Bessel operators on Jordan pairs and small representations of semisimple Lie groups

TL;DR

This paper constructs $L^2$-models for small unitary representations of semisimple Lie groups using Bessel operators on Jordan pairs, extending previous methods to more general parabolic subgroups and analyzing unitarizability conditions.

Contribution

It generalizes the construction of $L^2$-models to cases involving non-conjugate maximal parabolic subgroups using Jordan pairs and Bessel operators, and characterizes unitarizability via submanifold measures.

Findings

Bessel operators are tangential on certain submanifolds for small quotients.

Small quotients are unitarizable if submanifolds carry equivariant measures.

Spherical vectors are expressed using multivariable K-Bessel functions.

Abstract

We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier transformed picture of the degenerate principal series, the Lie algebra action is given in terms of Bessel operators on Jordan pairs. We prove that precisely for those parameters of the principal series where small quotients occur, the Bessel operators are tangential to certain submanifolds. Further, we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures. In this case we can express the spherical vectors in terms of multivariable K-Bessel functions, and some delicate estimates for these Bessel functions imply the existence of the $L^2$-models.