Intertwining Operators for Siegel Parabolics over Finite Fields

2026/08/02 by Nir Elber, Hahn Lheem
Mathematics · #math.RT #math.CO #math.NT #msc:20C33 #msc:11B65

paper · pdf

34 pages

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We consider degenerate principal series representations Ind_P^Gχ over finite fields, where G is a classical subgroup of GL_2n, and P is the Siegel parabolic subgroup. For example, we show that this representation is always multiplicity-free and irreducible for generic characters χ. We then discuss a particular intertwining operator I on Ind_P^Gχ and its related combinatorics. Firstly, this operator I produces families of diagonalizable antitriangular matrices with well-behaved eigenvalues. Secondly, applying I to a special vector in Ind_P^Gχ leads us to various matrix Gauss sums, whose evaluations imply an explicit equidistribution result of the trace and determinant of symmetric and alternating invertible matrices.

Citations

• Involutive random walks on total orders and the anti-diagonal eigenvalue property
• Orthogonal Matrices Over Finite Fields