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.

The new version concerns exclusively Riemannian symmetric pairs (G,K) of maximal rank. New section with (counter)examples added