On measures derived from orbital integrals

TL;DR

This paper develops a framework for deriving piecewise polynomial measures from orbital integrals on Lie groups, providing formulas for moments in the unitary case.

Contribution

It introduces a new method to compute measures from orbital integrals using polynomial transformations and applies it to the unitary group case.

Findings

Derived formulas for moments of orbital measures in the unitary group.

Established a framework for piecewise polynomial measures from invariant measures.

Connected orbital integrals with polynomial transformations on spaces with apolar inner product.

Abstract

The present work develops a framework to derive piecewise polynomial measures arising from invariant measures on adjoint orbits in the context of compact and semisimple Lie groups. These measures are computed from orbital integrals via transformations on spaces of polynomials endowed with the apolar inner product. In the case of the unitary group, we obtain a formula for the moments of the projection of an orbital measure.