Metaplectic forms. (English) Zbl 0559.10026

The authors prove a Siegel-Weil formula [A. Weil, Acta Math. 113, 1-87 (1965; Zbl 0161.023)] for the n-fold cover \(^{\sim}_ r({\mathbb{A}})\) of \(GL_ r({\mathbb{A}})\), where \({\mathbb{A}}\) is the ring of adèles of an A- field k containing all the n-th-roots of unity. Here n is a positive integer coprime to the characteristic of k. For \(^{\sim}_ 2({\mathbb{A}})\) and \(n=2\), the formula proves that the classical theta-functions are in fact residues of Eisenstein series [T. Kubota, On automorphic functions and the reciprocity law in a number field (1969; Zbl 0231.10017); and S. Gelbart and P. Sally, Proc. Natl. Acad. Sci. USA 72, 1406-1410 (1975; Zbl 0303.22010)]. In fact, generalized theta-functions are certain functions on \(GL_ r(k)^*\setminus^{\sim}_ r({\mathbb{A}})\). Using the language of group representations they may be considered as certain constituents of \(L^ 2(GL_ r(k)^*\setminus GL_ r({\mathbb{A}})).\)
More precisely, at every place v of k, there is always a reducible principal series representation of \(^{\sim}_ r(k_ v)\) which has a unique distinguished quotient (in the terminology of S. Gelbart and I. I. Piatetski-Shapiro, Invent Math. 59, 145-188 (1980; Zbl 0426.10027), i.e. it has a unique Whittaker model). This is in fact only true if \(r=n\) or n-1. The global representation which has these distinguished representations as its local components is then the generalized theta form, and the main result of this paper, Theorem II.2.2, proves that it is in fact the residue of an Eisenstein series, and consequently is automorphic. When \(r=2\), \(n=3\), and \(k={\mathbb{Q}}(\sqrt{-3})\), a similar result has been proved by the second author [J. Reine Angew. Math. 296, 125-161 (1977; Zbl 0358.10011) and 217-220 (1977; Zbl 0358.10012)] using classical language; and P. Deligne [Sémin. Bourbaki, Exp. No.539, Lect. Notes Math. 770, 244-277 (1980; Zbl 0433.10018)], using group representations.
As mentioned above, the methods used in the paper are those of the representation theory of metaplectic groups. These are developed in Chapter I. Among many interesting results proved in Chapter I is a formula for the dimension of the space of Whittaker functionals of any irreducible admissible representation of \(^{\sim}_ r(k_ v)\) in terms of its character (Theorem I.5.3). Finally at the end of Chapter II, global methods are used to answer a number of local questions whose proofs are otherwise not available (Theorem II.2.5 and Corollary II.2.6).
Reviewer: F.Shahidi


11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F12 Automorphic forms, one variable
11R56 Adèle rings and groups
Full Text: DOI Numdam EuDML


[1] H. Aritürk, On the composition series of principal series representations of a 3-fold covering group of SL (2,k),Nagoya Math. J.,77 (1980), 177–196. · Zbl 0405.22014
[2] H. Bass, J. Milnor andJ.-P. Serre, Solution of the congruence subgroup problem for SL n (n 3) and Sp 2n (n 2),Publ. Math. IHES,33 (1967), 59–137. · Zbl 0174.05203
[3] I. N. Bernstein andA. V. Zelevinski, Induced representations of reductivep-adic groups,Ann. scient. Ec. Norm. Sup.,10 (1977), 441–472.
[4] A. Borel andHarish-Chandra, Arithmetic subgroups of algebraic groups,Annals of Math.,75 (1962), 485–532. · Zbl 0107.14804 · doi:10.2307/1970210
[5] P. Deligne,Notes on metaplectic groups (Unpublished).
[6] P. Deligne, Sommes de Gauss cubiques et revêtements de SL(2).Sém. Bourbaki, Juin 1979, exposé no 539;Springer Lecture Notes,770 (1980), 244–277.
[7] Y. Flicker, Automorphic forms on covering groups of GL(2),Inv. Math.,57 (1980), 119–182. · Zbl 0431.10014 · doi:10.1007/BF01390092
[8] S. Gelbart, Weil’s representation and the spectrum of the metaplectic group,Springer Lecture Notes,530 (1976). · Zbl 0365.22017
[9] S. Gelbart, R. Howe andI. I. Piatetskii-Shapiro, Existence and uniqueness of Whittaker models for the metaplectic group,Israeli Math. J.,34 (1979), 21–37. · Zbl 0441.22015 · doi:10.1007/BF02761822
[10] S. Gelbart andI. I. Piatetskii-Shapiro, On Shimura’s correspondence for modular forms of half-integral weight,Proceedings, Colloquium on Automorphic Forms, Representation Theory and Arithmetic, Bombay, 1979, 1–39.
[11] S. Gelbart andI. I. Piatetskii-Shapiro, Distinguished representations and modular forms of half-integral weight,Inv. Math.,59 (1980), 145–188. · Zbl 0426.10027 · doi:10.1007/BF01390042
[12] I. M. Gelfand, M. I. Graev andI. I. Piatetskii-Shapiro,Representation Theory and Automorphic Functions, Nauka (1966) (Russian), Saunders (1969) (English).
[13] I. M. Gelfand andD. A. Kazhdan, Representations of the group GL(n, K) where K is a local field,Lie Groups, Budapest, Ed. I. M. Gelfand, 1974. · Zbl 0348.22011
[14] I. M. Gelfand andM. I. Naimark, Unitary Representations of the Classical Groups,Trudy Mat. Inst. Steklov,36, 1950 (Russian), Akademie-Verlag Berlin, 1958 (German).
[15] G. Harder, Minkowski’sche Reduktionstheorie über Funktionenkörper,Inv. Math.,7 (1969), 107–149. · Zbl 0242.20046 · doi:10.1007/BF01418773
[16] G. Harder, Chevalley groups over function fields and automorphic forms,Annals of Math.,100 (1974), 249–306. · Zbl 0309.14041 · doi:10.2307/1971073
[17] Harish-Chandra andVan Dijk, Harmonic Analysis on Reductivep-adic Groups,Springer Lecture Notes,162 (1970).
[18] Harish-Chandra,Admissible invariant distributions on reductive p-adic groups, Princeton, Preprint, IAS. · Zbl 0433.22012
[19] Harish-Chandra, A submersion principle and its applications,Geometry and Analysis, Papers dedicated to the Memory of V. K. Patodi, Springer-Verlag, 1981, 95–102. · Zbl 0512.22010
[20] E. Hecke,Gesammelte Werke, Vandenhoeck u. Ruprecht, 1970.
[21] R. Howe, The Fourier transform and germs of characters (case of GLn over ap-adic field),Math. Ann.,208 (1974), 305–322. · doi:10.1007/BF01432155
[22] H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley,Bull. Soc. Math., France,95 (1967), 243–309. · Zbl 0155.05901
[23] H. Jacquet andR. P. Langlands, Automorphic Forms on GL(2),Springer Lecture Notes,114 (1970). · Zbl 0236.12010
[24] M. Kneser, Strong Approximation,Proc. Symp. Pure Math., IX (1966), 187–196. · Zbl 0201.37904
[25] T. Kubota, Topological covering of SL(2) over a local field,J. Math. Soc. Japan,19 (1967), 114–121. · Zbl 0166.29603 · doi:10.2969/jmsj/01910114
[26] T. Kubota,Automorphic Forms and the Reciprocity Law in a Number Field, Kyoto Univ., 1969. · Zbl 0231.10017
[27] T. Kubota, Some results concerning reciprocity and functional analysis,Actes du Congr. Int. Math à Nice, 1970, t. I, 395–399.
[28] T. Kubota, Some number-theoretical results on real-analytic automorphic forms,Several Complex Variables II, Maryland, 1970,Springer Lecture Notes,185 (1971), 87–96.
[29] R. P. Langlands, On the Functional Equations satisfied by Eisenstein Series,Springer Lecture Notes,544 (1976). · Zbl 0332.10018
[30] R. P. Langlands, The volume of the fundamental domain for some arithmetical subgroups,Proc. Symp. Pure Math., IX (1966), 143–148. · Zbl 0218.20041
[31] R. P. Langlands, On the Classification of Irreducible Representations of Real Algebraic Groups,Lecture Notes, Princeton, IAS. · Zbl 0741.22009
[32] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés,Ann. scient., Ec. Norm. Sup., (1969), 1–62. · Zbl 0261.20025
[33] J. Milnor, An introduction to algebraic K-theory,Annals of Mathematics Studies,72, 1971. · Zbl 0237.18005
[34] C. Moen,Ph. D. Thesis, Univ. of Chicago, 1979.
[35] C. C. Moore, Group extensions ofp-adic and adelic linear groups,Publ. Math. IHES,35 (1968), 157–222.
[36] L. E. Morris, Eisenstein series for reductive groups over global function fields,Canad. Math. J.,34 (1982), 91–168. · Zbl 0499.22021 · doi:10.4153/CJM-1982-009-2
[37] S. J. Patterson, A cubic analogue of the theta series,J. für die r.u.a. Math.,296 (1977), 125–161, 217–220. · Zbl 0358.10011
[38] S. J. Patterson, The constant term of the cubic theta series,J. für die r.u.a. Math.,336 (1982), 185–190. · Zbl 0484.10018
[39] S. J. Patterson, On the distribution of general Gauss sums at prime arguments,Recent Progress in Analytic Number Theory, Vol. 2, Academic Press, 1981. · Zbl 0463.10027
[40] I. I. Piatetskii-Shapiro, Euler subgroups,Lie Groups, Budapest, Ed. I. M. Gelfand, 1974.
[41] F. Rodier, Modèle de Whittaker et charactères de représentations.Non-Commutative Harmonic Analysis, Proceedings, 1976,Springer Lecture Notes,466 (1977), 117–195.
[42] J.-P. Serre andH. Stark, Modular forms of weight 1/2,Modular Forms in One Varible VI, Bonn, 1976,Springer Lecture Notes,627 (1977), 27–67.
[43] F. Shahidi, Functional equation satisfied by certain L-series,Comp. Math.,37 (1978), 171–207. · Zbl 0393.12017
[44] F. Shahidi, Whittaker models for real groups,Duke Math. J.,47 (1980), 99–125. · Zbl 0433.22007 · doi:10.1215/S0012-7094-80-04708-0
[45] J. Shalika, The multiplicity one theorem for GL(n),Annals of Math.,100 (1974), 171–193. · Zbl 0316.12010 · doi:10.2307/1971071
[46] A. J. Silberger, Introduction to Harmonic Analysis on Reductivep-adic Groups,Princeton Math. Notes,23, 1979. · Zbl 0458.22006
[47] R. Steinberg, Lectures on Chevalley Groups,Lectures Notes, Yale Math. Dept., 1968. · Zbl 1196.22001
[48] A. Weil,L’intégration dans les groupes topologiques et ses applications, Hermann, 1940. · Zbl 0063.08195
[49] A. Weil, Sur certains groupes d’opérateurs unitaires,Acta Math.,111 (1964), 143–211. · Zbl 0203.03305 · doi:10.1007/BF02391012
[50] A. Weil,Basic Number Theory, Springer, 1967. · Zbl 0176.33601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.