Kirillov-Frenkel character formula for Loop groups, radial part and Brownian sheet

TL;DR

This paper establishes a connection between conditioned Brownian motions in affine Weyl chambers and the radial part of Brownian sheets on Lie algebras, extending classical formulas to loop groups.

Contribution

It proves a Kirillov-Frenkel character formula for loop groups and relates conditioned Brownian motions to radial parts of Brownian sheets in Lie algebra settings.

Findings

Conditioned Brownian motion in affine Weyl chambers matches the radial part of a Brownian sheet.

Extension of classical character formulas to loop groups and affine Lie algebras.

New probabilistic interpretation of the coadjoint action in the context of loop groups.

Abstract

We consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and prove that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber - which can be seen as a space time conditioned Brownian motion - is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra.