On measures derived from orbital integrals
Martin Miglioli
TL;DR
The paper develops a framework to obtain piecewise polynomial (Duistermaat–Heckman type) measures from Harish-Chandra orbital integrals for compact semisimple Lie groups. It translates orbital-measure computations into finite-dimensional polynomial spaces endowed with the apolar inner product and leverages a suite of operators (translations, degree-truncated multiplications, projections, and creation/annihilation/division/antiderivative maps) to produce explicit characterizations of pushforward measures to the Cartan algebra and of radial parts of orbital convolutions. Key contributions include a precise formula for the pushforward measure $\mu_a$ in terms of $I_\Delta$ and $P_{\mathrm{sym}}$, a rigorous characterization of the radial convolution $\nu_{a,b}$, and illustrative SU(2) examples that demonstrate the resulting piecewise densities. The work links harmonic analysis on Lie groups with algebraic polynomial methods, providing computable descriptions of these measures and offering a computational route to Duistermaat–Heckman-type phenomena.
Abstract
In this article we propose a novel framework to derive piecewise polynomial measures which result from invariant measures on adjoint orbits in the general context of compact and semi-simple Lie groups. The measures are computed from orbital integrals by a series of transformations on spaces of polynomials endowed with the apolar inner product.