Journal of the Ramanujan Mathematical Society

Volume 29, Issue 3, September 2014 pp. 253–272.

Generalized matrix coefficients for infinite dimensional unitary representations

Authors: Hongyu He
Author institution:Department of Mathematics, Yale University and Department of Mathematics, Louisiana State University

Summary: Let (π, H) be a unitary representation of a Lie group G. Classically, matrix coefficients are continuous functions on G attached to a pair of vectors in H and H*. In~this note, we generalize the definition of matrix coefficients to a pair of distributions in (H-∞, H*)-∞). Generalized matrix coefficients are in D'(G), the space of distributions on G. By~analyzing the structure of generalized matrix coefficients, we prove that, fixing an element in (H*)-∞, the map H-∞ → D'(G) is continuous. This effectively answers the question about computing generalized matrix coefficients. For the Heisenberg group, our generalized matrix coefficients can be considered as a generalization of the Fourier-Wigner transform.

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