s, ) is concentrated on the Borel subgroup B . Proof The proof is expanded from that m [K-RlJ, and modified to our situation. First of all, we show that E is negligible along G. E transforms ac- cording to the central character e given by e([±l2n, c]) = c for [±l2n, c} E Z def Z(G(A)). Suppose then that J is a cusp form on G(A) with central character e . By a standard unfolding argument, < E,J >G(A) J E(g)f(g) dg Z·G(k)\G(A) j L (,g,s)!c,g)dg Z·G(k)\G(A) -yEPn(k)\G(k) J (g )f(g) dg Z·Pn(k)\G(A) J J (g1g)f(g1g)dg1dg ZNn(A)Mn (k)\G(A) ZNn(k)Mn (k)\ZNn(A)Mn(k) J (g) j f(ng)dndg ZNn(A)Mn(k)\G(A) Nn(k)\Nn(A) J ) = j L 4>(1ng,s)dn N(k)\N(A) "YEPn(k)\G(k) = L j L 4>(w1ng,s)dn. w N(k)\N(A) "YEPn(w- 1 Pnw)\P Now defining M:}, = Mn w- 1 Pnw and N:}, = N n w- 1 Pnw , we see that the sum splits up into a sum over the k -points of M:},\M and N~\N . So we have Ep(g,s,4>) = L L j 4>(wn11 g,s)dn. w "Yt EM::,\M N::,(k)\N(A) Thus far, we have only used the fact that P =MI>< N , and that the change of variable n' = ,n,-1 on N(A) has modulus one for I E M(k). Next, note that the lemma above gives wN~(A)w- 1 = wN(A)w- 1 n Pn(A) C N B(A), all taking place in G(A) . But we must check that this also works in G(A) , given the embeddings N:},(A) C N(A) C N8 (A) ~ G(A), and w E G(k) ~ G( A) . In other words, writing for n" E N:},(A) , we must check that t: = 1 . Letting p = wn"w - 1 E N B(A) 77 (by the lemma), c = .X(w).X(w-1 ),B(w,n11 ),B(wn11 ,w-1 ) _ '( )'( - 1)A(W)A(n") A(Wn11 )A(w-l) rr- ( 11)- ( 11 -1) -AWAW .X(wnll) .X(p) CvW,n CvWn,w V ( 4.3.2) = II ( X ( W), X ( n 11 )) v Cv (pw, w - l) = II Cv ( w, w - l ) = 1 V V using Corollary 1.5.3, Definition 1.6.7, and the proof of Proposition 1.8.1, to- gether with the fact that x(p) = 1 for all p E NB(A) C Pn(A). Having proven that wN~(A)w-1 C N 8 (A) in G(A), and given that (g,s) is left NB(A) -invariant, we may write ( 4.3.3) Ep(g,s,)=L L w(,1g,s) where w "fiEM~\M w(g,s)= j N~(A)\N(A) Now, we note that for m E M::i(A), g E G(A) , ( wng, s) dn. M::i\M~(A) . Setting U(A) deJ M(A) n w- 1 N 8 (A)w, T(A) · U(A) is a Borel m M(A) , and T(A) · U(A) c M::,(A) c M(A) . Hence M::,(A) is a parabolic of M(A) . Note also that Z(M(A)) C T(A) C M::,(A) , so the function m 1--t w(mg) has central character µwlz(M(A)). If M::, = M, then the term in Ep( mg, s, ) corresponding to w is a constant times a character of 78 I , ., ' ,;•.1: .. . . , ' .... , ·,• M(A) , and hence is perpendicular to cusp forms. If M::, ~ M, then the term corresponding to w is an Eisenstein series on M(A), and so by an argument analogous to that given at the beginning of the proof, it is perpendicular to all cusp forms. This proves that E is negligible along P for B ~ P ~ G , and completes the proof that E is concentrated on the Borel. O The same short proof used for Corollary 3.4 of [K-Rl] goes over verbatim to give us: COROLLARY4.3.3. E(g,s,) and Ep)g,s,) ha.vethesamesetofpoles with the same orders. The proof that I(g, 'P) is also concentrated on B parallels that given in section 2 of [K-Rl] (for m even) so closely that there is little point in repeating it here. The only computation which needs some comment is the derivation (in Lemma 2.4) of the expression for the Fourier coefficient E(J(g, s, ), where /3 E Symn ( k) . A cocycle calculation is required at one point in the proof of that lemma, but this amounts to virtually the same check as that performed in equation ( 4.3.2) in Proposition 4.3.2 above. We shall therefore take it for granted that I is concentrated on B . Summarizing the results of the last two sections, we have: THEOREM 4.3.4. Subject to the conditions stated in Theorem 3.3.1, sup- pose that (1) Ep"(g,s,cp) isholomorphicat s=s0 (m,n),a.nd 79 Then Theorem 3.3.1 is proven. In other words, (1) and (2) hold without the P n subscript. §4.4 The constant term of E and the intertwining operators. For the next few chapters, we will study the analytic properties of the constant term of E(g, s, ) with respect to i\, and an intertwining operator which appears in it as a summand. Setting P = Pn in the proof of Proposition 4.3.2, note that { Wj lJ=o gives a different set of coset representatives for W Mn\ W/W Mn than used there, but one which still satisfies Wn-rN~,r(A)w;:~r C Ns(A) in G(A). Hence equation ( 4.3.3) remains valid and results in n ( 4.4.1) EPJg,s,) = LEk (g,s,) r=O where E,;__ (g,s,) = """" r(,g,s) and Pn L.- -yEQr \Mn r(g,s) = j (wrng,s)dn. N~_r(A) We note that since N~ = l, N~ = Nn , and Qo = Qn = Mn , we have E'L = and E'!!:..(g,s,)=n(g,s,)= J (wnng,s)dn. Pn Pn 80 For the middle n - l terms, however, the summation over Qr \Mn does not disappear, and these terms may be regarded as Eisenstein series on Mn (A) ~ GL(n, A) . They will be studied in chapter 8. Now we focus on the term 4>n . For convenience, until further notice we write P = MN in place of Pn = MnNn, and let w = Wn. Notice that for n E N(A), m E M(A) , and g E G(A) , cl> satisfies ( 4.4.2) cI>(nmg,s,cp) = x(m)la(m) ls+p"4>(g,s,c.p), where we let Pn = nil . In addition, if cp = 0 ~'Pv E S(V(Ar) is factorizable, then we may write ( 4.4.3) cI>(g,s,cp) = Q9'cI>v(gv,s,cpv), V choosing 9v E Gv such that p(IJv 9v) = g and setting Recall the discussion of p: rr: Gv---+ G(A) in §1.8. We develop this more generally. Let µ : M (A) ---+ C x be a continuous quasi-character which is genuine (i.e. is not the pullback via ?T of a character of M(A) ). Pull back µ to µ: IJ: Mv ---+ C and write µ = 0 ~µv for quasi- characters µv of Mv , as in Tate [T]. These µv are then genuine, since µ was. So it is apparent that µ(m) = ITv µv(mv) for any choice of mv E Mv with p(IJv mv) = m . Rather than keeping track, we will write µ = ®~µv , regarding both sides as functions on M(A) . 81 Now, for each place v E Ek , let I(µv) = { 4>v : Gv -+ C smooth / 4>v(nmg) = µv(m)/a(m)/~"cl>v(g) Vn E Nv, m E Mv}, . l d the usual notion where "smooth" means locally constant at the firute Paces, an · f Gv, where at the archimedean places. This defines a group representat10n ° the action is by right translation on functions. DEFINITION 4.4.1. We will call a character µv : Mv-+ T unramified if (1) v is a place of k at which Kv <-+ Gv , and If v is a place at which µv is unramified, then we may define a spherical vector 4>~ E J(µv) by 4>~(k) = 1 for all k E Kv <-+ Gv. This allows us to define the global induced G(A) -space f(µ) =®I f(µv), V by taking the restricted tensor product with respect to the vectors { cl>~} . Since the characters µ and µv were genuine, we may identify the resulting space of functions, nominally defined on rr: Gv , with a space of smooth functions 4> : G(A)--), C satisfying 4>(nmg) = µ(m)/a(m)/P" cI>(g) for n E N(A), m E M(A). 82 j .... , ',,..,: , .. . ' . ' Returning now to our situation, ( 4.4.2) and ( 4.4.3) above, together with the fact that cpv(s, cp~) = cp~(s) E !(xvii~) for any place v tt, Sk,V (see Lemma 2.4.1), lead us to conclude that = J P(A)nK\K A section ( s) is a K -finite section if 8<1> def spanc{k · 1 , ... , cI>r} for 8<1> and writing the functions ai : C ---t C are bolomorphic (respectively meromorphic). Note that all of the above is also valid in the local situation, with Iv( s) replacing I( s) , K v replacing K , and so forth. We will denote the set of K -finite sections of I by I K , and the subspace of K -finite vectors of I( s) by I(s)K. It is clear then that cI>(c.p) E IK for rp E S(V(At)K. Now, for Re( s) > Pn , the formula M( s )cI>(g, s) = j cl>( wng, s) dn N(A.) defines a G(A) -intertwining operator M(s): I(s)i< - I(-s)K. The usual theory of Eisenstein series, extended in Morris [M], tells us that the intertwining operator above has a meromorphic continuation. In other 84 words, given any holomorphic section iP( s) E J( s) K , M( s )iP( s) will be a meromorphic section taking values in I(-s) for any s E C ,...., { the pole set} . Now, the measure dn on N(A) which gives vol(N(k)\N(A)) = 1 equals Tii~i dbii for n = n(b) , where dbii is the measure on A constructed in Tate's thesis. As explained there, we may also write dn = Tiv dnv , so that for Re( s) > Pn and factorizable q, E J K , we have M(s) Pn, so that M(s) = @'Mv(s) V with the identification explained above. To begin to see where the poles of the operator M( s) lie, we must first determine what happens to the standard spherical sections iP~(s) E Iv(s). 85 5. THE LOCAL INTERTWINING OPERATOR: SPHERICAL SECTIONS §5.1 The method of Gindiken and Karpelevich. For the remainder of the chapter, we fix a place v ~ Sk for which Xv is unramified, and frequently omit the subscript v. Since it is easily seen that M(s) is unramified, ) 0 () av(n,s)a:.o( ) Mv(s O} where we set ,\* = - I:~=l x; . For any similarly defined half-space R , let 'En be the set of negative roots lying in R , and :Et be the set of positive roots in R (the closure of R ). Let N(R) equal the subgroup of Ns ~ Gv generated by the root groups N °' , where -a ranges over :ER . We then set - 1 -N(R) = [w, l]N(R)[w, 1] - c Gv . This is different from the situation in [Ll ], in that Langlands has no covering, and so he lets N(R) be the group generated by the roots in :En . Note that 87 conjugation by w changes the sign of all roots, so that 1r(N(R)) is exactly Langlands' group. With our initial choice R. , it is easily seen that and N(R.) = Nn, so that (5.1.1) Z ( S) = J cJ> 0 ( X, S) dx. N(R.) The idea of the Gindiken-Karpelevich method is to "remove" one root at a time from ~ R. by changing R. , and in so doing, to write Z ( s) as a product of one-dimensional integrals which are easily evaluated. As developed in Langlands, the method proceeds as follows. Given an open half-space R ( at some intermediate step) determined by ..\o E IJR , R = { v E IJR I ..\o( v) > O}, another element ,\1 E IJR is found which satisfies the following properties: (1) 88 ( a will always stand for a root) . (2) There exists a unique negative root -a0 such that A1 (-ao) = 0 , and so ER= {-ao} 11 {a< 0 I A1(a) > O}. Now, defining S = { V E l)R / A1 ( V) > O} , and taking Es and Et as before, note that Although it isn't explicitly noted in Langlands, one may also show easily that (5.1.2) and So beginning with R* ER- and E+R we obtain a succession of half-spaces ' . ' . ' R*, ... ,R,S, ... , each time removing a negative root - ao from ER to get Es ' and adding ao to Eh to get Et . Specializing this to our situation, we find that there is some order to the removal of the roots ( which will be needed later). 89 LEMMA 5.1.2. At some stage in the above process, suppose we are removing -ao from ~R with A1 (l) If -ao = - 2xr, then we have already removed all roots -(xi+ xr) with 1 :=:; i < r. (2) If -ao = -(xr + x.,) with r < s , then we have already removed all roots of the form -(xi + x.,) with l :=:; i < r, and -(Xi + X r) with l :=:; i < s. Proof. First of all, note that for all i with 1 :=:; i :=:; n - I , Xi - Xi+i E ~t C ~t , as ~t only grows at each step. So writing Ai = I:~=i aixi uniquely for ai E R , we have Ai (xi - Xi+i) > 0 for all i :=:; n - l , which says that (5.1.3) ai > a2 > · · · > an Now suppose that -a0 = -2xr is being removed by Ai , so that Ai (ao) = -4ar = 0 . If 1 :=:; i < r , then by equation (5.1.3). This says that -(xi + Xr), which was in En. , is no longer in ~R = { a < 0 I Ai (a) 2:: O} , and hence must have been removed at an earlier step. 90 Suppose that -ao = -(xr + x.,), (r < s), is being removed by ,\ 1 . Then >.i(-(xr + x.,)) = 0 ===> ar =-a.,. First, let 1 ~ i < r. Then since ai > ar by (5.1.3), and so -( Xi + x.,) has been removed. Finally, we consider -(xi+ Xr) with 1 ~ i < s. Since ½..\1(-(xi + xr)) = -ai _ ar, we have i < s ==} ai > a., = -ar ==> 0 > -ai - ar , and we are done. D This will be useful in the following sense. If [ ( ~ ~) , E] represents a variable in N ( S) , then we have the following diagrams for B = t B from the lemma: r r s 0 0 0 r 0 0 r 0 0 0 s O 0 (5.1.4) Oo = Xr + Xs In other words, if a r/:. ~s (if it has been removed), then the entry bij = bji of B corresponding to a must be zero. By induction, it is also easy to see that any entry lying above or to the left of a zero entry is also zero. So at any stage in the process, B representing N(S) has zeros above a stepped line proceeding from lower left to upper right. 91 One further bit of notation. Set ryz = spanz { Oj I 1 $ i $ n} where the { ai} are the simple roots as before, and define A : ryz -+ Hom(T, k;) in the obvious way: for t = diag( t 1 , • .• , tn, t11 , • •• , t;; 1 ) E T , write for all i, and extend via A(I: aixi) = L aiA(xi) , using additive exponential notation. A defines a map into Hom(T, RX) by further applying llv . We then tensor ryz and Hom(T, Rx) with C over the integers to obtain a map which associates to every µ E rye a mapping m Hom(T, C X) . So µ = L CjXi gives ltA(µ) Iv d.;j n;=l lti ,~i for Cj E C . This also extends trivially to Tv by first applying 7r to (t, t:) E Tv . In order to apply the method, it will be convenient to realize 0 E I(xvldet 1S) as lying in Ind~"(x vA(µ)). 1n other words, findµ E rye B,, such that t E Tv, n E NB, and g E Gv will give q,o(ntg) = x (t)jtA(µ+pn),O a . This requires that I det(t)js+Pn = jtA(µ+pn )I. An easy check shows that PB = I:;=I (n-i + l)xi , and so we need (s + nf) I::;=l xi = µ + L~1 (n - i + l)xi , which yields n µ = L)s - Pn + i)xi. i=I 92 ...... - -.-:• --.:-~.~ r;- -:---:;:->-__ ~-r~ r-r .. _:~~--~~ 1::.7,.~_,;:.. - ;,:=. ;. • ._ -- - ··.;. - ~.;.;...:...:::.- - -·--.;..s.&.::::,..;:::,.~-"'9-"-:,,'- Now · 'we write N° C Nn for the one-parameter subgroup generated by the root vector Xa 0 E g , and define N° d:_! [w, l ]N°[w, 1]- 1 C Gv, Then at any stage with ~R = {-ao} 11 ~s as above, we clearly have N(R) = No -- . N ( S) · Define an embedding and 'Pao tp00 : SL(2)v --+ Gv by ( exp(a) 1r, ( l 0) = exp(xX-a0 ) rao X 1 ) = exp(aaX) exp(-a) for x, a E kv , and a small. Now j cf> 0 (n,s)dn= j j 4> 0 (n2n1,s)dn2dn1, N(R) N° N(S) Where We take rii(x) de/ [w,l][cpao (~ ~x) ,l][w,1]-1, x E kv, as a variable of· integration for N° . Note that if x E Ov , then ( 1 0) - n1(x)= ['Pao X l ,l] EIo i's r1'ght K O h th h d since v -invariant. n t e o er an , We rnay Write (5,1.5) n1 (x) = n1 (x )a1 (x )k1 (x) 93 , ,, ,,,. ,: , ' ., •'' , , ' ;I .1, with ( ) { [ cp ( l X - l ) 1 1] for X O o>O This is done by Langlands on the Gv -level, but in our situation we must check that the cocycles work out correctly. This is essential, because although 0 is 94 left NB -invariant, it is certainly not invariant by NB on the left. If we show that we can write then it is easily seen that n2 1-+ n; is a bijection of the set N(S) , with a change of variables given by dn2 = II 1a1(-a) I dn; .X1(a)O as in Langlands. Hence, we will have (5.1.7) J 4> 0 (fi,s)dfi= J J 4> 0 (a1,s)4> 0 (al1nlJn,2nlal,S)dfi2dX N(R) kv N(S) = (14> 0 (a 1(x),s) II la1(-a)Jdx) ( J

.(n'(B))c(wn(-B),w-1 ). As befo re, We have taken n(b) = (~ :) and n'(b) = (l ~) for b= tb E M(n) · On th e Gv -level ' "(nJ'n,n,) ~ G -;E) G ~) (~ x:) = c-;EB ;x;:::) 97 '' I" ' ;i·· ,, , We wish to write this as p · 7r(n;) for p E N+(S) and n; E N(S), which we know is possible by Langlands' work. To see what is needed, let 7r(n;) = n'(B) and consider that if and only if B = (1 + xBE)B . Taking the transpose and rearranging, this amounts to requiring that B = B(l + xEB)-1 • We need the following lemma: LEMMA 5.1.5. With notation as above, EB has zeros on and below the diagonal. It satisfi.es (EB)2 = 0 in case I, and (EB) 3 = 0 in case II. Proof In case I, E = Err , and so (EB)ij = 0 unless i = r . But also (EB)rj = brj = bjr = 0 if r;::: j by Lemma 5.1.2. This shows the first claim. Next, (EBE)ij = L CjkbktC[j k,l is clearly zero for (i,j) i= (r,r). But (EBE)rr = brr= 0 by reference to diagram (5.1.4). Hence (EB) 2 = 0. In case 11, E = Ers + Esr , and we see easily that (EB)rj = bsj = 0 for r ;::: j and (EB)sj = brj = 0 for s ;::: j by Lemma 5.1.2 again. The other entries are zero also, showing that (EB)ij = 0 for i ;::: j . Now (EBE)ij = 0 98 ·- - - - ---- . ·--'---·· ·-·. __ . ____ .,.,. ___ - Unless i,j E {r,s} · We then check using diagram (5.1.4) that (EBE)rs = L erkbklels =bar= 0, k,I (EBE)ss = brr = 0, but (EBE)rr = bas =/. 0 in general. So EBE== b E ss rr in this case. But also k,I for all J ' and so we see that (EBE)(BEB) = (EB)3 = 0. 0 Now · , since (1 + N)-l = 1 _ N + Nz _ N3 + - . . . for N nilpotent, the le1n- -·~ila and th d' 'f e 1scussion preceding it imply that 1 we set - { B-xBEB B- B - xBEB + x 2 BEBEE in case II in case I _ (1-xEB+x 2(EB) 2 -x2EBE) ._ P- o 1 +xBE lies in N+(S) C NB, and satisfies 1r(n;1n2 n1) = p · 1r(n;) = P · n'(B) · It is an easy ch k ( h B has zero entries ec and will be seen in the work to come) t at Wherever B - )) does, or in other words, that 1r(n;) E 1r(N(S · F'ina11 aniounts t (5.1.8) Y, We must show that the cocycle computation works out, which 0 checking that [n11 , l][n'(B), cz(B)][ni, 1] = [p, l][n'(B), cz(B)] E Gv gg ,, I' .I €2 = .X( n'(B))c( wn( -B), w- 1 ) , and similarly for with (B) €2(B) . Equation (S.l.S) holds if and only if /J(n-;- 1 , n'(B))/3(n11n'(B), n1)€2(B) = f3(p, n'(B))cz(B) {=::} .X( n-;l.X( n' (B)) .X( n-;1n'(B) )A( ni) A( n'(B) )c( wn(-B), w-1) .X(n1 1 n'(B)) A(n11n'(B)n1) A( )A( I - = P n (B)) '( '(B))-( (-B) -1) .X(pn'(B)) A n c wn , w {=::} c(wn(-B),w- 1 ) = c(wn(-B),w-1) {=::} c(wn(-B),w-1) = c(wn(-B),w-1 ). van a e of integration represent mg n2 E N ( S) , we may change Since B is a . bl . - B for -B T . · he work we have done so far gives us: LEMMA 5 1 ·t · .6. With notation as above, we may wn e "'' Ith p E N+(s) and n; E N(S) if and only if, for all B representing n -2 E N(S) and all x-1 (/: Ov , we have "Where _ { B + xB EB in case I B = B + xBEB + x2 BEBEE in case II. Notice th t .f c(wn(B), w-1) is the Weil index a 1 B is non-degenerate, then of x N .,. · 11 amounts to 'f-'v( ½ < x, xB >) ' x E X(kv) . So the Lemma essentia y 100 proving that B and B have the same Weil indices. We shall prove this by showing that B and B are equivalent, under the GL(n) action Br-+ AB t A, to matrices having the same Weil indices. First of all, if n = l , then we already have, a one-dimensional integral, and the Lemma is not needed. Next, suppose that n > l , and that we have already removed all the roots -2x1 , -(x1 +x2 ), ••. , -(x1 +xn) in previous steps. Then B has b11 = b12 = b21 = · · · = b1n = bnl = 0, and the non-zero portion of B sits in a smaller dimensional block. As we must be removing a0 = -(xi +xi) with 1 < i::; j, we are completely reduced to a lower dimensional situation. So by induction, we have only two remaining cases to consider: first of all, we may suppose that -a0 = -( x 1 + x n) is the root currently being removed. Then as above, we know that bll = · · · = b1n = 0 · Hence (BEB)ij = I: bikek1b1j = bi1 bnj + binblj = 0, k,l smce bi1 = b1i = 0 = b1i , and so B = B in this case. Otherwise, we may assume that -(x1 + Xn) E :Es , so that except for a set of measure zero (in the space of integration of B ), b1 n -=I- 0 -=I- bnn . Of course, since we are removing some root -a0 , we know that b11 = 0, as -2x1 is the first root to go. We will describe a transformation taking B r-+ diag( x, y, z) , where x, z E k; and y is an (n - 2) x (n - 2) matrix (absent for n = 2 ). 101 Let A be the unipotent n x n matrix with ph row J (o 0 1 0 for 1::; j:::; n - 1, and with n th row (0, ... ,0,1). Let C = ABtA. A simple computation leads to: LEMMA 5.1.7. (1) Cij = bij - \nnb~n for 1 ~ i,j ~ n- l, and specificaJ]y Cll = -~::) 2 (2) Cnj = 0 for 1 ~ j ~ n - l . (3) Cnn = bnn · Next, let A' have pt row (1, 0, ... , 0) and Ph row J 0 1 0 o) for 2::; j :::; n - l, and n th row (0, ... ,0, 1) (here we use bin=/= 0 ). Set D = A'CtA'. LEMMA 5.1.8. (1) d _ -(b1n) 2 ll - . bnn (2) d1j = 0 for 2~j~n - l . (3) din= 0 for l~i~n-1. (4) dnn = bnn · 102 . -~. ;.. . ..:.. _;:::::__ .: -__ . -- Proof dij = :Z:kl aikCklaj1 for all i, j (omitting the ' on a). So = C11 = Next, 2 :S j :S n - l =;, for 1 :S z < n - l ) and so this 1s O ==} (3). dnn = I: ankCklanl Cnn = bnn ==} (4). Finally, 2 :S i,j :Sn -1 =? dij C1iC1j = b·. _ (btb;n )- (.=b.n_) (b . _ !i.rJ!in.) (b . _ binb;n) = b·. -(b;nbjn) + C11 IJ nn ~,;- }I bnn lJ bnn I) bnn bnn e~tlj) - bjn ( ~) - bin ( ~) + Ci;;nb~n) ==} (5). 0 Now we may relabel this last matrix, letting D denote the (n-2) x (n-2) matrix with entries labelled dij, 2 :S i, j :S n - l . Then we see that (A' A)B t(A' A)= diag(-ibi n), D, bnn)· nn Our goal is to show that when we similarly transform B to obtain D , we have D = D (the other two entries will not matter). Unfortunately, this only holds for ao = x 1 + x j, l :S j :S n - l . If ao = Xr + Xs, l < r :S s , then we must first apply a permutation matrix to B to switch the pt and r th row and column. Let P be the n x n permutation matrix representing the transposition (1, r) E S11 , so that n P = E1r + Er1 + L Eii· 103 i=2 i;i!:r Note that diagram (5.1.4) shows that the upper left-hand r x r block of B consists of zeros. Setting B' = PB t P , we then see that (1) b~j = brj for 1::;j::;n, (2) b~j = b1j for 1::;j::;n, (3) b~j = bij in the case that either l 0 (a1(x),s) II jaf(-o) jdx. k .X1(o)O As in Langlands, this reduces to kv with µ = L: 1 ( s - Pn + i)x i • Simplifying the expression above yields: LEMMA 5.2.1. Let qv = IOv/Pvl and define (v(s) d~ (1 - q;;s) - 1 for s E C and V ¢:. sk . Then Z ( ) = (v(2( S - Pn) + i + j) 00 8 (v(2(s-pn)+i+j+l) 108 where ao = Xi+ xi for 1 ::::; i ::::; j ::::; n . Proof Although the answer looks uniform with respect to the two types of root ao , the computations are different. In case I, with ao = 2xi , we have ( ) { [cp00 ( x - i ) , 1] = [m(t), 1], for X = 0~ v E I K , we have M(s)(g,s) = Q9'Mv(s)v(gv,s) V for p(Ilv 9v) = g E G(A) . Let S ::> sk be a finite set of places of k such that v = ~ for v r/:. S. With the results of the last chapter, we may then write M(s)(s) = ( II av(n,sj) · (@Mv(s)v(s)) @s(-s) bv(n,s 5 v 'I. S vE where s = ® ~ V 't. 5 as in [K-Rl]. We use the following notation for the zeta function of the field k. DEFINITION 6 .1.1. For any place v E ~k , define the local zeta function (v(s) ofthefield k via ifv 1. Thus normalized, the global zeta function (( s) has the "nice" functional equation (( s) = (( 1 - s) . Extend the local functions av and bv defined as in Theorem 5.1.1 to global functions via a( n, s) = flv av( n, s) , and similarly for b( n, s) . These functions then have well known poles and at least partially known zeros. This suggests that we write (6.1.1) M( s )(s) = a(n, 8 ) (@ bv(n, 8 ) Mv(s )v(s )) 0 s( - s ). b(n,s) 5 av(n,s) vE In this chapter, we will find the poles of the operator bv ( n,s) M ( s) for av(n,s) v v < oo . In fact, the following theorem will be proven. THEOREM 6.1.2. Let v be a finite place of k. For any holomorphic section v E J Kv , is also holomorphic. Moreover, for any given value s 0 , there exists a section 4>v such that the expression above does not vanish at So . An analogue of this theorem is proven in [PS-R2], but in that situation the function av(n, s) (and hence the operator Mv(s)) has poles which are more or less shifted by ½ unit on the real axis, reflecting some inherent difference between the "half-integral weight" and "integral weight" situations (in reference to classical Siegel modular forms). 114 §6.2 Reduction. For the rest of the chapter, fix a place v < oo . The proof begins with a reduction of the analytic properties of the intertwining operator to those of a certain zeta function similar to those studied by Igusa in [Il]. We begin with a result from [PS-R2]: LEMMA 6.2.1 [PS-R2]. Let A be the set of holomorphic sections P E IKv which have support(4>(s)) C Pvwi\ for all s E C . Then the analytic properties of the family of functions {Mv(s)4>(w,s) I 4> EA} coincide with the analytic properties of the family of sections { Mv(s)4>v(s) I 4>v E IKv' holomorphic}. In other words, if the operator Mv( s) has a pole of a certain order, then this pole will occur in a function s t--t M v ( s )P( w, s) for some section P E A . The proof of this lemma given in [PS-R2] goes over almost unchanged, noting that PvwPv = PvwNv in Gv , and that in the statement above as well as in the proof, w = [w, 1] E Gv . This now allows the problem to be reduced to studying a concrete class of zeta functions , rather than the more difficult operator A1v( s) . LEMMA 6.2.2. All sections in A may be written as finite linear combina- tions with holomorphic coefficients of sections of the form 4>(n1mwn(b),s) = la(m)ls+p"Xv(m)(b) 115 for n1, n(b) E Nv and m E Mv, where E S(Symn(kv)) is a Schwartz- Brubat function on Symn ( kv) which is independent of s . Sketch of proof. First note that E A has support which may be written as supp((s)) = PwC8 where C 8 is a compact open subset of N which may possibly depend on s. Next, by K -finiteness, it is possible to prove that there exist compact open subgroups C :::) C' (independent of s ) such that (1) (s) is C'-invariantontherightforall s,and (2) supp( ( s)) C PwC for all s . Let ting Ag, be the set of all such sections, we see that A = Uc,C' Ag, . For each s , we also have a map A-+ S(Symn(kv)) t-+ ( s) where (b,s) def (wn(b),s) , which clearly takes Ag, to a similarly defined space S(Symn(kv))g, which is finite dimensional. It is then easy to show the claim. D Now fix a function E S(Symn(kv)) and let be the section in A associated to x t--+ ( - x) as above. Let Symn ( kv) x be the set of symmetric matrices with non-zero determinant. 116 LEMMA 6.2 .3. For Re(_s) > Pn ' Mv(s)IP(w,s) = ((-l)\(-l)(m-1)/2det(V))v. zF(s ,I,.) rv((-l)n,17) ,'f'' Where zF(s, ,I,.) de/ I 'f' I ( X) dx' Symn(k.,) arid where F S : Ymn ( kv) x -+ T is a function given by F(x) = (det(x),~)v hv(x). t'v(det(x), z¢v) liere 8 ( :::: -l)(m+I)/2 det(V), and hv(x) is the Hasse invariant of x viewed as a d qua ratic form on k: . Proof For Re(s) > Pn, Mv(s)IP(w,s) = J IP(wn(x)w,s)dx. N Symn(k.,) ote that th e support condition on cp means that we are actually integrating 1r(wn(x)w) = ( ~l ~l) E fln = PvwNv {=:? det(x)-/- 0. Nowb .. y Writing (-1 0) = (-x-1 1) ( O 1) (1 -x-1), X -l Q -X -1 0 0 1 Wes ee that wn(x )w = p(x)wn(-x-1), where p(x) = ( (-~-i ~x) '€(x)) ~(x) . Is defined by ,Bv(w,n(x)) ,Bv(wn(x),w) = €(x) f3v(p,w) f3v(pw,n(-x- 1 )). 117 and I, . )• ,, :,,I (writing p for 1r(p(x)) ). We must simplify this last equality. It reduces quickly to E(x) = cv(wn(x),w) cv(p,w). Now for ease of notation, write 17 = ½1v (in Rao's notation, this takes Y H '!fv ( ½ y) ) and let 17 o x stand for the character of second degree z H 'T/( < z,z · x >) defined on Xv= spank.,(e1 , ... ,en). Then we see that - ( ( ) ) mv(wn(x)) mv(w) ( ( ) ) Cv Wn X , W = --'--:........:....:.....____;~ Cv Tn X T mv( wn(x )w) ' _ bv((-l)n, 77)-l 'Yv(17) - n]2 1 (ry O x) - / v ( det ( X), 'T/ )- 1 / v ( 'T/ ) - n v = 'Yv('T/)-n 'Yv((-lf, ry)-2 'Yv(det(x), TJ) 'Yv('T/ Ox) and also that cv(p,w) = (det(-x),(-lt)v, giving us Here, we use an identity for hv(x) from the appendix of [R]. Next, we note that the measure dx x def I det(:) !Pn is invariant under the action x H ax ta of a E GL(n, kv) , and furthermore, that it remains unchanged under the change 118 of variable x f--+ x- 1 • We then have Mv(s)(w,s) = J l-x-1 j8 +Pnxv(m(-x-1 ),t(x)) (x-1 )dx Symn(k.,) J lxl-s xv(m(-x- 1 ),t(x)) (x- 1 )dxx Symn(k.,) J lx lsxv(m(-x),t(x- 1 )) (x)dxx. Symn(k.,) Since x = x(x- 1 ) tx , we see that x and x-1 are in the same GL(n) orbit, and so hv(x) = hv(x-1 ), giving t:(x- 1 ) = t(x). Also note that ,v(det(-x),77) = ((-It,det(x))v ,v((-lr ,11) ,v(det(x),77). Hence Mv( s )( w, s) J I ls-pn (det(x),( -l)n+ 1 ) ., h.,(x) (det(-x),(-l)(m-t)/ 2 det(V))., "'(x) dx X (det(x),( -l)n) ., 1.,((-l)n,f/),.,(det(x),'7) 'f' as claimed. D _ ((-1)",(-l)(x)dx §6.3 Zeta integrals. So the problem of determining the poles of the op- erator M v ( s) for v < oo is reduced to finding the poles of the zeta integral zF(s,) = J lxl S-PnF(x)(x)dx Symn(k.,) with F as in Lemma 6.2.3. This is closely related to a family of zeta integrals studied in detail by lgusa in [11]. 119 Letting Y = Symn(kv) x and G = GL(n , kv) for the remainder of this section, we will write g[d] = gdtg for g E G, d E Y. This gives a group action of G on Y which splits up Y into a finite number of disjoint open orbits: h Y= II~ i=l where, for each i , we may choose d; E Y diagonal such that G[d;] = Y; . Suppose, as in Igusa, that c: k; --+ T is a continuous character, and set Zf(s , Pn to a holomorphic function of s ( actually to a rational function in q-s ) which has a meromorphic continuation to the s -plane, and satisfies a system of functional equations: h Zf(s, J) = L ,fj(s)Zf\Pn - s, . Here the Fourier trans- form ¢ of if> is defined using the character T 1--t 'IPv(tr(T)) of Symn(kv) . In our situation, F : Y --+ T is not induced from a character of k; , but does have the following nice properties: (1) F(gdtg) = F(d) for any g E G, d E Y, (2) F ( i ~) = F(A)F(B) for symmetric, non-degenerate matrices A and B, and 120 (3) in dimension 1, F is a character of second-degree on k; /(k ; )2 • Nevertheless, it is easy to see that Z F still satisfies a functional equation: we have Z{(s, ) = F(di) J lxl s-p"(x)dx Y, by (1) above, and so zF(s,J) = LF(di) Zl(s,J) I ij = LF(di) F(dj)- 1 ,ij(s) Z[(Pn -s,). ij We then define meromorphic functions cdi (s) = L F( di) F( dj)- 1 ,ij(s) I so that Z F has the functional equation h (6.3.1) F ~ ~ F z (s,) = ~Cdj(s) zj (Pn -s,). j=l In order to determine the poles of Z F ( s) ( applied to arbitrary ), it will be necessary first to explicitly solve for the factors cd( s) . In [PS-R2], the analogous factors appearing in the functional equation are cj( s) = Ei 'Yij( s) · Although a formula for these functions is stated in the appendix to section 4 of that paper, no derivation is given there. It should be noted that the method presented in the next section also allows the functional equation in [PS-R2] to be computed with very little additional effort. 121 §6.4 Computation of the functional equation. Since we will compute cd( s) by induction on n , it will often be denoted by c( n, s, d) to emphasize the dimension. We will actually be interested only in the poles and zeros of cd( s) , so we will feel free to discard any multiples of holomorphic non-vanishing functions, denoting this by cd( s) = ( .. . ) . We begin with the following com- putation: PROPOSITION 6.4.1. Taking all notation as m the previous section, for Re(s) > Pn we have c(n,s,d) = jdj 8 F(d)- 1 j lxls-p"F(x)1j,(tr(xd))dx, C where C = any open compact subset of Symn(kv) such that supp(Jd) CC, and where N - bi (assuming that N 1 2 N ), with the same conclusion. Next, by similar reasoning, we see that for any fixed ( i , j) with 1 ~ i < j ~ n - 1 , So this proves that (B, c) EA<==> ordv(c) 2 max{N, N1 - 2ordv(Bi) I 1 ~ i ~ n - 1} , which proves the lemma. D 127 _ __ __,_ . ...,_ ...... _ _ ... _ .. !:,aL- - - ~--~-~-...,,~ ~-- Using the lemma then, we see that (6.4.5) I( n-1, s, d,, d2 ) = j icl'+l!!.;-!J F( c) ,t,( cd2 ) D, [ J 1j;( cdii) BJ) dBj] dx c, Pt' J- cB 2 EP:'1 J writing d1 = diag(dP), ... , din-I)) . For our next step, we must recognize the integral in brackets as a Weil index. LEMMA 6.4.4. For any non-archimedean local field kv , let 1/Jv be our fixed additive character, and let D E Z be maximal such that 1/Jv = I on p-D UT.. V • n'Tlte 20 = pe V V • Then for any a, c E k; , we have J for any integer M such that Ms; -(ordv(a) + D + 2e + 1). Proof. We compute this similarly to a calculation in Perrin [P]. First, we treat the case where a= I and c E Uv , writing ( *) for the integral above. Then taking a = I and c = u , we have ( *) = Jb 2 EPM 1j;( ub2 ) db. Noting that ord(b2 ) ~ M {=} ord(b) ~ [MtI] d~ L, we have (*) = f XPL ( b )1j;( ub2 ) db , where [ J is the greatest integer function, and XA is the characteristic function of A for any set A . Next, it is easy to see that J { m+(p-(D+e+L)) if b E pL 1j;(2uby) dy = 0 if b rf_ pL' and so setting A= p-(D+e+L) , we have XpL(b) = m+(A)-1 J 1j;(2uby)dy. A 128 Substituting this into the expression for ( *) , we obtain (*) = m+(A)-I f j 'ljJ(ub2 + 2uby)dydb. A Note that the order of integration is fixed here Wen t 1 t th . ex comp e e e square, observing that Thus M ~ -(D + 2e + 1) [M + 1] -¢=:} 2L = 2 2 ::; -D - 2e -¢=:} -2(D + e + L) ~ -D -¢=:} uy2 E p-D for all y EA= p-(D+c+L). ( *) = m+(A)-1 J J XA(-y) t/J( u(b + y)2 ) dy db = m+(A)- 1 j j XA (x - y) t/J(uy2 ) dy dx. But now this is in the form needed to apply Weil's Corollary 2 to Theorem 2 [Wl). Following Weil's notation, we let J(y) = t/J(uy2 ), so that given the standard dual pairing on kv , < x, y >= t/J(xy), this gives us f( x + y )f( x )-1 J(y )-1 =< x, 2uy > , and so the symmetric mapping p associated to f is y ~ 2uy . Weil's Corollary 2 then gives us and so using Rao's notation now, we reach the conlusion (*) = l2ul- ½,v(ul/J), as desired. 129 Next, we consider the case where a = 1 and c = u1r for u E Uv . A calculation entirely similar to the one above yields (*) = J XpL(b)'lj;(u1rb2 ) db, where L = [ l-:J] . Setting B = p-(D+e+L+l) , we have (*) = m(B)-1 ff 'lj;(u1r(b2 + 2by))dydb. B As before, M ::; - D - 2e - 1 ~ 1 - 2(D + e + L + l) 2: -D ==} we may complete the square. This then gives us (*) =m(B)-1 ff XB( -y)'lj;(u1r(b+y)2 )dydb = m(B)- 1 ff XB(x - y)f(y) dy db, where f (y) = 'lj;( u1ry2 ) , and hence p = 2u1r = 2c . Another application of Weil's corollary then yields ( *) = l2cl-½ ,v( c'lj;) in Rao's notation. So far, we have proven the lemma for a= 1 and c E Uv U 1rUv . Now let c and a E k; be arbitrary, and suppose that M::; -(ord(a) + D + 2e + 1). Write ca= 1r2 ix for x E Uv U 1rUv . Then we finally have f 1Pv(cab2 )db= f 'lj;(1r2 ixb2 )db cb2 E'P~ 7r2i xb2 Ea'PM by our previous results. But now this last expression equals l2cai-½,v(ca'lj;v), since the Weil index is defined on square classes of k; . D In order to finish the simplification of equation (6.4.5), we need the following computations. 130 PROPOSITION 6.4.5. Let kv be any non-archimedean local field, and for a E k:;- , let H(a) stand for the Hilbert symbol character x ~ (x, a)v . For a character c of k; , let (v( s, c) denote the ftmction if c is ramified, if c is unramified. Fix d E k; . Then for M « 0 depending only on ordv ( d) , the following equations hold. (6.4.6) (6.4.7) J I I s .i.( ) ( d) ax = (v(s, H(d)) X 'f' X X' V X - ( V ( 1 - s' H ( d)) pM " JI Is x _ (v(2s) (v(½ - s ,H(d)) x '1/y(x),v(-xd,'lj))d X=(v(l-2s)(v(½+s,H(d)) pM " Note that = m eans "equal up to a multiple of a holomorphic non-vanishing function " as before, and that we only require that M be sufficiently negative, where this is determined by d . Since the proof of this proposition is technical and quite long, it will be omitted. We now have enough information to solve for I ( n - l , s ) in terms of zeta functions. LEMMA 6.4 .6. Let the notation be as in Proposition 6.4.2. Then we have _ ( v(s, H((-l)k28d1)) I(2k + l, s, d1 , d2) = (v(l _ s, H((-l)k2'5di)) fork 2: 0, and 12k s d d = (v(2s ) (v(½-s,H((-ll28d)) k ( ' ' 1 ' z) - (v( l - 2s ) (v(½ + s, H ((- l)k28d)) for 2: l , 131 where d and d1 stand for their respective determinants where appropriate, and 8 is the constant appearing in the deflnition of F. Proof. Using the results of Lemma 6.4.4, equation (6.4.5) simplifies to n-1 I(n - l,s,d1,d2) = J lei"+~ F(c)¢(cd2) J1 [12cdij)l-½,v(cdij)1P)] dxe. pN J=l V Now, writing 'r/ = ½¢ , from the basic facts about the Weil index we compute that and so continuing: We now need to specialize to n - l even or odd. Suppose that n - l = 2k + 1 2: 1 . Notice that we have an even number of Weil indices, and that for constants a, b E k; , ,'v ( ac, 'Ip) ,'v (be,¢) = ,v ( abc2 , 'Ip) ( ae, be )v = ,v(ab,¢) (a,b)v(e,abc)v = ,v(ab,¢) (a,b)v(e,-ab)v- 132 Discarding all the constant terms and temporarily writing aj = dij) , k I(2k + 1) = / icl 8 1P( cd2 )( c, 28a2k+1 )v n ( c, -aiai+k)v ax C pN 1=1 tJ = J lcl 8 1P(cd2) (c, (-l)k28d1)v axe pN tJ -J lxl 8 tp(x)(x,(-1t2od1)vdXX. Pf:' This case is then done by equation (6.4.6) above. Next, suppose that n - l = 2k 2:: 2 . In this case, we see that k I(2k) = j icl 8 1P(cd2) ,v(-28c,1j,) IT(c,-aiai+k)vdxc pN i=l tJ = j jxj 8 tp(x),v(-28d2x,1j,)(x,(-I)kd1)vdxx, Pf:' making the change of variable x = ca2 , and letting d2 pN = pN' . Noting that we see that up to a constant, where d stands for det( d) = det ( di d 2 ) • Substituting in the expression above, we obtain I(2k) = J jxj 8 1j,(x),v(-x(-ll2od,1j,)dxx, Pf:' 133 which finishes the proof by equation (6.4.7) of the previous proposition. D Having (more or less) explicitly solved for I(n - 1,s), we may now finish the computation of cd( s) . THEOREM 6.4.7. For any integer n 2'.: 1, and any GL(n) representative d E Symn(kv)x, we have the following formula: 1 { av(n,s) (v(1-s,H(6)) c(n, s, d) = bv(n,-a) (v(i+a,H(6)) av(n,s) bv(n,-s) if n is odd if n is even, where 6 = (-1).,-l · 2b det(d) , and av and bv are as in Theorem 5.1.1. Before starting the proof, we state the following inductive formulas for av and bv , which are easily checked. LEMMA 6.4.8. For n > 2 - ' ( ) { (v(2s)av(n-1,s-½) ifn is odd av n, s = av(n -1,s - ½) ifn is even, and b ( { (v(l - 2s)bv(n -1, -(s - ½)) if n is odd v n, -s) = bv(n - 1, -(s - ½)) if n is even. Proof of Theorem 6.4. 7. Beginning with n = 1 , we compute from Propo- sition 6.4.1 that c(l, s, d) = j /els-IF( c )1/,( cd) de pN for N sufficiently negative. But exactly as in the proof of Lemma 6.4.6, this reduces to pN' (v(2s) (v(½-s,H(28d)) - (v(l - 2s) (v(½ + s, H(28d))' 134 this last by use of equation (6.4. 7). This proves the theorem for n = l . Now suppose the theorem is true for all n < n 0 , where no 2: 2 . If no = 2k + l , then with d = ( di d 2 ) , we have c(no,s,d) = c(2k,s -½,d1)I(2k,s,d1,d2) -( av(2k,s-½)) ( (v(2s) (v(½-s,H((-ll28d))) = bv(2k,-(s-½)) (v(l-2s)(v(½+s,H((-l)k28d)) _ av(n 0 , s) (v( ½ - s, H((-ll28d)) = bv(n 0 , - s) (v(½ + s,H((-l)k28d)) using Proposition 6.4.2 and Lemmas 6.4.6 and 6.4.8. If, on the other hand, n 0 = 2k + 2 , then c(n 0 , s, d) = c(2k + l,s - ½,d1)I(2k + l,s,d1,d2) _ ( av(2k + 1, s - ½) (v(l - s, H((-ll28d1))) (v(s, H((-lt28d1)) - bv(2k + l, -( s - ½)) (v(s,H(( -l)k28d1 )) (v(l - s, H((-l)k28d1)) = av(n 0 ,s) - bv(no, -s) · This finishes the proof. O §6.5 Poles of the zeta integral. With the functional equation of Z F ( s) at our disposal, we may now prove the analogue of Piatetski-Shapiro and Rallis' theorem (p.206 [PS-R2]): THEOREM 6.5.1. For any ) results from a ¢> with OE supp(). Hence, we may write zF(s,(x)=c on pN (i.e. choose N largeenough). But then, the last integral equals c' ;~, j lxl',,(-;6 ,f,) dxx 261r•U and letting x = -2i51riu , we have 136 -- _____ . __ ,.,._ ----- ~--"'~_,.-__,...,. .,.. taking N' even without loss, and for appropriate constants c', M1, M2 . So we see that (v(2s) zF(s, 0 = Char(U2e+1) , so that smce F(x, 1) = 1 for x E U 2 . If, on the other hand, (v(2s 0 )-1 = 0, then So = 7 0 'ftl51 for some l E Z . We may then apply the functional equation as in the beginning of the proof of Proposition 6.4.1: let E S(Symn) I supp() C Sym~)} 137 I - is a subspace of S(Symn) . Fix, for the sake of argument, a point s0 E C at which zF(s) has a pole. Write the Laurent series zF(s, ) = L f.k( A . If this is true, then f.A 1st = 0 , which says that i*(f.A) = 0, and so supp(f.A) C Symn "' Sym~t) by the exact sequence above. But since A < 0 , it is also easy to see ( as in the n = l case) that supp( f. A) C { x E Symn I det( x) = 0} : in other words, if supp() C Sym~n), then zF(s,) is entire. Next, consider that if we define 9(x) = (gx'g) for g E GL(n) and x E Symn, then zF(s,9) = 138 and E S(Syrnn) : ,eA is called a homogeneous distribution. So it is clear that supp(fA) is a GL(n) -invariant subset of {x I det(x) = O} n (Symn ,.._, Sym~t)) and consists of a finite union of G L( n) orbits, since this is true of { x I det( x) = O} . Specializing to the case t = I now, it is an easy computation to show that the set Symn ""Sym~l) = { ( t~ ~) E Symn I U E Symn_ 1 , VE M(n - I, 1)} contains no non-zero GL(n) orbits. Thus ,eA is a homogeneous distribution with supp(fA) = {O} . Since we are at a finite place v , and S(Symn) consists of finite linear combinations of characteristic functions of compact open sets, we see that lim fA(Char(E)) E",.O E compact open exists, and is non-zero. Hence ,e A ( ( 0) for some constant c E C x . But now by the homogeneity of fA , we see that So E ( r:,'QJ) · Z. Thus we have proven: if s0 is a pole of zF(s) of order -A, and if k1 > A , where k1 is the smallest integer k such that fk(S1) -/- 0, then So E ( k>i) = ~cd(s)Zd (Pn -s,¢>). d 139 Since Re(pn - So) = Pn , we see that Z I (Pn - s, A . Next, suppose that Z F ( s) has a pole at s0 , but that RA -=/ 0 on S1 (i.e. k1 =A). Then zF(s, 1 0 2 0 ¢3 according to the isomorphism above. Then for Re( s) > Pn , zF(s, ) = j lxl S- PnF(x)(x)dx Sym~1 ) =ff f /Ajs-pn jcj-'-Pn+n-l F(A) F( c) 1 (B) 2(A) 3( c) dA dB de = ( f IAl (s-½) - Pn - 1 F (A)2(A)dA) ( J q,,(B)dB) ymn-l \M(n-1,1) x V lcl'+-'f-' F(c)ef,3(c)dc). As noted before, supp( ¢>3 ) C k; means that the last integral is entire, and so where f(s) is entire. Now we use the induction hypothesis, so that 1 F -----1 Z (s,) av(n - 1,s - 2 ) is entire. Referring back to Lemma 6.4.8, this last implies that a"(~,s) zF(s, ) is also entire, as desired. Having shown that we can force (1 ) zF(s, ) to be non-zero at a general av n,s point s 0 in the case n = 1 , we may then use the inductive argument above to construct a function ¢> 0 for general n, such that zF(s, 0 ) -/: 0 in case av(n,s 0 )-1 -/=- 0. If s0 is a pole of av(n,s), then this is even easier, and we may use the functional equation applied to the characteristic function of GL(n, Ov)[ln] as in the n = l case. This finishes the proof. 0 141 Proof of Theorem 6.1.2. By Lemmas 6.2.1 and 6.2.2, we must only prove that 1 ( )Mv(s) E S(Symn(kv)) . But by Lemma 6.2.3, and we have just proven that this last is entire. D §6.6 Final results. Resuming now the discussion of §6.1, we were inter- ested in the poles of the operator b.,~n,s~Mv(s) for v < oo. The added zeta av n,s functions in bv may now introduce some poles. We keep track of these only at the special value s 0 ( m, n) = ;: - nf 1 • LEMMA 6.6.1. { -1 ord bv ( n, s) = Bo(m,n) 0 Proof. Recall that [.!!.:fl] ifl~m~n ifn+l~m bv(n,s) = IT (v(2s+n+2-2k) k= l 142 and that v < oo implies that (v(z) has poles on the real axis only at z = 0 . If n is even, then a pole occurs at s 0 ( m, n) n ¢:::::} 2s 0 + n + 2 - 2k = 0 for k = 1, 2, ... , 2 n ¢:::::} m + 1 = 2k fork= 1,2, ... , 2 ¢:::::} 1 < m+l < !: - 2 - 2 ¢:::::}l~m~n-1. If n is odd, then bv has a pole at s 0 l< m+l n+l ¢:::::} - 2 < -2- ¢:::::} 1 ~ m ~ n. D In order to control the simple pole which may possibly occur at So in the term !:f:::~Mv(s)v(s), we need to take advantage of the special properties of Weil-Siegel sections. Following [K-Rl] , we will use this term to denote Kv -finite sections v(s) whose values at s0 lie in the image of S(Vvn) under the Gv -intertwining map note the mapping above composed with 143 Note that T. - v is Gv -intertwining, and also that with the action of Hv = O(V)v on S(Vvn) given in §2.3, we have Tv(wv(h)tpv) = Tv(tpv) for any 1Pv E S(Vvn) · Hence Where 1 · J h c 1 is the trivial representation of Hv . But now as in [K-Rl , t e 10 - lowin .. g Proposition applies: PROPOSITION 6.6.2. Let l be the dimension of a maximal isotropic sub- space of V. d I v , an et hv(m, n, l) = dime Home- H (S(Vt),Iv(-so) ® l). vX v (I) if l Sm Sn, then hv(m, n, l) = 0 · (2) If n + l S m S 2n - l , then { 1, hv(m, n, l) S O (3) If 2n + l S m , then { 1, hv(m, n, l) ~ O ifm - l ~ n + 1 otherwise. ifn ~ l otherwise. 'rhe Proof is essentially the same as that in [K-Rl]. As a r 1 esu t, we have 144 " •" COROLLARY 6.6.3. Let v E :Ek be a finite place. If 4>v E IK 11 1s a Weil-Seigel section, then bv(n,s) Mv(s)tf>v(s) av(n,s) is holomorphic at s = so(m, n). Moreover, if n + 1 ~ m and hv(m, n, l) = 0 in Proposition 6.6.2, then the expression above has a zero at So • 145 7. THE LOCAL INTERTWINING OPERATOR: ARCHIMEDEAN PLACES §7.1 Coverings of the unitary group. In order to finish our analysis of the global intertwining operator, we still must analyse the analytic properties of bv(n,s) Mv(s)v(s) av(n,s) when kv is archimedean. For the time being, fix a place v for which kv ~ R . The properties of the expression above when applied to Weil-Siegel sections will turn out to be governed by those of the operator ~Mv(s) when applied to av \SJ a certain section given by a character of Kv . Hence we must study this last group. Igusa's book [12} contains many details of the real metaplectic group. First of all, he shows that Kv = Sp(n,R) n 0(2n,R) is a maximal compact subgroup of Sp( n, R) , and that any maximal compact subgroup is a conjugate of K v . The group I( v is also naturally isomorphic to the nxn unitarygroup U(n)={gEGL(n,C)lgtg=ln} via Kv ~ U(n) U(n) is a compact connected Lie group with fundamental group 1r1(U(n)) ~ Z . This may be shown as follows: U(n) acts transitively on S 2n-l = {x E 146 C n (column vectors) I t xx = l} , and so defining the map U(n) .:!__.. s2n-1 b ~ Y g ._ g · en, we see that the stabilizer of en in U(n) is a copy of U(n - 1) (for n 2: 2) and that U(n)/U(n - 1) is homeomorphic to s2n-1 . Now U(l) ~ S1 is connected, and by induction, we may assume that U(n - 1) is also. Since r.p is an open mapping onto a connected space and has connected fibers ( '.:::'. U ( n - l) ) , it is easy to show that U ( n) is connected also. The mapping r.p above also defines a principal fiber bundle with fiber U( n- l) . Attached to any such bundle is a long exact sequence of groups · · · --+ 1r1(U( n - l)) --+ 1r1(U( n )) --+ 1r1(S2n-i) ~ 1r1_1 (U( n - 1)) --4 .. . called the homotopy sequence of the bundle (we omit base points). See Steenrod [St]. In our situation, this yeilds 0 = 1r2(S2n- 1)--+ 1r1(U(n -1)) -::'...t 1r1(U(n)) ~ 1r1(S2n-i) = 0 for n 2: 2. Hence 1r1(U(n)) '""1r1(U(n -1)) ~ · · · ~ 1r1(U(l)) ~ Z. Now by the usual theory of covering spaces (see [G-H]), for every subgroup H of 1r1 (U(n), 1), there exists a (connected) covering space (XH, x 0 ) .!.+ (U(n), 1) which is unique up to equivalence, and such that H = p*(1r1(XH,xo)). There is also a unique structure of a topological group which may be placed on X H 147 so that x 0 is the identity element and p is a group homomorphism. The covering will be [1r1(U(n)): H ] to one. We are of course interested in two-fold coverings, and so the preceding guarantees us that U(n) has a unique connected two-fold covering group (up to equivalence of coverings), since Z has a unique subgroup of index two. While Kv will turn out to be a connected covering, it will also be helpful to consider another model. Let U(n) de f {(x,z) E U(n) X TI det(x) = z 2 }. This is a topological group as a subgroup of U(n) x T , and it is easy to prove that it is path connected. Hence - p U(n) - U(n), p(x, z) = x is a model of the unique two-fold cover of U(n). This model is useful to us because projection on the second factor naturally gives a character c of U(n) with the property that c(g )2 = det(p(g)) . For this reason, we will call this character det ½ . PROPOSITION 7.1.1. All characters of the group U(n) defined above are of the form Cm for m E Z , where In other words, det½ generates the group of characters of U(n) . This is easily proven by the Weyl character formula or otherwise. 148 --1, Next, we state the facts concerning Kv , with a sketch of their proof. PROPOSITION 7.1.2. Given 1r : Mp(n, R)--+ Sp(n, R) as defined in §1.8, Kv = 1r-1 (Sp(n, R) n 0(2n, R)) is a connected group, as is Mp(n, R) itself Hence, making the identification Kv ~ U(n) , there is a covering isomorphism of Kv with U(n) . Sketch of Proof. Weil defines a cover of Sp(n, R) by the circle (in [Wl]) via M = {(g, 0 E Sp(n, R) X Un(L2(Rn)) I (-1 o U(h) o ( = U(h 9 ) for all h EHR} analogously to the definition of Mp(n, R). He proves that M ~ Sp(n, R) is a continuous open surjection, and so it is easy to see that M is con- nected, and hence path connected. Wallach, in [Wa], proves that Mp(n, R) = [M, M] . This can be used to show that Mp(n, R) is connected. Letting L be the connected component of the identity in Kv , one can then show that Mp(n,R)/L--+ Mp(n,R)/Kv is a [Kv: L]-to-one covering map. But 1r1 (Mp(n, R)/ Kv) = 1r1 (Sp(n, R)/ Kv) {1} , and so [Kv, L] = l , proving that K v is connected. D §7.2 Gaussians. In this section, we will develop certain special vectors in the Weil representation called Gaussians. These are eigenvectors for the action of the maximal compact subgroups of the metaplectic group. We do this first for the general metaplectic group, and then make modifications necessary for the dual pair situation. 149 To begin with, we drop most subscripts v and continue to work over the real numbers. In order to use results from other sources, it will be convenient to assume that our fixed character of (R, +) is given by ¢( x) = exp(21rix) . In this case, Rao states that the Weil index of x ~ ¢( ax2 ) is given by (7.2.1) so that ,( a'ljJ) = exp [ :i sign( a)] , ( l..1,) - ( .!,) d~ ,(a¢) - { 1, , a, 2 'f' - , a 'f' - - ' ,(¢) -i, if a> 0 if a< 0. Taking advantage of our fixed basis for W , and identifying W = X EBY with Rn EB Rn , it is proven in [12] that if F0 E S(Rn) is the function given by Fo(x) = exp(-1rxtx), then taking g = ( ~/3 !) EK= Sp(n,R)n0(2n) with det(,B) f- 0 , we have See Lemma 11 on p.37 of (12] . Here r is the same as Rao's unnormalized projective representation ( or Weil 's, for that matter), and we must be careful in choosing the second square root. This is done by declaring that I I det(i- 1 (r - ,a-1a))2 -+ det(Im(r))z > 0 as Re(r)-+ ,a-1a, where r E {z E M(n , C) I z = tz and Im(z) > O}. In any case, the important point is that we can use this as a starting point to prove: 150 ) / PROPOSITION 7.2.1. Let K C Mp(n, R) be the usual maximal compact subgroup, and let p: Mp(n, R) -t Un(L2 (Rn)) denote the Weil representation associated to 'I/; , given here by p(g, e) = ( . Then there is a continuous group homomorphism (character) such that p(g )Fo = D(g) · F0 for all g E K, and D(g )2 = det( 1r(g)), where we identify K = 1r(K) with U(n) v1a We will denote this character by det ½ . (1) Note that we are using the Weil representation used in the definition of Mp(n, R), and not the representation coming from a larger metaplectic group in a dual-pair situation. 1 ( 2) Also note that this is the "same" character as the det 2 defined on U(n), and that it in fact gives us a covering isomorphism K ~ U(n) over U(n) via g 1-t (1r(g), D(g)). Sketch of proof. By lgusa's result above, for g = ( _013 ! ) E nn n K , we have r(g )Fo = A(g )Fo, 151 where A(g) E T is a scalar given by Here, m(g) = ,(x(g), ½1/i)-11 (½1/i)-n, and it is easily shown that x(g) det( -/3) . Hence with no ambiguity about sign A(g ) 2 = 1 ( det( -/3), ½1/i )-2 1 ( ½1/i )-2 n I det( - /3) 1-1 det( 1 + i/3-1 a )-1 = ( - 1, det( - /3))R (e¥)-2n sign(det(-/3))det(-,8-ia)-1 = (i)-n det(-iln)-1 det(a - i,8)- 1 = det(a + i,8), which is at least a good start. Next, consider K = {(g, OE Sp(n, R) x Un(L2 (Rn)) I g E 0(2n) and ( = ±r(g)} to be an actual subset of Mp( n , R) . Define a map 1) : K - L2(Rn ) by D(g,0 = ( · Fo (= ±r(g)Fo). This is certainly a continuous mapping by definition of the topology on Mp(n, R), and we know that V(g ,0 = D(g,()Fo for all (g, 0 E 1r-1 (iln) n K , where D(g, O is a scalar in T satisfying D(g, 0 2 = det(g) by the above. It is not hard then to show that in fact D has an extension to all of K such that V( x) = D( x )Fo for all x E K . That D is a continuous homomorphism satisfying the required properties follows easily from the analogous properties of the Weil representation. D lgusa also proves (in a slightly different form) that for each maximal com- pact subgroup L of Mp(n, R), there is a unique one-dimensional subspace 152 , - J , , of L2 (Rn) which is L -invariant. Since all such L are conjugates of K, these subspaces will be generated by translates of the vector Fo above, which is called the Gaussian associated to K. It is uniquely determined by the condition that F0 (0) = 1 . Returning to the dual pair H(R) x G(R) in Sp( mn, R) , let Wv denote the Weil representation of Mp( mn, R) , and let Wv = Wv o Av be its pull back to the group Mp(n, R) "" G(R) as in chapter 2. Write K and K for the standard maximal compact subgroups of G(R) and Sp( mn, R) , respectively. We wish to find an eigenvector for the representation wv/i< . The problem here is that Av(K) ) . There is a similar situ- ation for the orthogonal group H(R) of (VR, (, )) , except that here we must define Xv = { i E AutR(VR) I i2 = 1 and (-, ·i) is positive-definite and symmetric}. It is easy to check that for each choice of ( i, j) E Xv x Xw , we get an element of Xv© w by taking i ® j = Av ( i, j) . In our case, we wish to find an eigenvector for K v , so we will fix j = J as above. Note that whatever i we choose, we will have Av(l V, k)Av(i, J)Av(lv, k - 1 ) = Av(i, J) = i ® J for all k E K , and so Av(K) c Ki©J , which must then be a conjugate of K = K J ( with J = ( _ 0 lmn) ) . The choice of the element i determines - lmn 0 which maximal compact of H(R) will have nice properties with respect to the Gaussian we choose below. Although it is not necessary, to be concrete, suppose that we have fixed a basis { V1, .. • , Vm} for V as in §2.1, with respect to which the inner product (,) has matrix Q = ((vr,vs)). Fix some i E Xv, and let it have matrix I. 154 Then with respect to the { er.,, e;.,} basis of W v described in §2.1, Av(I, J) has matrix ( 0 QI0 ®1)· (QJ) - 1 0 (-1) The matrix QI is positive definite, and so we may find a matrix d E GL(n, R) with QI = d td, so that Now we see that if the Gaussian associated to I{ is given by F1_(x) = exp(-1r < x, xJ > ), then the Gaussian associated to m( d) Km( d- 1 ) is just a multiple of r( m( d), 1 )F.1.. , which yields F(I,J)(x) def exp(-1r < xm(d),xm.(d)J >) in the L2 (Xv) model. Adapting this to the L2(Vvn) model, we write Vn~x V V n x = (xl,···,xn)t-+ LXr ® er r=l and see that this reduces to F(I,J)(x) = exp(-1rtr(x,xl)), where (x, xI) represents the m x m matrix with entries (xr, x.,I). Changing notation, let <.p~ denote the Gaussian associated to I and J , which we assume 155 to be fixed. We then know that wv(k)cp~ is a multiple of cp~ for all k EK. This multiple may be determined by writing Av(k) = [m(d) , 1] !£. [m(d), 1J-1 for ls. E K , and noting that if Wv ( k )cp~ = c( k) · cp~ , then on the one hand, wv([m(d), 1) ls. [m(d), 1)-1 ) wv([m(d), lJ)Fo 1 = det2(!£.) wv(m(d))Fo = c(k) wv(m(d))Fo. - 1 Ontheotherhand, c(k) isacharacterof Kv,andhence c(k)=[det2(k)t for some r E Z . By diagonalizing Q and I and equating ~( s ) E Iv( s) via identifying K v = U ( n) . This is a Weil-Siegel section, since ~(g , S) = v(9 , s, v, we have mtri ord Mv( s )v( k, s) ~ ord Mv( s )~(l , s ). k EK., so(m,n) s 0 (m,n) The proof is the same as that in [K-Rl], with obvious modifications. One point worth mentioning is that - - lnd~(n\det 1 I- ) ~ det1 ®lnd~(n)(l) O(n) O(n) O(n) a s U( n) -modules, and thus the theorem in [B-G-G] which asser ts multiplicity- one for Ind~ ~:~(1) still holds. In any case, we are reduced to looking at Mv(s)~(l, s ). Since M v(s )~ (gk , s ) = det 1(k) Mv(s )~(g ,s) 157 for all g E Gv and k E Kv , we see that Mv( S )~([w, lJ-1 , s) = dn,v(s, l) deC1([w, 1]) . During the proof, let S = Syrnn(R) => s+ = {x ES Ix> 0 ( x is positive definite)}. We must simplify J ~([w, 1] [n(-x), 1] [w, 1i-1 ,s) dx. s First of all, note that X E s ===> 1 + x 2 E s+ ===> (1 + x2 )-1 E s+ ' and so we may choose a matrix a = a( x) E s+ such that a ta = (1 + x2 )-1 . This allows us to write 158 so that ( 1 0) ( a( x) * ) ( ta X l - Q a( X) ta · X t ) - a• X de/ ta = p(x)k(x) E Pv · Kv. Hence, writing (7.3.1) n(x) def [w, 1] [n(-x), 1] [w, 1]-1 = [p(x), 1] [k(x), t:(x)], we have Now we compute that where a E s+ implies that all determinants are positive real numbers, and there is no ambiguity about signs of the square roots. Also, we recall that ( ( ) ) _ (a, (-1)¥ det(V))v = l Xv a x , l - ( .!..i,) , 'Yv a, 2 '+' agam smce 0 . This leaves us with ~( n( x ), s) = det(l + x 2 )-( !±fa) det1( k( x ), c( x) ), where the factor c( x) is defined by equation (7.3.1 ). Next, we extend the function Logo det : s+ -+ R to an open convex neighborhood of s+ in S(C) d~ Symn(C) as follows. The set -iHn = 159 {y + ix E S(C) I y > O} is open, convex, and contains s+ , so by analytic continuation we can define a function logdet : -iHn -+ C so that it equals Log(det(y)) for y E s+ . Now use this function to define a branch of z 1---+ det(z )c on -iHn for any c E C , so that det(Int = I . Define f(x) = det(l + x2 )-(!If11-) det(a(x))1 det(l - ix)1 using the branch constructed above. Considering that x r-+ ~ ( n( x ), s ) 1s a smooth function of x E S for fixed s , and set ting F( x) = ~ ( n( x), s) · f(x)- 1 , it is easy to see that F(x) 2 = I by Proposition 7.2.1. Thus F: S-+ { ±I} is a continuous function on a connected set , forcing F( x) = I . This proves that tl>~( n( x ), s) = det(l + x 2)-( !±fa) det( a( x ))1 det(I - ix )1 = det(l + x 2 )-( ~) det(l - ix )1, where the only term requiring care in its definition is det(l - ix )1 = exp(! · logdet(l - ix)) . It is then easy to show that Log(det(l + x2 )) = logdet(l +ix)+ logdet(l - ix), which then reduces our problem to one of solving for (7.3.2) I( o:, (3) def J det(I - ix )- 0 det(l + ix )-fi dx, s 160 where a = "+1" -I, /3 = "+1" +I , and Re( a), Re(/3) ~ 0 . Some tools for simplifying this type of integral are given by Shimura in [Sh], and the proof from this point on is due to Kudla [K2]. We begin with the classical formula J e- tr(zx)det(z)-'-Pn dz= rn(s)det(x)-s s+ valid initially for Re( s) > Pn - 1 and x E s+ , and then for x E -iHn by continuation. For this, see (1.16), p.273 [Sh]. This yields det(l - ix)-o = rn(a)- 1 (21rt 0 J e-2,rtr(t(l-ix))det(t)o-pn dt, s+ which we may substitute into (7.3.2) to obtain I(a,/3) = r n(a)- 1 (21rt 0 • J e-z.t,(t) det( t)"-'" [! e( tr( tx)) det(l + ix )-P dx] dt, writing e( d) def exp(21rid) . Shimura also proves ((1.23) p.274) that for any b Es+, n n-1 J { e-tr(ib) det(t)s-pn r n(s) 2~ e(tr(tx)) det(b+21rix)-s dx = 0 s Using this, for t E s+ if t Es+ if t rf; s+. J e(tr(tx)) det(l + ix)-/3 dx = r nun-I 2-~ (21rtf3 e-tr(2 ,rt) det(t)f3-Pn s 161 and so, setting Gn(s) = rn(s)(2rr)-n", I( a, /3) = Gn( a)- 1 Gnun- 1 2-~ J e-tr(41rt) det(o+/9-Pn)-Pn (t) dt s+ = Gn( a)-1 Gn(/3)- 1 2-~ r n( a+ {3 - Pn) ( 41r)-n(o+/1-Pn) = Gn( a+ {3 - Pn) 2-n(o+/9-1) Gn(a)Gn(f3) ( ) n(n+l) r n(s) = 2n l-s 7r 2 r n ( "+1n+l) r n ( s+1n-l) = dn,v(s, l) det-1([w, l]). The result is completed by noting that if F0 ( x) = exp( -1rx t x) , x E R, then r(r)Fo(x) = fRn F0 (y)e(ytx)dy = Fo(x), and r(-l2n)Fo(x) = Fo(-x) = Fo(x), so that -.!!.ti { 1 = e 4 z if n is even if n is odd by equation (7.2.1). O We now must compute ord., (m n) b.,~n,s~dn v(s, l), for which we need the o , av n,s , following information. LEMMA 7.3.3. (1) (2) ord av(n,s)- 1 = 0 s 0 (m,n) { -[n~m]-1 ord bv(n, s) = so(m,n) 0 if l ~ m ~ n + l, if n + l < m. 162 (3) { - [ ¥] if 1 :S m :S n + 1, ord r n(s) = m;-1 - n if n + 1 < m < 2n, s 0 (m,n) 0 if2n < m. ( 4) Let a = 1; ± l , so that a = p or q . Then ord rn (s+pn±z)-1 = { [n+~-a] so(m,n) 2 Q if O :S a :S n + 1, if n + 1 < a. Proof Recall that f(z) has simple poles at {O, -1, -2, ... } and no zeros. Again using the notation f ( s) = g( s) if the quotient f ( s) / g( s) is holomorphic non-vanishing, we have [~] ["t 1 l av(n, s) - IT f(s - Pn + k) and bv(n, s) = II res+ nt2 - k). k=l k=l (1) At s - Pn + k = r;: - ( n + 1) + k E ½ + Z , which shows that av has no poles. (2) Here s 0 + nt2 - k = m{I - k . Since m is odd, k contributes a pole <=:> (m{l :S k and 1 :S k :S [nt1]) {=:?- mt1 :S k :S [ntI] . It is easily checked that #{k E Z I m{l :S k :S [nt1]} = #{ r E Z I O :S r :S [ n+l - m+ I]} = { 0 if n + 1 < m 2 2 [ n-;m] + 1 if 1 :S m :S n + l. (3) r (s) - nn-l re k) and So - _2k -- m-(n:l+k) . The number of k n = k=O S - 2 which contribute poles is N = #{k E Z IO :S k :Sn - 1 and m - (n + 1) :S k and n = k (mod 2)}. 163 If 2n < m , then n - l < m - ( n + l) , and so N = O . If n + l < m < 2n , then 0 < m - ( n + l) < n - l , and so the number of k with m - ( n + 1) S k S n - 1 is 2n - m + 1 , which is even. Hence N = ½(2n - m + 1) = n - m;1 . If 1 S m S n + 1 , then m - ( n + 1) S O , so N = #{ k I O S k S n - 1 and n = k (mod 2)} , which is easily seen to be [n/2] . ( 4) Set ting a = r:; ± l , we see that s 0 + Pn ± l = '; ± l = a , so we wish to find ords=a r n(s/2)- 1 , where a~ 0. As before, the number of terms contributing a zero 1s a-k N = #{k E Z I - 2- SO and OS k Sn - 1 and k = a (mod 2)} = # { k E Z I a S k S n - 1 and k = a ( mod 2)} if OS a Sn+ 1, if n + 1 < a. D Now we may collect all these facts together with Proposition 7.3.1 and Lemma 7.3.2 to give us: PROPOSITION 7.3.4. Let v be a real place of k , and suppose that p > q , where (p, q) is the signature of (Vv, (, )) . Then for any Kv -finite Weil-Siegel section 4> v , bv(n,s) ord ( )Mv(s)v(s) ~ s 0 (m,n) av n, S 0 [9] [n+~-q] 0 164 if p Sn+ 1, if q S n + 1 < p and m < 2n, if q S n + l < p and 2n < m, if n + l < q. If q < P (since m = p + q is odd, p =f. q ), then the result above holds with p and q interchanged. Notice that the lower bound on the order is actually attained by v = ~ , which is Weil-Siegel, and that !:f :::~ Mv( s )v( s) is always holomorphic at ( ) S 0 m,n , Proof C 'd h ~ . · ons1 er t e iollowmg ranges: (i) : 1 Sm Sn+ l (ii) : p s n + l < p + q = m (iii) : q s n + l < p and m < 2n (iv): qsn+l~(s) = v(s,cp~), this section is Kv -invariant on the right, so that the method of Gindiken- Karpelevich may be applied. LEMMA 7.4.1. M ( );r,.o( )=2nav(n,s)o(-) V S '*'v S b ( ) V S • V n, S Sketch of Proof For (: ~) E SL(2, C) , we may write (~ ~) (~ za ) ( a -za), a-1 za a 166 where a = a(z) def (zz + 1)-½ E R, noting that the last matrix above lies in Sp(l) = SU(2). Referring to the notation of §5.2, and recalling that Xv = l, we have Za 0 (s) = J la1( zl(µ+¥)lvdz, k., where we may take a 1 (z) = 0 , c and so Zao(s)= c-2~ . =C· 2~r(2(s+pn).+i.+j) 2(s+pn)+i+J r(2(s+pn)+z+;+l) (v(2( s + Pn) + i + j) = C . _.:...._.:__;_ _ _;__:_ __ _..;_.....,.. (v(2(s + Pn) + i + j + 1) { 2 i = j for ao = Xi + x j ( i :S j) and c = ' . . 1, z # J See Definition 6.1.1. We are then finished by the proof of Theorem 5.1.1. D The analogue of Proposition 7.3.1 holds again in this case, and so for a Kv -finite Weil-Siegel section v min ordMv(s)v(k,s) ~ ordMv(s)~(I,s), kEK., S o So which gives us 167 PROPOSITION 7.4.2. For a complex place v E ~k , let v be a Kv -finite Weil-Siegel section. Then is holomorphic at s 0 (m, n) . Proof. Lemma 7.4.1 and the preceding remark. D 168 8. THE MIDDLE TERMS §8.1 The intertwining operators. By equation ( 6.1.1) and the work of the last three chapters, we have compiled fairly complete information about the order of the poles of the global intertwining operator M(s): I(s)K -t I(-s)K when applied to Weil-Siegel sections E I K . This will be analysed in chapter 9. Now we turn our attention to the middle terms in the sum (4.4.1). We must study the analytic properties of the terms where 1 S r S n - l and r(g,s) = J (wrng,s)dn. N~_r(A) First of all, consider the standard parabolic subgroup given by P = Qr· Nn , and notice that r satisfies: LEMMA 8.1.1. For n E Nn(A) and m = [m ( ai : 2 ), t:] E Qr(A), Let Ir(s) be the space of smooth functions on G(A) satisfying the equal- ity of the lemma. This is actually a G(A) -space induced from the parabolic 169 P(A). As with the operator M(s) studied previously, we may define a G(A) - intertwining operator Mr( s) : I( s) ---t Ir( s) for Re( s) > Pn by Mr(s)(g,s) d~ j (wrng,s)dn, N~_r(A) so that Mr(s)(s) = r(s). To analyse the analytic properties of Er:... ( s) , we first consider the prop- Pn erties of the operator Mr( s) above, which will be obtained easily from those of M( s) . We will then find the poles introduced by the Eisenstein series on Mn(A) . It will be possible to derive information on this last item by tensoring E!:.., (m,s) with a character, which will allow it to be considered as an Eisen- Pn stein series on Mn(A) ""GL(n, A) . Such series are studied carefully in section 5 of [K-Rl]. As explained in [K-Rl], if we consider a K -finite Weil-Siegel section , and write p(k)(s) = Lcj(k)j(s) J for a finite number of K -finite standard Weil-Siegel sections j ( p represents right action by K ), then the analytic properties of the sections Mr(s)(s) are determined by those of the functions Mr( s )j ( 1, s) . For the rest of this section, write Gr(A) = Sp(r, A) , and add superscripts r and n to indicate 170 objects associated to Gr(A) or Gn(A) . The map io : Gr(A) - Gn(A) i. (: ! ) = c-r : ln-r : ) described in section 4.4 of [K-Rl] admits an extension to Gr(A) . LEMMA 8.1.2. The map i : Gr(A) -+ Gn(A) defined by i([g, 1:]) = [io(g), €] is an injective homomorphism. It induces i* : Jn(s) -+ Ir(s'), where s' = s + Pn - Pr . The corresponding local statements also hold. Gr(A) . Omitting superscripts where no confusion will result, for any place V E :Ek ' by Theorem 1.4.5. It is routine to check that and so, that f3v(91, g2) = f3v(i 0 (g1 ), i 0 (g2)) for all places v . This gives the local result, and the global result follows by taking /3 = IT f3v . Next, let n E N;(A) , m = [m(a) , 1:] E M;(A) , and g E Gr(A) . Then for 171 a section 4>(s) E In(s), 4>(i(nmg),s) = xn(i(m)) la(i(m))l"+Pn 4>(i(g),s) = x(det(a),€) lal"+Pn 4>(i(g),s) = Xr(m) ja(m)(+Pr 4>(i(g),s) and so we may define a section i"'4> by ( i"'4> )(g, s') = 4>( i(g ), s) , where s' = s + Pn - Pr. D Given the above, the operators M;!( s) and M;( s') are related by the following commutative diagram (see [K-Rl]): Jn(s) M;:(s) l;.'(s) i· 1 1 i. Ir(s') r( - s') M;(s') Hence (8.1.1) M;1(s)4>(1,s) = M;(s')(i*4>)(1,s'), where the analytic properties of the right-hand side are well-known by the work of the previous three chapters. It should also be noted that i*: In(s) ~ rr(s') takes Weil-Siegel sections to Weil-Siegel sections on the smaller group Gr(A) , and that the special value s = s0 ( m, n) corresponds to s' = so( m, r) , which is the value at which we have full knowledge of M;(s') applied to Weil-Siegel sections. The appropriate local analogue of (8.1.1) also holds. 172 §8.2 Degenerate Eisenstein series. Since the previous section gives us information about the order of the term r(s) at s0 (m,n), we now concen- trate on the degenerate Eisenstein series E':..,(g,s,)= ~ r(,g,s), gEG(A). Pn ~ ,EQr \Mn Notice that if n E Nn(A) , m E Mn(A) , and k EK, then Eli (nmk,s,) = Er;_ (m,s,p(k)). n Pn Using the K -finiteness of the section ( s) again, the analytic properties of E':.., (s, ) may be determined from the restriction of E':., (s, i) to Mn(A) , ~ ~ letting i vary over a finite collection of "nice" sections (as before). It appears, therefore, that we need to know about Eisenstein series on Mn(A) "' GL(n, A) (the latter cover being defined via the Hilbert symbol). But it is easily seen that any poles or zeros of Er;_, ( m, s, ) are preserved by Pn the functions on Mn(A) ,..._, GL(n , A) given by m r--t x([m, 1r1 ) Er;_, ([m, 1], s, ), m E Mn(A). Pn We develop notation for this. Let for all m' E Mn(A) and m = m ( ai : 2 ) E Qr(A)} and consider the map 173 which is defined by (i) restricting to Mn(A) , (ii) tensoring with x-1 , and (iii) further restricting to Mn(A) x {1} C Mn(A). Then setting m E Mn(A), the following lemma holds: LEMMA 8.2.1. The analytic properties of Ek (s, ) may be obtained by studying those of Er:.., ( s, ) . Furthermore, this last is an Eisenstein series on Pn Mn(A),...., GL(n, A) of exactly the type studied in section 5 of {K-Rl}. Unfortunately, the properties of these Eisenstein series are known not in terms of the standard sections of Yr( s) ( those whose restriction to the standard maximal compact of GL(n, A) is s -independent), but in terms of certain special sections F( s) E Yr( s ) . Specialized to our situation, we briefly describe the facts concerning these sections. Define quasi-characters of Ax/ k via µ1 = I ls-~+Pn, µ2 = I 1- s+Pr, and w = µ 1µ21 1 Ii= I 12s+n - r. Let f E S(M(r,n,A)) be a Schwartz function in the usual sense. Then identifying Mn(A) with GL(n, A) , we may define a section F( s) E Ir( s) via F(g,s,f) = µ1(g) Jg j~ J J(h- 1 (0, lr)g) w- 1(h)dxh GL(r,A) for g E G L( n, A) and s > r - "I . Note the fact that our space Yr( s) 1s exactly the same as that defined in (5.29) of [K-Rl], except that our µ1 and 174 µ2 involve no characters other that those of the form 11 z • This has no effect on the results: see section 5 of (K-Rl] for details. The following proposition describes the situation: PROPOSITION 8.2.2 (K-Rl). The series E( m, s, f) deJ L F( 1 m, s, f), m E Mn(A), converges for s > -¥ , and has a meromorphic analytic continuation. Writing s' = s + Pn - Pr , the only possible poles occur at the points 2s' = 0, 1, ... , r - 1 ( ascending poles) and 2s' = n, n - l, ... , n - r + 1 ( descending poles) counted with multiplicities if the two series overlap. Furthermore, suppose that f = ®vfv , and that there is a place v E ~k such that supp(fv) C {x E M(r,n,kv) I rank(x) = r}. Then the ascending poles do not occur. Remark. This is a straightforward combination of Proposition 5.3 and Lemma 6.3 of (K-Rl]. There is nothing new to our situation, and virtually nothing to be checked. In view of Lemma 8.2.1 and Proposition 8.2.2, the goal now is to express the section r(s) as a linear combination (with meromorphic coefficients) of 175 sections of the form F(s, Ji) for Ji E S(M(r, n, A)) . First of all, it is natural to express the space Ir( s) as a restricted tensor product of similarly defined local spaces Ir,v( s) with respect to the spherical sections '11~( s) E Ir,v( s) defined for v < (X) by w~(g,s) = 1 Now fix

S k be a finite set of places such that if v ( s ) = ( s, v(s) = ~(s), and also that the following two properties hold: (1) for m E Mn,v(Ov), ~~,v(m,s) = Xv([m, 1])-1 Mr,v(s)~([m, 1],s) = Mr,v(s )~(l, s) = ~,v(l, s ), and (2) ~~,v(l, S) = M;,v( S )~·n(l, S) = M;,v(s')( i*~·n)(l, s') = Mr (s')o,r(l, s') = av(r, s'). r,v v bv(r, s') This last is by Theorem 5.1.1 and equation ( 8.1.1). These two facts imply that ~o (s) = av(r,s')wo(s). r,V bv(r, s') V 176 On the other hand, any J = ®v Jv E S(M(r, n, A)) must have Jv = Ji for almost all v , where Ji = Char(M(r, n, Ov)) for v < oo. Let Then a trivial modification of Lemma 5.4 of [K-Rl] shows that F; ( s) = Cv ( r, s) '1t ~ ( s), where r - 1 (8.2.1) cv(r,s) = IJ (v(2s +n- r - k). k=O Hence we see that (8.2.2) ;j;r(s) = 1 a(r,s? (@cv(r,s)bv(r,s?~r,v(s)) 0 FS,o(s) c(r,s)b(r, s ) av(r,s) vES where a, b , and c are the products of all the local factors, and F 5 •0 ( s) = ®v rt sFi(s) . So it remains to express a finite number of factors ( bv(r, s') ~ ) Cv r,s) ( ')cI>r,v(s, av r, s vES in terms of sections F( s, JJ) E !r,v( s) . This is done by a straightforward mod- ification of Propositions 5.7, 5.10, and 5.11 of [K-Rl). We will limit ourselves to stating the collected results. 177 PROPOSITION 8.2.3. For each place v E S, there exists a finite collection of functions {ft} C S(M(r, n, kv)) , and corresponding meromorphic functions /3l(s) which are holomorphic at s 0 (m,n), such that ( ) bv(r , s') - ( ) ~ j( ) ( fl) Cvr,s ( )rvS =~f3vsFvs, . av r, s' ' . J We have the following additional facts: (l) if v isa finiteplaceand b.,tr'"'.)Mrv(s)v(s) has a zero at so(m,n), av r,s , then either J3t(s 0 ) = 0 for all j (this will always be the case if m < r + l or m > 2r ), or each Ji is supported in the matrices of rank r m M(r,n,kv) (c.f Proposition 8.2.2). (2) If v is a real place, then each f3t may be written as J3 j( ) ( , ) bv(r, s')) f3l 0 ( ) v S = dr,v(s , l ( ') v' S av r,s where each J3t· 0 (s) isholomorpbicat s 0 (m,n). Thenotationisasin § 7.3, and the order at s 0 of the expression in parentheses is given by Proposition 7.3.4. If v is a complex place, then nothing further may be said. 178 9. BOOKKEEPING: HOLOMORPHY OF THE CONSTANT TERM §9.1 The global intertwining operator. Finally, we collect all the in- formation in chapters 5 through 8 together and find lower bounds for the or- der at s 0 (m, n) of each summand in the expression for the constant term of E(g, s, ): n E1\(g,s,) = LEk(g,s,) r=O The bookkeeping here is much easier than in [K-Rl] because our a( n, s) and b( n, s) involve only products of ordinary zeta functions, and not zeta functions depending on a character, as in that paper. For the remainder of this chapter , fix a function cp = 0 v'Pv E S(V(A)")K, and let (s) = (s,cp). We begin by solving for the order at s 0 of the factor (see §4.4 and equation (6.1.1)). While we have investigated the term in paren- theses completely, we have yet to write down the contribution of the factor a(n,s) ~. LEMMA 9.1.1. ord a(n, s) = s = so b(n, s) +l if l ~ m < n + l, 0 ifm=n+l, - l if n + l < m ~ 2n + 1, 0 if2n+3~m. 179 Sketch of Proof Note first that the global zeta function (( s) defined in Definition 6.1.1 has no zeros at integer points, and its only poles are simple poles at s = 0 and 1 . Hence ((2s + [) will have no zeros at half-integer points for any l E Z . The rest is a simple counting argument using the definitions of a and b in Theorem 5.1.1. D It was proven in Corollary 6.6.3 and Propositions 7.3.4 and 7.4.2 that the term is at least holomorphic at 80 for all v E S . Hence by the lemma above, it might appear that n(s) has a pole when m and n fall in the range n + l < m ~ 2n + 1 . To show that this is not the case, we must use the fact that (V(k), (, )) is anisotropic. Recall the following standard facts about quadratics modules. LEMMA 9.1.2. Let (V(k), q) be a non-degenerate quadratic module of dimension m over a global number field k . (Here q( x) = ( x, x) is a quadratic form.) Then (1) (Hasse-Minkowski) V(k) is isotropic (represents O non-trivially) if and only if Vv = V(kv) is isotropic for every place v E ~k . (2) Suppose that m = 3 and that Vv is isotropic for all places except at most one. Then all Vv are isotropic, and so V( k) is isotropic. (3) Suppose that m 2': 5. Then V(k) is isotropic if and only if Vv 1s 180 isotropic for all places v such that kv c./ R . For proofs of these facts, see Serre [Se], or any reference on quadratic forms. We now consider two cases: Case I: m > 1 and there exists a real place v 0 for which Vv0 is anisotropic. Case II: Either m=l (so that Vv is anisotropic at every place), or m > 1 and there exists no anisotropic real place. If m > 1 , then by Lemma 9.1.2 we see that m = 3 and that there must be at least two non-archimedean places at which Vv is anisotropic. Suppose now that we are in case I. Without loss of generality, assume that the signature of Vv 0 is (p, q) = (m, 0) since the results of Proposition 7.3.4 were symmetric in p and q . Then by that proposition, d bv0 (n, s) M ( ).if,. ( ) > { O[~J Or ( ) Vo S 'cl' Vo S _ 2 So av n, S 0 [ ntl] if m ~ n + l , if n + l < m < 2n, if 2n < m. Assuming only that the other v E S terms in (9.1.1) are holomorphic at s = s 0 , we combine the above with Lemma 9.1.1, yielding (9.1.2) +1 0 [m;-n] _ 1 [ntI] -1 [~] W e will organize this in a moment. if 1 ~ m < n + 1, if m = n + 1, if n + 1 < m < 2n, if m = 2n + 1, if 2n + 3 ~ m. Suppose we are in case II with m = 1 . Then Lemma 9.1.1 gives us ord a(n,s) = +1 and this is all we need to show that ords 0 n(s) 2: +1 · So~ ' 181 Next, suppose that m = 3 . If m < n + l , then we again obtain ords0 n( s) 2: +1 from the global :f;j . If m = 3 = n+ l, then ord :f;j = 0, and we get no assistance from Proposition 6.6.2, so the best we can do is ords 0 n( s) 2: 0 · In the range n + l < m ::; 2n + 1 , we see that m = 3 only occurs when n = l , and obviously 2n + 1 ::; m = 3 cannot occur at all. So if m = 3, n = l , then since we have at least two non-archimedean places v1 and v2 for which Vv; is anisotropic, we may use l = O in Proposition 6.6.2 for each of these places. This yields hv;(m,n,O) = 0 for i = 1,2, and so by Corollary 6.6.3, ord1(s ) 2: ord [a(l,s) IT bv;(l,s) Mv;(s)v;(s)] So So b(l,s) i=l av;(l,s) 2: -1 + 1 + 1 = +1 in this case. We collect these results: LEMMA 9.1.3. +1 ifl::;mn(s) is holomorphic at s0 (m,n). §9.2 The n1iddle terms. As noted in Lemma 8.2.1, the analytic properties 182 of E':... ( s) for 1 S r S n - l may be obtained by examining Pn Now by equation (8.2.2) and Proposition 8.2.3, we may write (9.2.1) -r ( ) 1 a(r,s') ~ . . EP m,s,4> = ( ) b( ') L-/P(s)E(m,s,f1) n C r, 8 r, S . } (see also Proposition 8.2.2). Now we know that all the j3i(s) are holomorphic at s = s 0 , but we will need the extra information given in Proposition 8.2.3 in many cases. First of all, note that by definition (see (8.2.1)) and a simple calculation, 0 if 1 ~ m < r + 1, r-1 +1 if m = r + 1, ord c(r,s)- 1 = ord IT ((2s1 - k)- 1 = s=s 0 (m,n) s'=s 0 (m,r) k=O +2 if r + 1 < m < 2r, +1 if m = 2r + 1, 0 if 2r + 1 < m. It is also easy to see from Proposition 8.2.2 that the Eisenstein series E( s, Ji) has a possible ascending pole at s 0 only if r + 1 ~ m < 2r and a possible descending pole at s 0 only if (9.2.2) n+2Sm~n+r+l. For convenience, we record the order of c(r,s)-1 times a possible ascending pole: { 0 if 1 ~ m ~ r + 1, (9.2.3) ord (asc. pole) > +1 if r + 1 < m ~ 2r + 1 , So c(r,s) 0 if 2r + 1 < m 183 Now suppose we are in case I. Then by Proposition 8.2.3, every function j3i ( s) will equal d (s' m) bvJr, s') . j3i(s) r,v 0 , ( ') * , avo r,s where f3i ( s) 1s holomorphic at s O ( m, n) , and v O is our anisotropic real place (here assumed to be positive definite without loss). Hence in this case, :[;;;:j j3i ( s) has the same estimate on its order that we obtained for n( s) in equation (9.1.2), with r in place of n (since s = s 0 (m, n) corresponds to s' = so(m, r) ). Combining this estimate with equation (9.2.3), Ord (asc. pole) a(r, s') . ---- ---'----'-/3) ( s) > so c(r,s) b(r,s) - + 1 if 1 :S m < r + 1, O if r + 1 :S m :S 2r + 1 and m = r + 1, + 1 if r + 1 :S m :S 2r + 1 and m = r + 2, r + 3, +2 if r + 1 :Sm :S 2r + 1 and m ~ r + 4, +1 if 2r + 1 < m and r = 1, 2, +2 if 2r + 1 < m and r ~ 3. We write e(m, n, r) for ord80 E~" (s, ). By equation (9.2.1), all we have left to consider is the possible descending pole from E(s, Ji). Since a descending pole may occur only if m > r + 2 ( r < n ===} r + 2 < n + 2 :S m ), we see from the chart above that e( m, n, r) ~ 0 for all m, n , and r . The only combinations which allow the possibility that e( m, n, r) = 0 in case I are the 184 following: m = r + 1, m = n + 2 and r = 1, 2, or n - l, (9.2.4) m = n + 3 and r = 2. (the only overlap being m = n + 2, r = 2 = n-1 ==} (m,n,r) = (5,3,2) ). Next, suppose we are in case II with m = 1 . Then by Lemma 9.1.1, d a(r,s') · (9 2 2) d or s 0 ~ = + l . The term /3J ( s) has no pole, and by equations . . an (9.2.3), we hit neither poles of E( s, Ji) nor zeros of c( r, s )-1 . Hence in this case, e(l, n, r) 2: +1 . Finally, suppose we are in case II with m=3, so that there are at least two anisotropic non-archimedean places v1 and vz . (1) If m = 3 < r + 1 (i.e. r > 2 ), then we again have e(3,n,r) 2: +1 from Lemma 9.1.1 as in the m = 1 case above. (2) If m = 3 = r + 1 (i.e. r = 2 ), then one may check that Proposition 6.6.2 fails to help. We hit no descending pole by (9.2.2) , and (9.2.3) yields ord(asc. pole)c(r,s)-1 2: 0, So so the best we can do is to note that ords 0 :g:;:J JJj ( s) > 0 ==? e(3,n,2) 2: 0. (3) If m = 3 > r+ 1, then only r = 1 occurs. But now m = 2r+ 1, and so by Proposition 6.6.2, each place Vi ( i = 1, 2) has hv;(m, n, 0) = 185 0, and so Proposition 8.2.3 assures us that (3l(s) has a double zero. E ( a(r,s') h quations 9.2.2) and (9.2.3) contribute another zero, while b(r7) as a pole. So finally e(3, n, 1) ~ +2. Note that in all cases, we have established that E':., (s, cl>) with 1 ~ r ~ n -1 Pn is holomorphic at s 0 (m, n). §9.3 Diagrams. Before organizing the preceding material into a chart to show the vanishing of various terms, we must mention a factor which we have ignored until now: the matching of the central characters of the terms Er:., (m,s,cl>) ,consideredasfunctionson Mn(A). Itisclearthat E':., (m,s,cl>) ~ ~ has the same central character as cl>r(m, s), and by Lemma 8.1.1, this is given by (9.3.1) a) , ,] ,___, x(t) lal'•-2r)(•+p")+r(r+l). Then Lemma 3.2 of [K-Rl) applies: LEMMA 9.3.1 [K-Rl). For O ~ r < r' ~ n, the central characters of cl>r and r, , considered as functions on Mn(A) , coincide at the critical value s 0 (m,n) if and only if r + r' = m-1. In the following chart, the top row of each case lists the value of r , 0 ~ r ~ n , while the bottom row contains either O or * to represent the vanishing or possible non-vanishing of E':., (g, s, cp) at s 0 ( m, n) . Possibly non-vanishing Pn terms Er;_, whose central characters match at s 0 will be marked in pairs as Pn *' *' and *" *" . 186 PROPOSITION 9.3.2. The vanishing of the terms Er;._, (g,s,) at s0 (m,n) Pn is given by m = 1 m=3 n=l n 2: 2 m=5 n=l n =2 n=3 n 2: 4 m 2: 7 n+3 2 = 0. Fixing m = 5, by Lemma 9.1.2 we know that there must by at least one real place v 0 for which Vv 0 is anisotropic. We claim the following: either ( 1) there exists another anisotropic real place v1 , or (2) there exists some other place v1 (finite or not) such that the dimension lv 1 equals 1. We will write Ev( Q) for the local Hasse invariant of the matrix Q ( choosing a basis for V( k) , etc., as in §2.1) to avoid confusion with hv( m, n, l) . Suppose now that lv = 2 for all places v -=/= v 0 (notice that l = 0, l , or 2 are the only possibilities by considering a Witt decomposition of Vv ). Then for v -=/= Vo , the matrix Q is similar to 1 1 a -1 -1 over kv for some a E k; by splitting off 2 hyperbolic planes. For a diagonal matrix D = diag(d1 , ... , dm), the local Hasse invariant is given by Ev(D) = ITi(s) E l(s); is a WeiJ-s· iege] section, then M(0)'1>(0) = '1>(0) . Flem.ark T . . Mv(0)'1>v(O) = µv . 'Pv(O) for · he proof hmges on showmg that some h constant ~ t" g that µv = 1 w enever µv and for all places v E LJk , no Ill g>v ::::: cf>o d v • The details of the proof carry over almoS t unchange · CoRot . i·n all cases for which m = LARY 10.1.2. Theorem 3.3.11s proven n-/- 1 191 ---- Proof By Proposition 9.3.2, if m = n + l then EP" (s, ) l.,=o = (s) + M(O)(O) = 2(O). D §10.2 The cases m = n + 2 and m = n + 3. Referring back to the notation of §4.1, consider the parabolic Pi , which has a Levi factor M 1 ~ GL(l) x Sp(n -1) . The idea in this section is to compute the constant term of E(g, s, ) with respect to the parabolic subgroup Pi (A) n .Pn(A) in two ways: one may first take the constant term with respect to Pi (A) (integrating over N1(A)) and then with respect to the maximal (Siegel) parabolic .P;:~/(A) C Sp(n - 1, A) , or one may take the constant term with respect to Pn(A) first, and then with respect to the parabolic On-i(A) C Mn(A). See the following diagram (the diamond of [K-Rl]): (10.2.1) (Levi of .Pi) GL(l) x Sp(n - 1) Sp(n) GL(l) xGL(n - 1) (Levi of Pini\) GL(n) (Levi of Pn) We begin with the P1 (A) side of the diamond, reverting to the notational convention of §8.1 in which a superscript of n or n - l is used to denote whether an object is associated to Sp(n) or Sp(n - 1) , respectively. 192 LEMMA 10.2.1 [K-Rl]. Let i: Sp(n-l,A)-+ Sp(n,A) be the map given in Lemma 8.1.2, and fix a section E IK. Then for g E Sp(n - 1,A) C M1(A), (10.2.2) E-1\ (i(g), s, )= En- 1(g, s + ½, i*) + En- 1 (9 , s - ½, i*(U(s) )) for Re( s) large, where U ( s) is the operator defined by U(s)(g,s) = J (wng,s)dn. NU(f\) Here Nu(A) = { (l 1":-, ~ : ) I y EA, x E An-I} C NB(A) <-+ Sp(n , A ) 0 0 - X ln-1 with w = ( ~ 1 1" - 1 ~ ) E Sp( n, k) ln-1 and En-I denotes the Eisenstein series on Sp(n - l, A) with respect to the parabolic i5;:~l(A) . Remark. The proof is straightforward by the comments in [K-Rl], and will be omitted. Now from [A] and [M], the functional equation of E(g , s, ) is given by E(g,s,) = E(g,-s ,M(s)(s)), by which we mean that the meromorphic analytic continuation of E(g, s, ) equals the continuation of the series L M( s )(,g, s ). ;EPn \G 193 For Sp( n - l) , this says that and if U( s) can be continued to a meromorphic section, then using this equality on the last term of (10.2.2) would yield: (10.2.3) Again, from [K-Rl] we have: LEMMA 10.2.2 [K-Rl]. ( 1) U ( s) has a meromorphic analytic continuation. (2) Mn-l(s - ½) o i* o U(s) = i* o Mn(s). Using these two facts, equation (10.2.2) then becomes (10.2.4) EP1 (i(g), s, cl>)= En-l(g, s + ½, i*cI>) + En-1(g, ½ - s, i* 0 Mn(s)cI>). Now if m = n + l , then writing n' = n - I , we have m = n' + 2 , and 8 0( m, n') = ½ . Hence we wish to evaluate the right-hand side of (10.2.4) at s = 0 · The proof of the proposition used to simplify the right-hand side of (10.2.4) in [K-Rl) depends on the following conjecture, which has not yet been proven in the metaplectic case: CONJECTURE 10.2.3. Fix any constant s 0 = r;: - nil 2 0 , and let v be a finit e place of k . For a non-degenerate symmetric space Vv of odd dimension 194 I. I m over kv , let R(Vvn) be the image of the map 'P 1---+ {g ~ wv(g)r.p(O)} where Iv(s 0 ) is the induced space on Sp(n, kv) introduced in §4.4. Then dim(V.,)=m where Vv is allowed to vary over all spaces of dimension m . In other words, for a fixed s 0 as above, the space Iv(s 0 ) will be generated by Weil-Siegel sections coming from different symmetric spaces. Given this conjecture, the proof of the following proposition works just as in [K-Rl]. PROPOSITION 10.2.4. Suppose that Conjecture 10.2.3 is true. If m ~ 5 and m = n + l , then for a Weil-Siegel section E I K , we have E n-le 1 "* Mn( ),T,.) I En- 1( 1 "*,T,.) g, 2 - s,z O S '¥ s=O = 9,z,Z '¥, for all g E Sp(n - 1, A) . In particular, by equation (10.2.4), Using the n' = n - l notation, we then see that if we take the constant term of EF'i with respect to Pn, C Sp(n'), we have EP- P- (i(g),O,) = 2EP!i (g , ½,i*). 1n n n' 195 .. I Now taking the constant term of E along the other side of the diamond (10.2.1), by the Weil-Siegel formula for m = n + l we have smce (g, 0, r.p) is invariant on the left by the unipotent radical of Qn-1 (A) . Hence E(, (g, ½, i*) = w( i(g) )r.p(O) for g E Sp(n-1,A). Butnow,givena i(n-I -finite function ) = w(g)(O). Pn, This then gives us: PROPOSITION 10.2.5. Given Conjecture 10.2.3, Theorem 3.3.1 holds for m=n+2. Trying to prove Weil-Siegel for m = n' + 3, we see that m = 5, n' = 2 is the first case of this sort with m odd, and it has been done by Proposition 9.3.2. As mentioned in the statement of Theorem 3.3.1, the case ( m, n) = (7, 4) remains unproven, so we now assume that m > 7 with m = n + 2. For convenience, we still write n' = n - l , so that m = n' + 3 . 196 Returningnowtoequation(l0.2.4),write E 1 for En- 1(g,s+½,i*'P) and E~ for En- 1(g,½-s,i*0Mn(s)'P) asin[K-Rl]. Further taking the constant term with respect to i\, C Sp(n'), (E1 )p- splits up into a sum of n' +1 = n n' terms, each having central character with repect to Mn,(A) given by XI l(n-1-2r)(s'+Pn-1)+r(r+l) where s' = s + ½ and O :Sr :Sn - 1 (see equation (9.3.1)). At so(m, n), the exponent becomes m ( n - 1 - 2r )- + r( r + 1) 2 Similarly, (EDP", is a sum of n terms with exponents (n -1- 2r)(½ - s + Pn-1) + r(r + 1) which at s 0 ( m, n) yields m ( n - 1 - 2r )( n + 1 - - ) + r( r + 1) 2 ( (Ef )p", exponent). On the other hand, taking the constant term along the right-hand side of the diamond, we have EPn (i(g), s0 (m, n), 'P) = w(i(g))cp(O) by the Weil-Siegel formula for m = n + 2, and so again we obtain (10.2.5) EP- P- (i(g),s 0 (m,n),'P) =w(i(g))cp(O), 1n n 197 Which has central character X / /(n - I)!f- by Proposition 2.4.2. Now we will use the fact th at functions on Mn,(A) transforming with different central charac- ters 1. are inearly . d d . . in epen ent (m a smtable space). By Proposition 9.3.2, we now t· no Ice that ( E1 )p- has only 3 possibly non-zero terms: these are given n' by r ::::: 0 2 d . h ' , an n . The r = O term has a central character wit expo- nent ( n - I) m • 2 at so(m, n), which matches that in equat10n (10.2.5) from the right h - and side of the diamond. The other two terms r = 2 and r = n have exponent ;2 - ~ n + I . If we can show that no term in (Ei)Pn, has this exponent, then the r = 2 and r = n terms must cancel each other at the Value s::::: so(m,n). . t f (E' )- if and There will be a matching exponen rom 1 P", OnJy if n 2 3 m) ( + 1) 2 - 2n+l=(n-1-2r)(n+1-2 +rr has a 1 . . t to asking for an integer so ution for r with O ~ r ~ n -1 . This amoun s Solution to r2 + (1 - n)r + (n - 1) = 0. tJ f, tion factors as n ortunately, for n = 5 (i.e. (m, n') == (7, 4) ), this equa ( r the gap in our r - 2)2 - . hich accounts ior - 0 , and so we do get matchmg, w 198 rd proof of the (7, 4) case. However, if m > 7, m = n + 2, then 11 < 2n and 5 < 9 ==} n2 - 8n + 16 < n2 - 6n + 5 < n2 - 6n + 9 ==} n - 4 < J(n - l)(n - 5) < n - 3 ==} J(n - l)(n - 5) = J(l - n) 2 - 4(n - 1) ~ Q and so the quadratic equation above has no solution in integers. Hence the r = 2 term in (E1 )p- cancels with the r = n term at s = s0 (m,n), which n' tells us that Finally, an argument like that preceding Proposition 10.2.5 gives us: PROPOSITION 10.2.6. Given Conjecture 10.2.3, the m = n + 3 case of Theorem 3.3.1 holds for (m, n)-/:- (7, 4) . §10.3 The cases 3 < m < n + l . Now we suppose that 3 < m < n + l , so that by Proposition 9.3.2, (10.3.1) using parentheses around the superscript m - 1 to indicate that E(m-l) is the r = m - 1 term in the expression for the constant term. The goal here is obviously to show that (10.3.2) 199 for all g E Sp( n, A) . First of all, note that since both of these are left N (A) - invariant, by an argument using the K -finiteness condition, it suffices to prove this equality for all g E Mn(A) "'GL(n, A) and all K -finite

O , and assume that the Weil-Siegel formula has been proven for all smaller k , where we are fixing m and V and allowing n to vary. The starting point for the induction, k = 0 , has already been proven by Corollary 10.1.2. Using the E 1 , E~ notation of the last section, by our induction assumption, at so( m, n) we have ((E1)pn_J ls=so = 2(i*)(g,s + ½)ls=so = 2(i(g),s 0 ) for all g E GL(n - l) (recall the shift caused by i* from Lemma 8.1.2). Suppose for now that we have proven LEMMA 10.3.1. For 5 ~ m < n + l, the function has a zero at s 0 (m, n) . 200 This will be the subject of the next section. Then (EP)Qn-l (g, 8 0, (i(g),s 0 ) for all g E G L( n - l) by equating the results from both sides of the diamond. On the other hand, by equation (10.3.1), (E- )- also equals Pn Qn-1 and so (10.3.3) g E GL(n) have constant terms with respect to Qn-l which are equal. Notice here that if g E GL(n) =Mn, then (g,s 0 (m,n)) = x(g) jgjT ( s) = L 1j( s )wj (-s ), j where Wj ( s) E I( s) are K -finite standard sections and the ,j( 8 ) are mero- morphic functions. This is possible by the work of §4.4. By Lemma 9.1.3, since n(s)=Mn(s)(s) vanishesat so(m,n) for m(s)) = L 1j(s) · En- 1(g,s",i*w(s" - ½)), j and so we will have proven Lemma 10.3.1 if we show that each of the Eisenstein series En- 1 (g,s",i*W(s" - ½)) is holomorphic at s~ = nt2 - ';. Following the conventions of section 8 of [K-Rl], we now renormalize: where we had an n before, we now use an n+ l, setting k = n+2-m 2 1. We also let m' = n + 2 + k = 2n +4- m, so that s 0 (m', n) = ni3 - '; corresponds to the special value s~ above (with n + l in place of n ). Write w(s) E In(s) for any of the sections given by i*wJ(s-½)- PROPOSITION 10.4.1. For m 2 5, and with all notation as above, the Eisenstein series E(g,s, w(s)), g E Sp(n, A) 202 -- are holomorphic at the special value s 0 (m', n). Hence Lemma 10.3.1 holds true by the work above. Proof The proof follows the proof of Proposition 8.1 of [K-Rl] in its essen- tials, differing mainly in the details of balancing poles and zeros of the various terms. First of all, noting that m' = m+2k with k ~ 1 , we add k hyperbolic planes to the space V , writing V' = VEBW for a split form W of dimension 2k . Then V' is a non-degenerate symmetric space of odd dimension m' . We define a mapping for any v E :Ek with v archimedean or v ~ Sk as follows. Let Sp(n, kv) act on S(W;) by composing the Weil representation of Sp(n, kv) ( dim(W) is even here: see §2.2) with the projection Sp(n,kv) - Sp(n,kv). We may then define a function v(s, ~) for some so c( r, s) - O if 2r + 1 < m' by equation (9.2.3) (these are the only possible cases), and d a(r, s) { -1 _!)Jso b(r,s) = O if r + l < m' ::; 2r + 1, if 2r + 1 < m', by Lemma 9.1.1, so that taken together, these two terms contribute no poles in any case. 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