Harish-Chandra modules with the unique embedding property

Collingwood, David H.

doi:10.1090/S0002-9947-1984-0719657-6

Harish-Chandra modules with the unique embedding property
HTML articles powered by AMS MathViewer

by David H. Collingwood PDF

Trans. Amer. Math. Soc. 281 (1984), 1-48 Request permission

Abstract:

Let $G$ be a connected semisimple real matrix group. In view of Casselman’s subrepresentation theorem, every irreducible admissible representation of $G$ may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.

References

Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
W. Casselman, Jacquet modules for real reductive groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 557–563. MR 562655
William Casselman and Dragan Miličić, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. 49 (1982), no. 4, 869–930. MR 683007, DOI 10.1215/S0012-7094-82-04943-2
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564. MR 180628, DOI 10.2307/2373023
Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813
Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26–65. MR 58604, DOI 10.1090/S0002-9947-1954-0058604-0
Henryk Hecht, The characters of some representations of Harish-Chandra, Math. Ann. 219 (1976), no. 3, 213–226. MR 427542, DOI 10.1007/BF01354284
Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112
Henryk Hecht and Wilfried Schmid, Characters, asymptotics and ${\mathfrak {n}}$-homology of Harish-Chandra modules, Acta Math. 151 (1983), no. 1-2, 49–151. MR 716371, DOI 10.1007/BF02393204

On the asymptotics of Harish-Chandra modules

Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
A. W. Knapp and N. R. Wallach, Szegö kernels associated with discrete series, Invent. Math. 34 (1976), no. 3, 163–200. MR 419686, DOI 10.1007/BF01403066
A. W. Knapp and Gregg J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math. (2) 116 (1982), no. 2, 389–455. MR 672840, DOI 10.2307/2007066
M. W. Baldoni Silva and H. Kraljević, Composition factors of the principal series representations of the group $\textrm {Sp}(n,\,1)$, Trans. Amer. Math. Soc. 262 (1980), no. 2, 447–471. MR 586728, DOI 10.1090/S0002-9947-1980-0586728-3

On the classification of irreducible representations of real algebraic groups

Asymptotic behavior of matrix coefficients of the discrete series

44

M. Primc, Representations of semisimple Lie groups with one conjugacy class of Cartan subgroups, Glasnik Mat. Ser. III 14(34) (1979), no. 2, 227–254 (English, with Serbo-Croatian summary). MR 646350
Wilfried Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47–144. MR 396854, DOI 10.1007/BF01389847
Wilfried Schmid, Vanishing theorems for Lie algebra cohomology and the cohomology of discrete subgroups of semisimple Lie groups, Adv. in Math. 41 (1981), no. 1, 78–113. MR 625335, DOI 10.1016/S0001-8708(81)80005-9
Wilfried Schmid, Two character identities for semisimple Lie groups, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977, pp. 196–225. MR 0507247
M. Welleda Baldoni Silva, The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one, Trans. Amer. Math. Soc. 261 (1980), no. 2, 303–368. MR 580893, DOI 10.1090/S0002-9947-1980-0580893-X
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
Ernest Thieleker, On the quasi-simple irreducible representations of the Lorentz groups, Trans. Amer. Math. Soc. 179 (1973), 465–505. MR 325856, DOI 10.1090/S0002-9947-1973-0325856-0

A character formula for the discrete series of a semisimple Lie group

David A. Vogan Jr., Gel′fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. MR 506503, DOI 10.1007/BF01390063
David A. Vogan Jr., Irreducible characters of semisimple Lie groups. I, Duke Math. J. 46 (1979), no. 1, 61–108. MR 523602
David A. Vogan Jr., Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures, Duke Math. J. 46 (1979), no. 4, 805–859. MR 552528
David A. Vogan, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math. 71 (1983), no. 2, 381–417. MR 689650, DOI 10.1007/BF01389104
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
Nolan R. Wallach, Square integrable automorphic forms and cohomology of arithmetic quotients of $\textrm {SU}(p,\,q)$, Math. Ann. 266 (1984), no. 3, 261–278. MR 730169, DOI 10.1007/BF01475577

Representations of semisimple Lie groups and Lie algebras

48

D. P. Želobenko, A description of the quasisimple irreducible representations of the groups $U(n,1)$ and $\textrm {Spin}(n,1)$, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 1, 34–53, 231 (Russian). MR 0450464
Gregg Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math. (2) 106 (1977), no. 2, 295–308. MR 457636, DOI 10.2307/1971097