Real loci of based loop groups

Abstract

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $\sigma$. This induces an involutive automorphism $\tau$ of the based loop space $\Omega(G)$. There exists a maximal torus $T\subset G$ such that the canonical action of $T\times S^1$ on $\Omega(G)$ is compatible with $\tau$ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat's convexity theorem. Namely, the images of $\Omega(G)$ and $\Omega(G)^\tau$ (fixed point set of $\tau$) under the $T\times S^1$ moment map on $\Omega(G)$ are equal. The space $\Omega(G)^\tau$ is homotopy equivalent to the loop space $\Omega(G/K)$ of the Riemannian symmetric space $G/K$. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in $\mathbb{Z}_2$ of $\Omega(G)$ and $\Omega(G/K)$. Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe.