Realization of square-integrable representations of unimodular Lie groups on 𝐿²-cohomology spaces

Rosenberg, Jonathan

doi:10.1090/S0002-9947-1980-0576861-4

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Realization of square-integrable representations of unimodular Lie groups on $L\sp{2}$ -cohomology spaces

Author: Jonathan Rosenberg
Journal: Trans. Amer. Math. Soc. 261 (1980), 1-32
MSC: Primary 22E45; Secondary 22E25
MathSciNet review: 576861
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An analogue of the ``Langlands conjecture'' is proved for a large class of connected unimodular Lie groups having square-integrable representations (modulo their centers). For nilpotent groups, it is shown (without restrictions on the group or the polarization) that the ${L^2}$ -cohomology spaces of a homogeneous holomorphic line bundle, associated with a totally complex polarization for a flat orbit, vanish except in one degree given by the ``deviation from positivity'' of the polarization. In this degree the group acts irreducibly by a square-integrable representation, confirming a conjecture of Moscovici and Verona. Analogous results which improve on theorems of Satake are proved for extensions of a nilpotent group having square-integrable representations by a reductive group, by combining the theorem for the nilpotent case with Schmid's proof of the Langlands conjecture. Some related results on Lie algebra cohomology and the ``Harish-Chandra homomorphism'' for Lie algebras with a triangular decomposition are also given.

[1] Nguyên Huu Anh, Lie groups with square integrable representations, Ann. of Math. (2) 104 (1976), no. 3, 431–458. MR 0432822 (55 #5802)
[2] Nguy\cftil{e}n H u’u Anh, Classification des groupes de Lie connexes unimodulaires possédant une série discrète, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 13, A847–A849 (French, with English summary). MR 551761 (80m:22022)
[3] Nguy\cftil{e}n H u’u Anh, Sur une conjecture de Wolf et Moore caractérisant les groupes de Lie ayant une série discrète, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 14, A919–A921 (French, with English summary). MR 520768 (80c:22015)
[4] Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 0463358 (57 #3310)
[5] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255–354. MR 0293012 (45 #2092)
[6] Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. MR 0089473 (19,681d)
[7] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480 (17,1040e)
[8] Pierre Cartier, Vecteurs différentiables dans les représentations unitaires des groupes de Lie, Séminaire Bourbakt (1974/1975), Exp. No. 454, Springer, Berlin, 1976, pp. 20–34. Lecture Notes in Math., Vol. 514 (French). MR 0460541 (57 #534)
[9] William Casselman and M. Scott Osborne, The 𝔫-cohomology of representations with an infinitesimal character, Compositio Math. 31 (1975), no. 2, 219–227. MR 0396704 (53 #566)
[10] Alain Connes and Henri Moscovici, The 𝐿²-index theorem for homogeneous spaces, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 4, 688–690. MR 532554 (80g:58046), http://dx.doi.org/10.1090/S0273-0979-1979-14670-6
[11] Jacques Dixmier, Sur les représentations unitaries des groupes de Lie nilpotents. III, Canad. J. Math. 10 (1958), 321–348. MR 0095427 (20 #1929)
[12] Jacques Dixmier, Enveloping algebras, North-Holland Publishing Co., Amsterdam, 1977. North-Holland Mathematical Library, Vol. 14; Translated from the French. MR 0498740 (58 #16803b)
[13] Jacques Dixmier, Polarisations dans les algèbres de Lie, Ann. Sci. École Norm. Sup. (4) 4 (1971), 321–335 (French). MR 0291374 (45 #467)
[14] Michel Duflo, Sur les extensions des représentations irréductibles des groupes de Lie nilpotents, Ann. Sci. École Norm. Sup. (4) 5 (1972), 71–120 (French). MR 0302823 (46 #1966)
[15] Hidenori Fujiwara, On holomorphically induced representations of split solvable Lie groups, Proc. Japan Acad. 51 (1975 suppl), 808–810. MR 0404528 (53 #8328)
[16] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 (French). MR 0075539 (17,763c)
[17] Harish-Chandra, Representations of semi-simple Lie groups. V, VI, Amer. J. Math. 78 (1956), 1-41, 564-628.
[18] -, Harmonic analysis on semi-simple Lie groups, Bull. Amer. Math. Soc. 76 (1970), 529-551.
[19] Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075 (34 #2933)
[20] Roger E. Howe, On a connection between nilpotent groups and oscillatory integrals associated to singularities, Pacific J. Math. 73 (1977), no. 2, 329–363. MR 0578891 (58 #28270)
[21] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001 (25 #5396)
[22] A. A. Kirillov, The method of orbits in the theory of unitary representations of Lie groups, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 96–98 (Russian). MR 0233930 (38 #2251)
[23] A. Kleppner and R. L. Lipsman, The Plancherel formula for group extensions. II, Ann. Sci. École Norm. Sup. 6 (1973), 103-132.
[24] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 0142696 (26 #265)
[25] Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 87–208. Lecture Notes in Math., Vol. 170. MR 0294568 (45 #3638)
[26] Ronald L. Lipsman, Characters of Lie groups. II. Real polarizations and the orbital-integral character formula, J. Analyse Math. 31 (1977), 257–286. MR 0579006 (58 #28295)
[27] -, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, preprint, 1979.
[28] Calvin C. Moore and Joseph A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445–462 (1974). MR 0338267 (49 #3033), http://dx.doi.org/10.1090/S0002-9947-1973-0338267-9
[29] H. Moscovici, A vanishing theorem for ${L^2}$ -cohomology in the nilpotent case, preprint, 1978.
[30] -, Representations of nilpotent groups on spaces of harmonic spinors (to appear).
[31] Henri Moscovici and Andrei Verona, Harmonically induced representations of nilpotent Lie groups, Invent. Math. 48 (1978), no. 1, 61–73. MR 508089 (80a:22011), http://dx.doi.org/10.1007/BF01390062
[32] Richard C. Penney, Canonical objects in Kirillov theory on nilpotent Lie groups, Proc. Amer. Math. Soc. 66 (1977), no. 1, 175–178. MR 0453922 (56 #12175), http://dx.doi.org/10.1090/S0002-9939-1977-0453922-7
[33] Richard C. Penney, Harmonically induced representations on nilpotent Lie groups and automorphic forms on nilmanifolds, Trans. Amer. Math. Soc. 260 (1980), no. 1, 123–145. MR 570782 (81h:22008), http://dx.doi.org/10.1090/S0002-9947-1980-0570782-9
[34] Neils Skovhus Poulsen, On 𝐶^{∞}-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87–120. MR 0310137 (46 #9239)
[35] L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) 4 (1971), 457–608. MR 0439985 (55 #12866)
[36] L. Pukanszky, Unitary representations of Lie groups with cocompact radical and applications, Trans. Amer. Math. Soc. 236 (1978), 1–49. MR 0486313 (58 #6070), http://dx.doi.org/10.1090/S0002-9947-1978-0486313-9
[37] R. W. Richardson Jr., Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math. 129 (1972), 35–73. MR 0299723 (45 #8771)
[38] Jonathan Rosenberg, Square-integrable factor representations of locally compact groups, Trans. Amer. Math. Soc. 237 (1978), 1–33. MR 0486292 (58 #6056), http://dx.doi.org/10.1090/S0002-9947-1978-0486292-4
[39] Hugo Rossi and Michèle Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Functional Analysis 13 (1973), 324–389. MR 0407206 (53 #10989)
[40] Wulf Rossmann, Kirillov’s character formula for reductive Lie groups, Invent. Math. 48 (1978), no. 3, 207–220. MR 508985 (81g:22012), http://dx.doi.org/10.1007/BF01390244
[41] I. Satake, Unitary representations of a semi-direct product of Lie groups on ∂-cohomology spaces, Math. Ann. 190 (1970/71), 177–202. MR 0296213 (45 #5274)
[42] Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1–42. MR 0286942 (44 #4149)
[43] Wilfried Schmid, 𝐿²-cohomology and the discrete series, Ann. of Math. (2) 103 (1976), no. 2, 375–394. MR 0396856 (53 #716)
[44] 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 (81j:22020), http://dx.doi.org/10.2307/1971266
[45] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. MR 0498999 (58 #16979)
[46] Joseph A. Wolf, The action of a real semisimple Lie group on a complex flag manifold. II. Unitary representations on partially holomorphic cohomology spaces, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathematical Society, No. 138. MR 0393350 (52 #14160)
[47] Joseph A. Wolf, Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1972/73), 611–640. MR 0311248 (46 #10340)
[48] Joseph A. Wolf, Partially harmonic spinors and representations of reductive Lie groups, J. Functional Analysis 15 (1974), 117–154. MR 0393351 (52 #14161)
[49] Joseph A. Wolf, Representations associated to minimal co-adjoint orbits, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 329–349. MR 519619 (80e:22020)
[50] A. A. Zaĭcev, The nontriviality of the space of a holomorphically induced representation of a solvable Lie group, Funkcional. Anal. i Priložen. 11 (1977), no. 2, 78–79 (Russian). MR 0460537 (57 #530)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|

Bibliography