Representation Theory

Author(s): Paul Mezo
Journal: Represent. Theory 5 (2001), 524-580.
MSC (2000): Primary 11F70; Secondary 11F72, 22E55
Posted: November 27, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: Let $\mathbf{A}$ be the adele ring of a number field containing the

th roots of unity, and let $\widetilde{\mathrm{GL}}(r,\mathbf{A})$ be an

-fold metaplectic covering of $\mathrm{GL}(r,\mathbf{A})$ . Under an assumption on

, we prove identities between all of the terms in Arthur's invariant trace formulas for $\widetilde{\mathrm{GL}}(r,\mathbf{A})$ and $\mathrm{GL}(r,\mathbf{A})$ . We then establish a correspondence between the automorphic representations of these groups.

1.: W. A. Adkins and S. H. Weintraub, Algebra, Springer-Verlag, 1992. MR 94a:00001
2.: J. Arthur, The trace formula in invariant form, Ann. of Math. 114 (1981), 1-74. MR 84a:10031
3.: -, On a family of distributions obtained from Eisenstein series II: Explicit formulas, Amer. J. Math. 104 (1982), 1289-1336. MR 85d:22033
4.: -, A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (1983), 1-89. MR 84k:22021
5.: -, On a family of distributions obtained from orbits, Canad. J. Math. 38 (1986), 179-214. MR 87k:11058
6.: -, The characters of supercuspidal representations as weighted orbital integrals, Proc. Indian Acad. Sci. (Math. Sci.) 97 (1987), 3-19. MR 90c:22052
7.: -, The invariant trace formula I. Local theory, J. Amer. Math. Soc. 1 (1988), 323-383. MR 89e:22029
8.: -, The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501-554. MR 89j:22039
9.: -, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223-293. MR 89h:22036
10.: -, Intertwining operators and residues I. Weighted characters, J. Funct. Anal. 84 (1989), no. 1, 19-84. MR 90j:22018
11.: -, Intertwining operators and residues II. Invariant distributions, Comp. Math. 70 (1989), 51-99. MR 90j:22019
12.: -, Towards a local trace formula, Algebraic analysis, geometry and number theory, 1989. MR 98h:22020
13.: J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Stud., vol. 120, Princeton Univ. Press, Princeton, NJ, 1989. MR 90m:22041
14.: I. N. Bernstein and A. N. Zelevinsky, Induced representations of $\mathfrak{p}$ -adic groups. I., Ann. Sci. Ecole Norm. Sup. 10 (1977), 441-472. MR 58:28310
15.: J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, J. Analyse Math. 47 (1986), 180-192. MR 88g:22016
16.: L. Clozel and P. Delorme, Le Théorème de Paley-Wiener invariant pour les groupes de Lie réductifs, Invent. Math. 77 (1984), 427-453. MR 86h:22015
17.: Y. Flicker, Automorphic forms on covering groups of GL(2), Invent. Math. 57 (1980), no. 2, 119-182. MR 81m:10057
18.: Y. Flicker and D. Kazhdan, Metaplectic correspondence, Inst. Hautes Etudes Sci. Publ. Math. 64 (1986), 53-110. MR 88d:11049
19.: Harish-Chandra, Discrete series for semisimple Lie groups I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241-318. MR 36:2744
20.: J.-S. Huang, The unitary dual of the covering groups of GL(n) over a local field, Harmonic Analysis in China, Kluwer Academic Publishers, 1995, pp. 103-124. MR 96g:22024
21.: H. Jacquet, On the residual spectrum of $\mathrm{GL}(n)$ , Lie group representations, II, Lecture Notes in Math., vol. 1041, Springer-Verlag, 1984, pp. 185-208. MR 85k:22045
22.: H. Jacquet and R. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., vol. 114, Springer-Verlag, 1970. MR 53:5481
23.: H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), 777-815. MR 82m:10050a
24.: D. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Etudes Sci. Publ. Math. 59 (1984), 35-142. MR 85g:22033
25.: -, Towards a generalized Shimura correspondence, Adv. in Math. 60 (1986), 161-234. MR 87m:22050
26.: M. Kneser, Strong approximation, Proceedings of Symposia in Pure Mathematics, vol. 9, 1965, pp. 187-196. MR 35:4225
27.: S. Lang, Algebraic number theory, Addison-Wesley Publishing Company, 1970. MR 44:181
28.: -, Algebra, Addison-Wesley, 1984. MR 86j:00003
29.: R. P. Langlands, Base change for GL(2), Princeton University Press, 1980. MR 82a:10032
30.: P. Mezo, Comparisons of general linear groups and their metaplecitic coverings I, To appear in Canad. J. Math.
31.: -, Matching of weighted orbital integrals for metaplectic correspondences, To appear in Canad. Math. Bull.
32.: -, A global comparison for general linear groups and their metaplectic coverings, Ph.D. thesis, University of Toronto, 1998.
33.: -, New identities of differential operators from orbital integrals on $\mathrm{GL}(r,\mathbf{C})$ , Proc. Amer. Math. Soc. (to appear).
34.: C. M $\oe$ glin and J.-L. Waldspurger, Le spectre résiduel de $\mathrm{GL}(n)$ , Ann. Sci. Ecole. Norm. Sup. 22 (1989), 605-674. MR 91b:22028
35.: G. Shimura, On modular forms of half-integral weight, Ann. of Math. 97 (1973), 440-481. MR 48:10989
36.: H. Sun, Spectral decomposition of a covering of : the Borel case, Preprint.
37.: M. Tadic, On characters of irreducible unitary representations of general linear groups, Abh. Math. Sem. Univ. Hamburg 65 (1995), 341-363. MR 96m:22039
38.: M.-F. Vignéras, Caractérisation des intégrales orbitales sur un groupe réductif p-adique, J. Fac. Sc. Univ. Tokyo, Sec IA 29 (1981), 945-961. MR 84b:22017
39.: A. V. Zelevinsky, Induced representations of reductive p-adic groups II, Ann. Sci. Ecole. Norm. Sup. 13 (1980), 165-210. MR 83g:22012

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-4165

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS