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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Square integrable representations of semisimple Lie groups
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by Juan A. Tirao PDF
Trans. Amer. Math. Soc. 190 (1974), 57-75 Request permission

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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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 190 (1974), 57-75
  • MSC: Primary 22E45; Secondary 32L05, 32M15
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0338270-X
  • MathSciNet review: 0338270