An invariant measure for the loop space of a simply connected compact symmetric space

TL;DR

This paper establishes the existence of an LU-invariant probability measure on the loop space of a simply connected compact symmetric space, contributing to the understanding of symmetries in infinite-dimensional geometric structures.

Contribution

It proves the existence and conjectures the uniqueness of an LU-invariant measure on the loop space of a symmetric space, a novel result in geometric analysis.

Findings

Existence of LU-invariant measure on loop space proven

Conjecture on the measure's uniqueness proposed

Advances understanding of symmetry in infinite-dimensional spaces

Abstract

Let X denote a simply connected compact Riemannian symmetric space, U the universal covering of the identity component of the group of automorphisms of X, and LU the loop group of U. In this paper we prove the existence (and conjecture the uniqueness) of an LU-invariant probability measure on a distributional completion of the loop space of X.