Nilpotent Orbits and Whittaker Functions for Derived Functor Modules of Sp(2, R)

Miyazaki, Takuya

Canadian journal of mathematics, Tome 54 (2002) no. 4, pp. 769-794

Résumé

We study the moderate growth generalized Whittaker functions, associated to a unitary character $\psi $ of a unipotent subgroup, for the non-tempered cohomological representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$ . Through an explicit calculation of a holonomic system which characterizes these functions we observe that their existence is determined by the including relation between the real nilpotent coadjoint $G$ -orbit of $\psi $ in $\mathfrak{g}_{\mathbb{R}}^{*}$ and the asymptotic support of the cohomological representation.

Détail

DOI : 10.4153/CJM-2002-030-8

Mots-clés : 22E46, 22E30

Bibliographie

[B-V] [B-V] Barbasch, D. and Vogan, D., The local structure of characters. J. Funct. Anal. 37 (1980), 27–55. Google Scholar

[Ba] [Ba] Barchini, L., Szegö kernels associated with Zuckerman modules. J. Funct. Anal. 131 (1995), 145–182. Google Scholar

[D] [D] Djoković, D. Z., Closures of conjugacy classes in classical real linear Lie groups. Algebra, Carbondale 1980, Proc. Conf., Southern Illinois Univ., Carbondale, Ill., 1980, Lecture Notes in Math. 848, 1981, 63–83. Google Scholar

[K] [K] Kawanaka, N., Shintani lifting and Gelfand-Graev representations. Proc. Sympos. PureMath. 47 (1987), 147–163. Google Scholar

[Ko] [Ko] Kostant, B., On Whittaker vectors and representation theory. Invent.Math. 48 (1978), 101–184. Google Scholar

[M-O-S] [M-O-S] Magnus, W., Oberhettinger, F. and Soni, R. P., Formulas and Theorems for the special functions of Mathematical Physics, Third Edition. Die GrundlehrenMath.Wiss. Einzeldarstellungen 52, Springer-Verlag, 1966. Google Scholar

[Ma] [Ma] Matumoto, H., C∞-Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets. Compositio Math. 82 (1992), 189–244. Google Scholar

[M] [M] Miyazaki, T., The generalized Whittaker functions for Sp(2, R) and the gamma factor of the Andrianov L-function. J.Math. Sci. Tokyo 7 (2000), 241–295. Google Scholar

[N] [N] Nöel, A., Nilpotent orbits and theta-stable parabolic subalgebras. Represent. Theory 2 (1998), 1–32. Google Scholar

[O] [O] Oda, T., An explicit integral representation of Whittaker functions on Sp(2, R) for the large discrete series representations. Tôhoku Math. J. 46 (1994), 261–279. Google Scholar

[S-V] [S-V] Schmid, W. and Vilonen, K., Characteristic cycles and wave front cycles of representations of reductive Lie groups. Ann. of Math. 151 (2000), 1071–1118. Google Scholar

[V1] [V1] Vogan, D. A., Associated varieties and unipotent representations. Harmonic Analysis on Reductive Groups, (eds.,W. Baker and P. Sally), Birkhäuser, Boston, Bassel, Berlin, 1991, 315–388. Google Scholar

[V2] [V2] Vogan, D. A., Gelfand-Kirillov dimension for Harish-Chandra modules. Invent.Math. 48 (1978), 75–98. Google Scholar

[W] [W] Wong, H.-W., Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations. J. Funct. Anal. 129 (1995), 428–454. Google Scholar

[Y1] [Y1] Yamashita, H., Finite multiplicity theorems for induced representations of semisimple Lie groups I. J. Math. Kyoto Univ. 28 (1988), 173–211; II: Applications to generalized Gelfand-Graev representations, ibid. 28 (1988), 383–444. Google Scholar

[Y2] [Y2] Yamashita, H., Cayley transform and generalized Whittaker models for irreducible highest weight modules. preprint. Google Scholar