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Journal of Lie Theory 33 (2023), No. 1, 149--168
Copyright Heldermann Verlag 2023

Mackey-Type Identity for Invariant Functions on Lie Algebras of Finite Unitary Groups and an Application

Cesar Cuenca
Department of Mathematics, Harvard University, Cambridge, U.S.A.

Grigori Olshanski
(1) Institute for Information Transmission Problems, Moscow, Russia
(2) Skolkovo Institute of Science and Technology, Moscow, Russia
(3) Faculty of Mathematics, HSE University, Moscow, Russia


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 $U(N, \mathbb{F}_{q^2})$. 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' previous paper [Advances Math. 395 (2022), 108087]. 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 $U(N, \mathbb{F}_{q^2})$.

Keywords: Finite unitary groups, branching graphs, Mackey's theorem.

MSC: 20C33, 22E65, 16T10.

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