Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



The Fourier-Jacobi map and small representations

Author: Martin H. Weissman
Journal: Represent. Theory 7 (2003), 275-299
MSC (2000): Primary 20G05, 22E50; Secondary 22E35, 22E10
Published electronically: July 28, 2003
MathSciNet review: 1993361
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the ``Fourier-Jacobi'' functor on smooth representations of split, simple, simply-laced $p$-adic groups. This functor has been extensively studied on the symplectic group, where it provides the representation-theoretic analogue of the Fourier-Jacobi expansion of Siegel modular forms. Our applications are different from those studied classically with the symplectic group. In particular, we are able to describe the composition series of certain degenerate principal series. This includes the location of minimal and small (in the sense of the support of the local character expansion) representations as spherical subquotients.

References [Enhancements On Off] (What's this?)

  • [B-S] Rolf Berndt and Ralf Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, vol. 163, Springer-Verlag, 1998. MR 99i:11030
  • [B-Z] I.N. Bernstein and A.V. Zelevinsky, Induced representations of reductive $p$-adic groups. I., Ann. Sci. École Norm. Sup. 10 (4) (1977), 441-472. MR 58:28310
  • [Del] Pierre Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206-226. MR 45:5137
  • [Dij] G. van Dijk, Smooth and admissible representations of $p$-adic unipotent groups, Compositio Math. 37 (1) (1978), 77-101. MR 58:11239
  • [E-Z] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics, vol. 55, Springer-Verlag, 1985. MR 86j:11043
  • [Gro] Benedict H. Gross, On the motive of $G$ and the principal homomorphism $SL_{2} \rightarrow \hat G$, Asian J. of Math. 1 (1), 208-213. MR 99d:20077
  • [F-H] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematica, vol. 129, Springer-Verlag, 1991. MR 93a:20069
  • [G-W] Benedict H. Gross and Nolan R. Wallach, On quaternionic discrete series representations and their continuations, J. Reine Angew. Math. 481, 73-123. MR 98f:22022
  • [How] Roger Howe, On a notion of rank for unitary representations of the classical groups, Harmonic analysis and group representations, 1982, pp. 223-331. MR 86j:22016
  • [K-S] David Kazhdan and Gordan Savin, The smallest representation of simply laced groups, Israel Math. Conf. Proc., vol. 2, (Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, 1990, pp. 209-223. MR 93f:22019
  • [KuS] Stephen Kudla and W. Jay Sweet Jr., Degenerate principal series representations of $U(n,n)$, Israel J. Math 98 (1997), 253-306. MR 98h:22021
  • [M-W] C. M\oeglin and J.L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (3) (1987), 427-452. MR 89f:22024
  • [MVW] C. M\oeglin, M.F. Vigneras, and J.L. Waldspurger, Correspondence de Howe sur un corp $p$-adique, Lecture Notes in Mathematics, No. 1291, 1987. MR 91f:11040
  • [M-S] G. Muic and F. Shahidi, Irreducibility of standard representations for Iwahori-spherical representations, Math. Ann. 312 (1) (1998), 151-165. MR 99g:22012
  • [RRS] Roger Richardson and Gerhard Röhrle and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (3) (1992), 649-671. MR 93j:20092
  • [Tat] John Tate, Number Theoretic Background, Proc. of Symp. in Pure Math., Vol. 33., 1979, pp. 3-26. MR 80m:12009

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20G05, 22E50, 22E35, 22E10

Retrieve articles in all journals with MSC (2000): 20G05, 22E50, 22E35, 22E10

Additional Information

Martin H. Weissman
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, University of California, Berkeley, 940 Evans Hall, Berkeley, California 94704

Keywords: Representation theory
Received by editor(s): March 7, 2002
Received by editor(s) in revised form: September 2, 2002, October 31, 2002, January 2, 2003, and April 23, 2003
Published electronically: July 28, 2003
Additional Notes: The author was supported in part by a NSF Graduate Research Fellowship during the preparation of this paper.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society