Embeddings of Harish-Chandra modules, 𝔫-homology and the composition series problem: the case of real rank one

Collingwood, David H.

doi:10.1090/S0002-9947-1984-0752491-X

Embeddings of Harish-Chandra modules, $\mathfrak {n}$-homology and the composition series problem: the case of real rank one
HTML articles powered by AMS MathViewer

by David H. Collingwood PDF

Trans. Amer. Math. Soc. 285 (1984), 565-579 Request permission

Abstract:

Let $G$ be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup $P = MAN$ and form the (normalized) principal series representations $I_P^G(U)$. In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands’ classification) all irreducible submodules and quotients of $I_P^G(U)$. As a corollary, all embeddings of an irreducible Harish-Chandra module into principal series are computed. The number of such embeddings is always less than or equal to three. These computations are equivalent to the determination of zero ${\mathfrak {n}}$-homology.

References

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
David H. Collingwood, Harish-Chandra modules with the unique embedding property, Trans. Amer. Math. Soc. 281 (1984), no. 1, 1–48. MR 719657, DOI 10.1090/S0002-9947-1984-0719657-6
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
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, Weyl group of a cuspidal parabolic, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 275–294. MR 376963, DOI 10.24033/asens.1288
Hrvoje Kraljević, On representations of the group $SU(n,1)$, Trans. Amer. Math. Soc. 221 (1976), no. 2, 433–448. MR 409725, DOI 10.1090/S0002-9947-1976-0409725-6
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

Composition factors of the principal series representations of the group

Asympotic behavior of the matrix coefficients of the discrete series

44

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
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., Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J. 49 (1982), no. 4, 943–1073. MR 683010

Representations of real reductive Lie groups

Description of the quasi-simple irreducible representations of the groups

and

11

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