Intertwining operators into cohomology representations for semisimple Lie groups II

Author:
Robert W. Donley Jr.

Journal:
Represent. Theory **2** (1998), 278-297

MSC (1991):
Primary 22E46

DOI:
https://doi.org/10.1090/S1088-4165-98-00044-2

Published electronically:
June 16, 1998

MathSciNet review:
1628039

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: One approach to constructing unitary representations for semisimple Lie groups utilizes analytic cohomology on open orbits of generalized flag manifolds. This work gives explicit formulas for harmonic cocycles associated to certain holomorphic homogeneous vector bundles, extending previous results of the author (*Intertwining operators into cohomology representations for semisimple Lie groups*, J. Funct. Anal. ** 151** (1997), 138-165). The key step shows that holomorphic discrete series representations and their limits are well-behaved with respect to restriction to certain submanifolds.

**[AR]**R. Aguilar-Rodriguez,*Connections between representations of Lie groups and sheaf cohomology*, Ph.D. dissertation, 1987.**[Ba]**L. Barchini,*Szegő mappings, harmonic forms, and Dolbeault cohomology*, J. Funct. Anal.**118**(1993), no. 2, 351–406. MR**1250267**, https://doi.org/10.1006/jfan.1993.1148**[BKZ]**L. Barchini, A. W. Knapp, and R. Zierau,*Intertwining operators into Dolbeault cohomology representations*, J. Funct. Anal.**107**(1992), no. 2, 302–341. MR**1172027**, https://doi.org/10.1016/0022-1236(92)90110-5**[BZ]**L. Barchini and R. Zierau,*Square integrable harmonic forms and representation theory*, Duke Math. J. (to appear).**[BE]**Robert J. Baston and Michael G. Eastwood,*The Penrose transform*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Its interaction with representation theory; Oxford Science Publications. MR**1038279****[Bo]**R. Bott,*Homogeneous vector bundles*, Ann. of Math.**66**(1957), 203-248. MR**19:681d****[CM]**William Casselman and Dragan Miličić,*Asymptotic behavior of matrix coefficients of admissible representations*, Duke Math. J.**49**(1982), no. 4, 869–930. MR**683007**, https://doi.org/10.1215/S0012-7094-82-04943-2**[Do]**R. W. Donley, Jr.,*Intertwining operators into cohomology representations for semisimple Lie groups*, J. Funct. Anal.**151**(1997), 138-165. CMP**98:06****[EH]**Thomas Enright, Roger Howe, and Nolan Wallach,*A classification of unitary highest weight modules*, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97–143. MR**733809****[GS]**Phillip Griffiths and Wilfried Schmid,*Locally homogeneous complex manifolds*, Acta Math.**123**(1969), 253–302. MR**259958**, https://doi.org/10.1007/BF02392390**[HC1]**Harish-Chandra,*Representations of semisimple Lie groups on a Banach space I*, Trans. Amer. Math. Soc.**75**(1953), 185-243. MR**15:100f****[HC2]**Harish-Chandra,*Representations of semisimple Lie groups V*, Amer. J. Math.**78**(1956), 1-41. MR**18:490c****[HC3]**Harish-Chandra,*Representations of semisimple Lie groups VI*, Amer. J. Math.**78**(1956), 564-628. MR**18:490d****[JV]**Hans Plesner Jakobsen and Michèle Vergne,*Restrictions and expansions of holomorphic representations*, J. Functional Analysis**34**(1979), no. 1, 29–53. MR**551108**, https://doi.org/10.1016/0022-1236(79)90023-5**[K1]**A. W. Knapp,*Representations of 𝐺𝐿₂(𝑅) and 𝐺𝐿₂(𝐶)*, 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. 87–91. MR**546590****[K2]**Anthony W. Knapp,*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239****[K3]**Anthony W. Knapp,*Lie groups, Lie algebras, and cohomology*, Mathematical Notes, vol. 34, Princeton University Press, Princeton, NJ, 1988. MR**938524****[K4]**A. W. Knapp,*Imbedding discrete series in*, in ``Harmonic analysis on Lie groups (Sandjerg Estate, August 26-30, 1991)," Copenhagen University Mathematics Institute, Report Series 1991, No. 3, pp. 27-29, Copenhagen, 1991.**[K5]**A. W. Knapp,*Introduction to representations in analytic cohomology*, The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992) Contemp. Math., vol. 154, Amer. Math. Soc., Providence, RI, 1993, pp. 1–19. MR**1246374**, https://doi.org/10.1090/conm/154/01353**[KO]**A. W. Knapp and K. Okamoto,*Limits of holomorphic discrete series*, J. Functional Analysis**9**(1972), 375–409. MR**0299726**, https://doi.org/10.1016/0022-1236(72)90017-1**[KV]**Anthony W. Knapp and David A. Vogan Jr.,*Cohomological induction and unitary representations*, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR**1330919****[KW]**A. W. Knapp and N. R. Wallach,*Correction and addition: “Szegő kernels associated with discrete series” [Invent. Math. 34 (1976), no. 3, 163–200; MR 54 #7704]*, Invent. Math.**62**(1980/81), no. 2, 341–346. MR**595593**, https://doi.org/10.1007/BF01389165**[KZ]**A. W. Knapp and Gregg J. Zuckerman,*Correction: “Classification of irreducible tempered representations of semisimple groups” [Ann. of Math. (2) 116 (1982), no. 2, 389–501; MR 84h:22034ab]*, Ann. of Math. (2)**119**(1984), no. 3, 639. MR**744867**, https://doi.org/10.2307/2007089**[Ko]**Toshiyuki Kobayashi,*Discrete decomposability of the restriction of 𝐴_{𝔮}(𝜆) with respect to reductive subgroups and its applications*, Invent. Math.**117**(1994), no. 2, 181–205. MR**1273263**, https://doi.org/10.1007/BF01232239**[Kd]**Kunihiko Kodaira,*Complex manifolds and deformation of complex structures*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao; With an appendix by Daisuke Fujiwara. MR**815922****[Ks]**Bertram Kostant,*Lie algebra cohomology and the generalized Borel-Weil theorem*, Ann. of Math. (2)**74**(1961), 329–387. MR**142696**, https://doi.org/10.2307/1970237**[La]**R. P. Langlands,*On the classification of irreducible representations of real algebraic groups*, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170. MR**1011897**, https://doi.org/10.1090/surv/031/03**[Mi]**Hisaichi Midorikawa,*Schur orthogonality relations for certain non square integrable representations of real semisimple Lie groups*, Tokyo J. Math.**8**(1985), no. 2, 303–336. MR**826991**, https://doi.org/10.3836/tjm/1270151217**[RSW]**John Rawnsley, Wilfried Schmid, and Joseph A. Wolf,*Singular unitary representations and indefinite harmonic theory*, J. Functional Analysis**51**(1983), no. 1, 1–114. MR**699229**, https://doi.org/10.1016/0022-1236(83)90029-0**[S1]**Wilfried Schmid,*Homogeneous complex manifolds and representations of semisimple Lie groups*, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 223–286. Dissertation, University of California, Berkeley, CA, 1967. MR**1011899**, https://doi.org/10.1090/surv/031/05**[S2]**Wilfried Schmid,*𝐿²-cohomology and the discrete series*, Ann. of Math. (2)**103**(1976), no. 2, 375–394. MR**396856**, https://doi.org/10.2307/1970944**[S3]**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****[TW]**Juan A. Tirao and Joseph A. Wolf,*Homogeneous holomorphic vector bundles*, Indiana Univ. Math. J.**20**(1970/71), 15–31. MR**263110**, https://doi.org/10.1512/iumj.1970.20.20002**[Vo1]**David A. Vogan Jr.,*The algebraic structure of the representation of semisimple Lie groups. I*, Ann. of Math. (2)**109**(1979), no. 1, 1–60. MR**519352**, https://doi.org/10.2307/1971266**[Vo2]**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**, https://doi.org/10.1007/BF01389104**[VW]**D. A. Vogan Jr. and N. R. Wallach,*Intertwining operators for real reductive groups*, Adv. Math.**82**(1990), no. 2, 203–243. MR**1063958**, https://doi.org/10.1016/0001-8708(90)90089-6**[Wa]**Nolan R. Wallach,*Representations of reductive Lie groups*, 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. 71–86. MR**546589****[Wo]**H. W. Wong,*Cohomological induction in various categories and the maximal globalization conjecture*, preprint, 1995.**[Zi]**Roger Zierau,*Unitarity of certain Dolbeault cohomology representations*, The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992) Contemp. Math., vol. 154, Amer. Math. Soc., Providence, RI, 1993, pp. 239–259. MR**1246388**, https://doi.org/10.1090/conm/154/01367

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (1991):
22E46

Retrieve articles in all journals with MSC (1991): 22E46

Additional Information

**Robert W. Donley Jr.**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Address at time of publication:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
donley@math.mit.edu

DOI:
https://doi.org/10.1090/S1088-4165-98-00044-2

Received by editor(s):
February 5, 1998

Received by editor(s) in revised form:
March 31, 1998, and May 13, 1998

Published electronically:
June 16, 1998

Additional Notes:
Supported by NSF Grant DMS 9627447.

Article copyright:
© Copyright 1998
American Mathematical Society