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Transactions of the American Mathematical Society
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0338270-X
PII: S 0002-9947(1974)0338270-X
Keywords: Bounded symmetric domains, homogeneous holomorphic vector bundles, square integrable holomorphic sections
Article copyright: © Copyright 1974 American Mathematical Society