Proc. Nat. Acad. Sci. USA Vol. 72, No. 4, pp. 1406-1410, April 1975
Intertwining Operators and Automorphic Forms for the Metaplectic Group (principal series representation/theta-functions/Eisenstein series/ noncuspidal automorphic forms ofhalf-integral weight) STEPHEN GELBART* AND PAUL SALLYt * Cornell University, Ithaca, New York 14850; and t University of Chicago, Chicago, Illinois 60637 Communicated by E. M. Stein, January 6, 1975
ABSTRACT The local results announced in this paper above. Let 0, denote the ring of integers of F, and set concern the analytic continuation of intertwining oper- ators for the metaplectic covering group of SL(2) over an K, equal to SL2(0,). Since a,, is chosen to be trivial on arbitrary local field and the decomposition of Weil's K, X K,, if the residual characteristic of F, is odd, we representation attached to the quadratic form q(x) = x2. can introduce the continuous two co-cycle a(g,g') = The global results concern Eisenstein series on the meta- plectic group and a Siegel-Weil type formula relating the fI av(g,,g,') on GA and use it to define a global meta- residues of these series to certain generalized theta func- V tions. plectic group GA. This group is a nontrivial topological central extension of GA by { A 1}. By quadratic reciproc- The purpose of this note is to announce some results ity, a is cohomologically trivial on GF, about intertwining operators for local metaplectic By B we denote the upper triangular subgroup of G; groups and their application to the global theory of A is the diagonal subgroup, and N the subgroup automorphic forms. The main results imply a kind of {[0 ] }. If H is any subgroup of Gv or GA, we denote its Siegel-Weil formula relating theta-functions on the complete inverse image in GO or GA by ft. If a is co- metaplectic group to residues of Eisenstein series. homologically trivial on H X H, then H is isomorphic to Many of these results were conjectured in a similar some subgroup of f which we again denote by H. In context in a preprint by the first named author.t A this case, ft = H X {I 1} . As already noted, important detailed treatment of the local theory presented here examples of such subgroups of G, are N, and K,(v odd); will appear in a joint paper (manuscript in preparation). A,, is not, since Hilbert's symbol, regarded as a two-co- The global theory pertains to theta-functions and Eisen- cycle on FZ, is nontrivial. stein series and extends earlier work of Kubota (refs. 1 The inverse image of { 1 } in Ov or GA will be denoted and 2), Shimura (refs. 3 and 4), and Siegel (refs. 5 and by Z2 and our interest will be exclusively in representa- 6). Proofs of this part of the theory will be given by tions of G, or GA which are nontrivial on Z2, i.e., repre- the first named author (manuscript in preparation). sentations which define genuine projective representa- Basic facts and notation tions of G,, or GA. We refer to such representations as genuine. Let G denote the algebraic group SL(2) and F a local field of characteristic zero. Weil's metaplectic group is 1. Intertwining operators for the metaplectic group (the then a topological central extension of GF by I1- }. p-adic case) This group is denoted by GF and is nontrivial as a group Since F, is fixed throughout the local theory, we shall extension iff F $ C. Actually, from ref. 7 it follows that drop the subscript v in F and the subscript F, in GF,. GF is the unique such extension. Also, elements of G will implicitly be viewed as elements To describe GF explicitly, we follow ref. 1 and use a of G via the lift g -- (g, 1). Finally, for simplicity of conveniently chosen two-cocycle a on GF. Thus, GF is exposition, we shall assume the residual characteristic realized as the set of pairs (g,~), with g E GF, r = + 1, of F is odd. Completely analogous results are valid in and multiplication is given by (g,¢) (g',~) = [gg',gst general. a(g,g') ]. If (a,b) denotes the Hilbert symbol of a and b in Let r denote a generator of the prime ideal P of 0 and FX, then at(g,g') = (aa') if g = [a-l] and g' = [0 (0)-1]. e a generator of the roots of unity in F. If x C Fx is arbi- Further details are given in the preprint mentioned trary, there exists an integer ord(x), completely deter- in footnote t. mined by x, with the property that x-ord(x) E ox. The Now suppose F is a global field. Let A(F) denote its genuine principal series representations of G are induced adele ring and v any place of F. For each v, the local from characters of A and these in turn are described by metaplectic group G, can be defined in terms of a,, as genuine characters of the double cover of Fx determined by Hilbert's symbol. t (1974) "Automorphic Forms on the Metaplectic Group," denote the set with x mimeographed notes, Cornell University Mathematics Depart- LEMMA 1. Let Px of pairs (x,¢) ment, Ithaca, N.Y. C FX, v = +1, and ?multiplication given by (x,¢) (x',V') 1406 Downloaded by guest on September 29, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Intertwining Operators and Automorphic Forms 1407
= (xx',(x,x')r'). Then each quasi-character g of PF non- p-adic gammafunction of ref. 9, and trivial on Z2 is of theform so(1) = 1, l p [(Xr) ] = r[sgn,(_-x) (TT) l/2][ -sgnf(x)1/2A(X) q - 1 _q-1/2(q - q8) for some quasi-character A of FZ. A q J(E) = Recall that sgn,(x) = (y,x) and IA(x) itself is of the - 1 + q'/2(q-8 - qs) form IxI.-8,*(x) for some s £ C and unitary character a* q of Ox (extended to Fx in the obvious way). We say that p is "unramified" iff , is; the correspondence ,i- = and (s,/. *) is one-to-one. Recall that g is nontrivial on Z2. = - Now set B equal to the inverse image in G of the upper ()(er) q28(1 q'++ q8-12 q'q-1+28/23\ triangular subgroup B of G. Since B = NA, we have q-1 q-l2-' B = NA, a semi-direct product. Thus, we can extend Let us write A (s,,u *) for A (,a). The operator A(s,,lu*) P(U[o °-j,]) = ,g(x,¢) to B and consider the induced then depends analytically on s if 0 < Re(s) < '/2 and representation defines a unitary operator when Re(s) = 0. Using Propo- sition 1 one can establish: pea) = Ind (B,G,p). THEOREM 1. The map s -- A(sju*), initially defined The natural inner product in the space of p(jz) is given for 0 < Re(s) < /2, extends to a meromorphic operator by an integration over K since 0 = NAK. From this it valued function in Re(s) > 0 whose only possible pole is at follows that p(g) is unitary iff 1 and hence p is. These S = 1/2; if /h* is nontrivial, i.e., if a is ramified, then are the genuine principal series representations of 0. A (s,. *) is actually entire for Re(s) > 0. [Recall A (s,,O*) intertwines representations of G nontrivial on Z2.] LEMMA 2. Let w denote the element [° I] of U and define This Theorem will be crucial for the global theory. 'aw(xXO ]r) = z((['OZ],¢)°w) = a([X'0,b). Then p(jiw) Analogous results for the real and complex fields are and p(Qu) are equivalent. already known (compare refs. 11 and 12) and will be By Bruhat's decomposition, G = B U BwN. Thus, recalled now for later use. each so: G -.o C in the space of p(p) is determined by its Suppose. first that F = R. Each genuine quasi-char- restriction +(x) = sp(w["f]). This means p(jz) may be acter g of FPisthenof theformg(x,¢) = Ixj8(ni)[1 -sgn(x) ]/2 realized in L2(F), with with s E C and v = 1. Furthermore, the correspon- dencep-uu(y) = 1- 1 onto the = 4(X + lxl8[sgn(x)][lI-gn(n)]/2is p(gffllb],)+) b) set of quasicharacters of Fx. So suppose p(s,,q) denotes and the obvious induced representation of 0F and Re(s) = O. Then p(s,77) and p(- s,7) are interwined by an operator p(jO)([ ])9(X) =|)(a2x). A(s,-q) which continues to a meromorphic function of s Using this noncompact realization of p(g) and the non- with poles at s = 1/2, 1 odd and positive. In particular, archimedean analysis of refs. 8 and 9, one can actually suppose f is in the space of p(s,7) and transforms under implement the equivalence of Lemma 2 explicitly as in S02(R) according to the character e"t '2, k odd. Then A (s,1)f has a pole at s = 1/2 if and only if k= 1(4) and Proposition 1 below. For simplicity, we assume g is unramified. k > 0; similarly, A (s,-1)f has a pole at s = 1/2 if and only if k = 3(4) and k < 0. In either case, if Api denotes PROPOSITION 1. Let x denote the canonical nontrivial the space of suchf, P('/2,1 1)/AA is equivalent to a additive character of F trivial on 0 but not on P'- (see so-called trash representation of SL2(R) (compare ref. 10) and let p'(,a) denote p(,q) realized in L2(F) via the Section 4.1 of the preprint mentioned in footnote I). x-Fourier transform F = f -*.f. Suppose 0 < Re(s) < 1/2. Now suppose F = C. In this case, Fx = FX X Z2,SO Then p'(p) and p' (p") are intertwined by the operator g(zD) = tjzj2s(z/JzJ)rn. The operator A(,a) = A(y4) = A (I) where A (p) = FA (,a)F-', A (s,m) then has a pole at each positive integer n and the pole-free subspace of p(s,m) at n is a quotient with A (,a)f(y) = y v)IvI-8-'K8(v)dv respect to the finite dimensional representation of SL2(C) of dimension n. and 2. Local components of Weil's representation K8(v) = '/4E E sgng (-av)r(I .I'+'sgn ). The assumptions and conventions of the previous sec- IaI-So(a) tion remain in force here. That is, F is a non-archi- Here f is assumed to belong to S(F), the Schwartz-Bruhat medean field with odd residual characteristic. Writing for we consider the in space of F; the summations are over {1,Tr,r, f } (a set of q(x) x2, operators L2(F) defined by representatives for the cosets of Fx modulo (F )2); r is the r{[ bf1p(x) = x[bq(x)]so(x)} [1] Downloaded by guest on September 29, 2021 1408 Mathematics: Gelbart and Sally Proc. Nat. Acad. Sci. USA 72 (1975) and BA'\GA such that 0(g) = ¢+(g) and fKA Io(k)()2dk < co If O is right KA-finite, the Eisenstein series on GA associ- r(w)
3. Eisenstein series on the metaplectic group COROLLARY 1. Let H(s) denote the space of R(ts). the of F is a number field. As usual, S,. will Let L2(GFVAA) denote space square-integrablefunc- In this section, = the denote the set of archimedean places of F and Gc. the tions on GF\GA satisfying y(g¢) Py(g). Then (a) product fI SL2(Fv). Let K. denote the standard direct integral VES0, maximal compact subgroup of G,,, and AO , the sub- H(s) d~s| Re(s) =O group of G. consisting of positive real diagonal matrices Im(8) >O at each place v E S,,. Set KA equal to KO,, K0 where Ko = K,. Then if g G GA, is equivalent to a GA-invariant subspace of L2(GF\OA) v finite whose orthocomplement, Ld2, decomposes discretely; in = nhdk (b) Ld2 is the direct sum of Lo2(GF\GA) (functions L2(GF\GA) satisfying the cuspidalconditionzfN,\NAIP(n?,g)dn
where n NA, = a E {([Oa-o],¢) C h, [o°-J, A",= = 0 for almost every g) and a space L02(E) consisting of jal = 1} and k E KA. The function H(g) = t is well residues p(g) = Res E(gi,4),s), Re(so) > O. defined. In particular, H([Oa- = log jal, and e2H(1b) S = 80 describes the modular function for BA. From familiar arguments in the theory of Eisenstein Let BA' denote the subgroup NAAFAco of BA. Let H series, it follows that any irreducible genuine representa- denote the Hilbert space of measurable functions on tion of GA occurring discretely in L2(GF\GA) outside the Downloaded by guest on September 29, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Intertwining Operators and Automorphic Forms 1409
= space of cusp forms imbeds as a quotient of some iy, has a pole at s '/2 if and only if (Iki + 1)/2 is odd. = our Theorem 4 implies precisely such a R(f2X so) with so C (0,1] a pole of some E(g,,,s). See ref. When F Q = Theorem 6 of Kubota 16 for the case 4(g) = 0(g). result. When F C we obtain (ref. 1). [Recall from Section 1 that A (s, i 1)f has a Remark 4: Observe that the correspondence +(g) pole at s = '/2 if and only if (IkI + 1)/2 is odd.] 4(g)e(s+l)H(#) provides a natural correspondence be- tween H and R(s), the space of R(x,s). This observa- COROLLARY 2. Suppose iF is an irreducible unitary tion, together with Theorem 3, makes it easy to prove genuine representation of GA which occurs discretely in that for Re(s) > 1, and v E H(s), L2(GF\OA) outside the space of cusp forms. Then i =- 0 ir, with each Tr, a quotient of P,('/2, 1) (compare the M(s)v = (wng) dn V notation of Section 1).
= IN,ib(wng) dn. Conclusion Our Siegel-Weil type formula is an identity between THEOREM 4. The only possible pole of E(g,4,s) to the the representation of GA generated by a certain theta- right of Re(s) = 0 is at s = 1/2. (Recall that 0(g) is non- function on GA and the representation generated by the trivial on Z2.) residues of certain Eisenstein series. More precisely, let Remark 5: The proof of Theorem 4 is not entirely ie denote the global Weil representation determined straight-forward but we can sketch it briefly below. by the local representations ;f, of Section 2, and TRE the Then we By Theorem 3, it suffices to analyze the poles of M(s). But representation generated by Res E(g,4O,s). M(s) (summed over all RA-types) is easily seen to inter- have: 8=1/2 twine R(2,s) with R(x, -s) and hence is the direct sum of operators M(s,X) intertwining R(G,X,s) with the THEOREM 5. f, and 7RE are equivalent. principal series representation R(t,X-',-s). Thus, it This Theorem is closely related to the Siegel-Weil suffices to analyze the poles of M(s,X). But R(G,X,s) is formula of ref. 17 via analytic continuation. More pre- m variables. unitary when Re(s) = 0. Hence, if A _ Qi*, we may cisely, suppose q is a quadratic form in = write R(t,X,s) as p(s,, ,,*) (x) in the notation of Section The Siegel-Weil formula asserts that Eg[1] Ig(4D) 1. (All tensor products are to be interpreted as ordinary if m > 5. Here IlQ() is a distribution on FA"I arising tensor products of the corresponding projective repre- from Weil's generalized theta-functions and E"(4) is sentations of A, or GC,.) defined by an Eisenstein-Siegel series which converges > = = On the other hand, if Re(s) > 0, let M(,u,*,s), for each only if m 5. But now when g (1,1), s (m/2)- v, denote the operator from the space of p(u,,*,s) to the 1 > 1, and 46 is appropriately chosen, E(g,4,s) coincides q space of p(J,,*-l,-s) defined by setting M(ju,,*,s)jp(g) with Eg(b) (the choice of depending on D and q). equal to Thus, Theorem 5 asserts that the Siegel-Weil formula is valid when m = 1 provided E9(4) is interpreted as the residue of some E(g,O,s) at s = -1/2 [recall E(g,O,s) = L(2s,/i,*) IN. (w9) E(g,M(s), -s) ]. Similarres ults should obtain for 2 where L(2s,M,*) is the local Eulerian factor for the global < m < 4 without always taking residues. L-function L(2s, ®spy). These operators are defined for s In case F is totally imaginary, Theorem 5 is implicit pure imaginary by analytic continuation (compare in ref. 1. Section 1). They are then scalar multiples of unitary The work of both authors is supported by grants from the operators and for given X are in fact unitary for almost National Science Foundation. all v. Thus, M(M,,*,s) can be defined on the space of p(s,,M*) for Re(s) = 0 and by Remark 4, 1. Kubota, T. (1969) Automorphic Functions and the Reciprocity Law in a Number Field (Kyoto University Press, Kyoto, M(s,X) = L(2s,X) M(Iu,*,s). Japan). v 2. Kubota, T. (1973) "A generalized Weil-type representa- tion," Technical Report TR 73-7, University of Maryland, If we finally compare M(,.*,s) (the intertwining opera- College Park, Md. tor for p(s,M,,*) normalized so as to leave the K-fixed 3. Shimura, G. (1973) "On modular forms of half-integral vector for p(s,,u,*) invariant whenever such exists) with weight," Ann. Math. 97, 440-481. 4. 8himura, G. (1974) "On the holomorphy of certain Dirichlet the intertwining operators A (s,s, *) of Section 1, series", mimeographed notes, Princeton University, Mathe- Theorem 4 can be made to follow from Theorem 1. [Keep matics Department, Princeton, N.J. "Die Funktionalgleichungen einiger in mind that L(2s, has a pole at s = 1/2 only if 5. Siegel, C. L. (1956) 0&s,,) GodA, Dirichletscher Reihen," Math. Z. 63, 363-373. is a principal quasi-character of A-. ] 6. Siegel, C. L. (1956) "A generalization of the Epstein zeta- function," J. Ind. Math. Soc. 20, 1-10. Remark 6: In ref. 4, Shimura proves that a certain 7. Moore, C. C. (1968) "Group extensions of p-adic linear classically defined Eisenstein series E'(z,k,s), z = x + groups," Publ. Math. I.H.E.S., No. 35. Downloaded by guest on September 29, 2021 1410 Mathematics: Gelbart and Sally Proc. Nat. Acad. Sci. USA 72 (1976)
8. Sally, P. J. (1968) "Unitary and uniformly bounded repre- 13. Shalika, J. A. (1966) "Representations of the two by two sentations of the two by two unimodular group over local unimodular groups over local fields," Thesis, Johns Hopkins fields," Amer. J. Math. XC, 406-442. University. 9. Sally, P. J. & Taibleson, M. (1966) "Special functions on 14. Weil, A. (1964) "Sur certaines groupes d'opdrateurs uni- locally compact fields," Acta Math. 116, 279-309. taries," Acta Math. 111, 143-211. 10. Tate, J. (1967) "Fourier analysis in number fields and 15. Langlands, R. P. (1967) "On the functional equations Hecke's Zeta-functions," Algebraic Number Theory (Thomp- satisfied by Eisenstein Series" (mimeographed notes, Yale son Book Co., Washington, D.C.). University Mathematics Department, New Haven, Conn.) 11. Sally, P. J. (1967) "Analytic continuation of the irreducible 16. Gelbart, S. (1975) "Automorphic forms on adele groups," unitary representations of the covering group of SL(2,R)," Annals of Mathematics Studies, No. 88 (Princeton Uni- Memoirs of the A.M.S., No. 69. versity Press, Princeton, N.J.). 12. Gelfand, I. M. et al. (1966) Generalized Functions (Academic 17. Weil, A. (1965) "Sur la formule de Siegel dans la th6orie des Press, New York), Vol. 5. groupes classiques," Acta Math. 113, 1-87. Downloaded by guest on September 29, 2021