Mackey-type identity for invariant functions on Lie algebras of finite unitary groups and an application

TL;DR

This paper establishes a Mackey-type identity for invariant functions on Lie algebras of finite unitary groups, and applies it to construct harmonic functions and invariant measures in an infinite-dimensional setting.

Contribution

It introduces a new Mackey-type identity for finite unitary groups and applies it to infinite-dimensional harmonic analysis and invariant measures.

Findings

Established a Mackey-type identity for invariant functions on Lie algebras of finite unitary groups.

Constructed positive harmonic functions on a new branching graph with a negative Hall-Littlewood parameter.

Proved the existence of an infinite-parameter family of invariant measures for an infinite-dimensional analogue of unitary groups.

Abstract

The Mackey-type identity mentioned in the title relates the operations of parabolic induction and restriction for invariant functions on the Lie algebras of the finite unitary groups $U(N, q^2)$. This result is applied to constructing positive harmonic functions on a new branching graph with a negative Hall-Littlewood parameter, as introduced in the authors' paper [Adv. Math. vol. 395 (2022), 108087; arXiv:2102.01947].This in turn implies the existence of an infinite-parameter family of invariant measures for the coadjoint action of an infinite-dimensional analogue of the groups $U(N, q^2)$.