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Transactions of the American Mathematical Society

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



Square integrable representations of semisimple Lie groups

Author: Juan A. Tirao
Journal: Trans. Amer. Math. Soc. 190 (1974), 57-75
MSC: Primary 22E45; Secondary 32L05, 32M15
MathSciNet review: 0338270
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Abstract: Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component $ {A_0}(D)$ of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle $ E \to D$ admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle $ E \to D$ must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of $ E \to D$ by defining a map $ \Phi :G \to {\text{GL}}(V)$ (V being the fiber of the vector bundle) which extends the classical ``universal factor of automorphy'' for the action of $ {A_0}(D)$ on D. Then, we study the space H of all square integrable holomorphic sections of $ E \to D$. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over $ G/Z$ (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in $ {L_2}(G/Z)$.

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  • [1] W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442-528. MR 35 #6870. MR 0216035 (35:6870)
  • [2] F. Bruhat, Travaux de Harish-Chandra, Séminaire Bourbaki, 9iéme année: 1956/57, 2nd ed., Exposé 143, Secrétariat mathématique, Paris, 1959, pp. 1-9. MR 28 #1090. MR 1610953
  • [3] Harish-Chandra, Representations of semisimple Lie groups. V, Amer. J. Math. 78 (1956), 1-41. MR 18, 490. MR 0082055 (18:490c)
  • [4] -, Representations of semisimple Lie groups. VI. Integrable and square-integrable representations, Amer. J. Math. 78 (1956), 564-628. MR 18, 490. MR 0082056 (18:490d)
  • [5] -, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241-318. MR 36 #2744. MR 0219665 (36:2744)
  • [6] -, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1-111. MR 36 #2745. MR 0219666 (36:2745)
  • [7] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl.. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [8] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, Interscience, New York, 1969. MR 38 #6501. MR 0238225 (38:6501)
  • [9] M. S. Narasimhan and K. Okamoto, An analogue of the Borel-Weil-Bott theorem for Hermitian symmetric pairs of non-compact type, Ann. of Math. (2) 91 (1970), 486-511. MR 43 #0419. MR 0274657 (43:419)
  • [10] J. A. Tirao and J. A. Wolf, Homogeneous holomorphic vector bundles, Indiana Univ. Math. J. 20 (1970/71), 15-31. MR 41 #7715. MR 0263110 (41:7715)

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Keywords: Bounded symmetric domains, homogeneous holomorphic vector bundles, square integrable holomorphic sections
Article copyright: © Copyright 1974 American Mathematical Society

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