Projections of orbital measures for classical Lie groups

TL;DR

This paper extends the computation of orbital measure projections to classical Lie groups like SO(2n+1), Sp(2n), and O(2n), expressing results via determinants of symmetrically arranged B-splines.

Contribution

It generalizes previous work on unitary groups to other classical Lie groups, providing explicit determinant formulas involving B-splines.

Findings

Derived explicit formulas for orbital measure projections for SO(2n+1), Sp(2n), and O(2n).

Expressed the results in terms of determinants of B-splines with symmetric knots.

Extended the mathematical understanding of orbital measures in classical Lie groups.

Abstract

In this paper we compute the radial parts of projections of the orbital measures for the compact Lie groups $SO(2n+1), Sp(2n)$ and $O(2n)$, extending the previous results for the case of the unitary group by Olshanski and Faraut. The answer is given in terms of determinants of the B-splines with the knots arranged symmetrically about zero.