Generalized Laguerre functions and Whittaker vectors for holomorphic discrete series

TY - JOUR

T1 - Generalized Laguerre functions and Whittaker vectors for holomorphic discrete series

AU - Frahm, Jan

AU - Ørsted, Bent

AU - Ólafsson, Gestur

PY - 2023

Y1 - 2023

N2 - 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 $L^2$-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.

AB - 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 $L^2$-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.

KW - Laguerre functions

KW - Whittaker vectors

KW - holomorphic discrete series

UR - https://www.heldermann.de/JLT/JLT33/JLT331/jlt33011.htm

M3 - Journal article

SN - 0949-5932

VL - 33

SP - 253

EP - 270

JO - Journal of Lie Theory

JF - Journal of Lie Theory

IS - 1

ER -