Poisson transforms on vector bundles

Yang, An

doi:10.1090/S0002-9947-98-01659-6

Poisson transforms on vector bundles
HTML articles powered by AMS MathViewer

by An Yang PDF

Trans. Amer. Math. Soc. 350 (1998), 857-887 Request permission

Abstract:

Let $G$ be a connected real semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $(\tau ,V)$ be an irreducible unitary representation of $K$, and $G\times _K V$ the associated vector bundle. In the algebra of invariant differential operators on $G\times _K V$ the center of the universal enveloping algebra of $\operatorname {Lie}(G)$ induces a certain commutative subalgebra $Z_\tau$. We are able to determine the characters of $Z_\tau$. Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to $\tau$ the trivial representation.

References

E. P. van den Ban, The principal series for a reductive symmetric space. II. Eisenstein integrals, J. Funct. Anal. 109 (1992), no. 2, 331–441. MR 1186325, DOI 10.1016/0022-1236(92)90021-A
E. P. van den Ban and H. Schlichtkrull, Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces, J. Reine Angew. Math. 380 (1987), 108–165. MR 916202
W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $G$, Canad. J. Math. 41 (1989), no. 3, 385–438. MR 1013462, DOI 10.4153/CJM-1989-019-5
Anton Deitmar, Invariant operators on higher $K$-types, J. Reine Angew. Math. 412 (1990), 97–107. MR 1079003, DOI 10.1515/crll.1990.412.97
Pierre-Y. Gaillard, Eigenforms of the Laplacian on real and complex hyperbolic spaces, J. Funct. Anal. 78 (1988), no. 1, 99–115. MR 937634, DOI 10.1016/0022-1236(88)90134-6
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Sigurdur Helgason, A duality for symmetric spaces with applications to group representations. II. Differential equations and eigenspace representations, Advances in Math. 22 (1976), no. 2, 187–219. MR 430162, DOI 10.1016/0001-8708(76)90153-5
Sigurdur Helgason, Some results on invariant differential operators on symmetric spaces, Amer. J. Math. 114 (1992), no. 4, 789–811. MR 1175692, DOI 10.2307/2374798
T. Kato, Perturbation theory for linear operators, 1980.
M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Ōshima, and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2) 107 (1978), no. 1, 1–39. MR 485861, DOI 10.2307/1971253
J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1–44. MR 346093, DOI 10.1090/S0002-9947-1973-0346093-X
John B. Lewis, Eigenfunctions on symmetric spaces with distribution-valued boundary forms, J. Functional Analysis 29 (1978), no. 3, 287–307. MR 512246, DOI 10.1016/0022-1236(78)90032-0
Katsuhiro Minemura, Invariant differential operators and spherical sections on a homogeneous vector bundle, Tokyo J. Math. 15 (1992), no. 1, 231–245. MR 1164198, DOI 10.3836/tjm/1270130263
Toshio Ōshima and Jir\B{o} Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), no. 1, 1–81. MR 564184, DOI 10.1007/BF01389818
Nobukazu Shimeno, Eigenspaces of invariant differential operators on a homogeneous line bundle on a Riemannian symmetric space, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 1, 201–234. MR 1049027
Wilfried Schmid, Boundary value problems for group invariant differential equations, Astérisque Numéro Hors Série (1985), 311–321. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837206
Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
Harmen van der Ven, Vector valued Poisson transforms on Riemannian symmetric spaces of rank one, J. Funct. Anal. 119 (1994), no. 2, 358–400. MR 1261097, DOI 10.1006/jfan.1994.1015
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
Nolan R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 287–369. MR 727854, DOI 10.1007/BFb0071436
Nolan R. Wallach, On Harish-Chandra’s generalized $C$-functions, Amer. J. Math. 97 (1975), 386–403. MR 399357, DOI 10.2307/2373718
Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999