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 of the group of all holomorphic diffeomorphisms of *D* onto itself. In this case, any *G*-homogeneous vector bundle admits a natural structure of *G*-homogeneous holomorphic vector bundles. The vector bundle must be holomorphically trivial, since *D* is a Stein manifold. We exhibit explicitly a holomorphic trivialization of by defining a map (*V* being the fiber of the vector bundle) which extends the classical ``universal factor of automorphy'' for the action of on *D*. Then, we study the space *H* of all square integrable holomorphic sections of . The natural action of *G* on *H* defines a unitary irreducible representation of *G*. The representations obtained in this way are square integrable over (*Z* denotes the center of *G*) in the sense that the absolute values of their matrix coefficients are in .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0338270-X

Keywords:
Bounded symmetric domains,
homogeneous holomorphic vector bundles,
square integrable holomorphic sections

Article copyright:
© Copyright 1974
American Mathematical Society