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.
cesar.a.cuenk@gmail.com
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
olsh2007@gmail.com
[Abstract-pdf]
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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