Homology of loops on a splitting-rank symmetric space via the Geometric Satake for real groups

TL;DR

This paper uses the Geometric Satake Equivalence to describe the equivariant homology of loops on splitting-rank symmetric spaces, revealing a potential duality and extending known results for compact Lie groups.

Contribution

It provides a new homological description of loop spaces on symmetric spaces via the Geometric Satake for real groups, connecting to dual groups and duality concepts.

Findings

Homology of loop spaces described via the relative dual group

Extension of Yun and Zhu's calculations to symmetric spaces

Hints at a duality for splitting-rank symmetric spaces

Abstract

Using Nadler's Geometric Satake Equivalence for real reductive groups, we obtain a description of the equivariant homology of the loop space of splitting-rank symmetric spaces in terms of the relative dual group of the space. The description is in line with Yun and Zhu's calculation for the loop space of a compact Lie group. The formula hints at a relative duality for these spaces.