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Journal of Lie Theory 33 (2023), No. 1, 253--270
Copyright Heldermann Verlag 2023

Generalized Laguerre Functions and Whittaker Vectors for Holomorphic Discrete Series

Jan Frahm
Dept. of Mathematics, Aarhus University, Denmark

Bent Oersted
Dept. of Mathematics, Aarhus University, Denmark

Gestur Olafsson
Department of Mathematics, Louisiana State University, Baton Rouge, U.S.A.

We study degenerate Whittaker vectors in scalar type holomorphic discrete series representations of tube type Hermitian Lie groups and their analytic continuation. In four different realizations, the bounded domain picture, the tube domain picture, the L2-model and the Fock model, we find their explicit K-type expansions. The coefficients are expressed in terms of the generalized Laguerre functions on the corresponding symmetric cone, and we relate the K-type expansions to the formula for the generating function of the Laguerre polynomials and to their recurrence relations.

Keywords: Laguerre functions, Whittaker vectors, holomorphic discrete series.

MSC: 22E46; 43A85.

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