Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An $ L\sp 2$-cohomology construction of unitary highest weight modules for $ {\rm U}(p,q)$


Author: Lisa A. Mantini
Journal: Trans. Amer. Math. Soc. 323 (1991), 583-603
MSC: Primary 22E45; Secondary 32L25, 32M15, 58G05
DOI: https://doi.org/10.1090/S0002-9947-1991-1020992-9
MathSciNet review: 1020992
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper a geometric construction is given of all unitary highest weight modules of $ G = \operatorname{U} (p,q)$. The construction is based on the unitary model of the $ k$th tensor power of the metaplectic representation in a Bargmann-Segal-Fock space of square-integrable differential forms. The representations are constructed as holomorphic sections of certain vector bundles over $ G/K$, and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics.


References [Enhancements On Off] (What's this?)

  • [A] J. Adams, Unitary highest weight modules, Adv. in Math. 63 (1987), 113-137. MR 872349 (88b:22014)
  • [B] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, I, Comm. Pure Appl. Math. 14 (1961), 187-214. MR 0157250 (28:486)
  • [BR] R. Blattner and J. Rawnsley, Quantization of the action of $ U(k,l)$ on $ {\mathbb{R}^{2(k + l)}}$, J. Funct. Anal. 50 (1983), 188-214. MR 693228 (85c:58044)
  • [D] M. Davidson, The harmonic representation of $ {\text{U}}(p,q)$ and its connection with the generalized unit disc, Pacific J. Math. 129 (1987), 33-55. MR 901255 (89c:22021)
  • [DS] M. Davidson and R. Stanke, Gradient-type differential operators and unitary highest weight representations of $ {\mathbf{SU}}(p,q)$, J. Funct. Anal. 81 (1988), 100-125. MR 967893 (90e:22025)
  • [Du] E. Dunne, Hyperfunctions in representation theory and mathematical physics, Integral Geometry, Contemp. Math., vol. 63, Amer. Math. Soc., Providence, R.I., 1987, pp. 51-65. MR 876313 (88f:22029)
  • [E] M. Eastwood, The generalized twistor transform and unitary representations of $ {\mathbf{SU}}(p,q)$, preprint.
  • [EPW] M. Eastwood, R. Penrose, and R. O. Wells, Jr., Cohomology and massless fields, Comm. Math. Phys. 78 (1981), 305-351. MR 603497 (83d:81052)
  • [EPt] M. Eastwood and A. Pilato, On the density of twistor elementary states, preprint. MR 1132385 (93e:22027a)
  • [EHW] T. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, Representation Theory of Reductive Groups, Progr. Math., vol. 40, Birkhäuser, Boston, Mass., 1983, pp. 97-143. MR 733809 (86c:22028)
  • [EP] T. Enright and R. Parthasarathy, A proof of a conjecture of Kashiwara and Vergne, Non-Commutative Harmonic Analysis and Lie Groups, Lecture Notes in Math., vol. 880, Springer Verlag, New York, 1981, pp. 74-90. MR 644829 (83c:17008)
  • [H] S. Helgason, The Radon transform, Progr. Math., vol. 5, Birkhäuser, Boston, Mass., 1980.
  • [J1] H. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385-412. MR 712588 (85a:17004)
  • [J2] -, On singular holomorphic representations, Invent. Math. 62 (1980), 67-78. MR 595582 (82e:22025)
  • [KV] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1-47. MR 0463359 (57:3311)
  • [M1] L. Mantini, An integral transform in $ {L^2}$-cohomology for the ladder representations of $ {\text{U}}(p,q)$, J. Funct. Anal. 60 (1985), 211-242. MR 777237 (87a:22029)
  • [M2] -, An $ {L^2}$-cohomology construction of negative spin mass zero equations for $ {\text{U}}(p,q)$, J. Math. Anal. Appl. 136 (1988), 419-449. MR 972147 (90c:22040)
  • [PR] C. Patton and H. Rossi, Unitary structures in cohomology, Trans. Amer. Math. Soc. 290 (1985), 235-258. MR 787963 (87g:22014)
  • [RSW] J. Rawnsley, W. Schmid, and J. Wolf, Singular unitary representations and indefinite harmonic theory, J. Funct. Anal. 51 (1983), 1-114. MR 699229 (84j:22022)
  • [We] R. O. Wells, Jr., Complex geometry in mathematical physics, Presses Univ. de Montréal, Montreal, 1982. MR 654864 (84f:32037)
  • [Z1] R. Zierau, Geometric construction of certain highest weight modules, Proc. Amer. Math. Soc. 95 (1985), 631-635. MR 810176 (87i:22044)
  • [Z2] -, Geometric construction of unitary highest weight representations, Ph.D. Dissertation, Univ. of California at Berkeley, December 1984.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E45, 32L25, 32M15, 58G05

Retrieve articles in all journals with MSC: 22E45, 32L25, 32M15, 58G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1020992-9
Keywords: Semisimple Lie groups, unitary representations, highest weight modules, metaplectic representation, integral transform
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society