Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

2015/07/27 by Szechtman, Fernando
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1507.07410

Abstract

Let A be a ring with 1≠ 0, not necessarily finite, endowed with an involution~*, that is, an anti-automorphism of order ≤ 2. Let H_n(A) be the additive group of all n× n hermitian matrices over A relative to *. Let \mathcal U_n(A) be the subgroup of GL_n(A) of all upper triangular matrices with 1's along the main diagonal. Let P=H_n(A)\rtimes \mathcal U_n(A), where \mathcal U_n(A) acts on H_n(A) by *-congruence transformations. We may view P as a unipotent subgroup of either a symplectic group Sp_2n(A), if *=1_A (in which case A is commutative), or a unitary group U_2n(A) if *≠ 1_A. In this paper we construct and classify a family of irreducible representations of P over a field F that is essentially arbitrary. In particular, when A is finite and F=\mathbb C we obtain irreducible representations of P of the highest possible degree.

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