The Fourier-Jacobi map and small representations
HTML articles powered by AMS MathViewer
- by Martin H. Weissman
- Represent. Theory 7 (2003), 275-299
- DOI: https://doi.org/10.1090/S1088-4165-03-00197-3
- Published electronically: July 28, 2003
- PDF | Request permission
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
- Rolf Berndt and Ralf Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, vol. 163, Birkhäuser Verlag, Basel, 1998. MR 1634977, DOI 10.1007/978-3-0348-0283-3
- I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
- Pierre Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206–226 (French). MR 296076, DOI 10.1007/BF01404126
- G. van Dijk, Smooth and admissible representations of $p$-adic unipotent groups, Compositio Math. 37 (1978), no. 1, 77–101. MR 492085
- Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
- Benedict H. Gross, On the motive of $G$ and the principal homomorphism $\textrm {SL}_2\to \widehat G$, Asian J. Math. 1 (1997), no. 1, 208–213. MR 1480995, DOI 10.4310/AJM.1997.v1.n1.a8
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Benedict H. Gross and Nolan R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996), 73–123. MR 1421947, DOI 10.1515/crll.1996.481.73
- Roger Howe, On a notion of rank for unitary representations of the classical groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 223–331. MR 777342
- D. Kazhdan and G. Savin, The smallest representation of simply laced groups, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 209–223. MR 1159103
- Stephen S. Kudla and W. Jay Sweet Jr., Degenerate principal series representations for $\textrm {U}(n,n)$, Israel J. Math. 98 (1997), 253–306. MR 1459856, DOI 10.1007/BF02937337
- C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (1987), no. 3, 427–452 (French). MR 913667, DOI 10.1007/BF01200363
- Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR 1041060, DOI 10.1007/BFb0082712
- Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99, DOI 10.1007/s002080050216
- Roger Richardson, Gerhard Röhrle, and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649–671. MR 1189494, DOI 10.1007/BF01231348
- J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607, DOI 10.1090/pspum/033.2/546607
Bibliographic 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
- MR Author ID: 718173
- Email: martinw@math.harvard.edu
- 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.
- © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory 7 (2003), 275-299
- MSC (2000): Primary 20G05, 22E50; Secondary 22E35, 22E10
- DOI: https://doi.org/10.1090/S1088-4165-03-00197-3
- MathSciNet review: 1993361