An integration formula for unipotent radicals

TL;DR

This paper derives a Weyl-type integration formula for the unipotent radical of a maximal parabolic subgroup in classical groups, based on a decomposition under the Levi subgroup's action, with specific dimension assumptions.

Contribution

It provides a new decomposition of the unipotent radical and establishes an explicit integration formula for local fields, extending classical harmonic analysis tools.

Findings

Decomposition of unipotent radical N under Levi subgroup action

Derivation of a Weyl-type integration formula for N over local fields

Applicable under specific dimension and parity conditions

Abstract

Let P be a maximal parabolic of a classical group over a field F. Then the Levi subgroup M is isomorphic to the product of a classical group and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W is less than or equal to dim X and that dim W is even. When F is a local field, we obtain a Weyl-type integration formula for N.