The trace formula and prehomogeneous vector spaces

TL;DR

This paper develops a method to express the geometric side of the Arthur-Selberg trace formula using zeta integrals related to prehomogeneous vector spaces, providing explicit formulas for weighted orbital integrals.

Contribution

It introduces a new approach connecting trace formulas with zeta integrals of prehomogeneous vector spaces, with explicit formulas for specific conjugacy classes.

Findings

Explicit formulas for weighted orbital integrals derived

Method works for classical groups of rank up to two

Implemented for principal unipotent conjugacy class

Abstract

We describe an approach to express the geometric side of the Arthur-Selberg trace formula in terms of zeta integrals attached to prehomogeneous vector spaces. This will provide explicit formulas for weighted orbital integrals and for the coefficients by which they are multiplied in the trace formula. We implement this programme for the principal unipotent conjugacy class. The method relies on certain convergence results and uses the notions of induced conjugacy classes and canonical parabolic subgroups. So far, it works for certain types of conjugacy classes, which covers all classes appearing in classical groups of absolute rank up to two.