Hua type integrals over unitary groups and over projective limits of unitary groups

TL;DR

This paper explores Hua type integrals over unitary groups and their projective limits, providing explicit measure transformations, evaluating matrix integrals, and generalizing classical Hua Loo Keng integrals.

Contribution

It introduces new measure maps from unitary groups, computes explicit integrals, and constructs inverse limits with Haar measure analogues, extending classical integral results.

Findings

Explicit formulas for measure images under group maps

Evaluation of matrix integrals as products of Gamma-functions

Construction of inverse limits of unitary groups with Haar measure analogues

Abstract

We discuss some natural maps from a unitary group U(n) to a smaller group U(n-m) (these maps are versions of the Livshic characteristic function). We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces (values of the integrals are products of Gamma-functions). These integrals generalize Hua Loo Keng integrals. We construct inverse limits of unitary groups equipped with analogues of the Haar measure and evaluate some integrals over these inverse limits.