Representation homology of simply connected spaces

TL;DR

This paper introduces the concept of representation homology for simply connected spaces, computes it using rational homotopy theory, and explores its algebraic properties and connections to the Strong Macdonald Conjecture.

Contribution

It provides the first computation of higher representation homology for simply connected spaces using algebraic models and relates these results to a major conjecture in representation theory.

Findings

Computed representation homology for simply connected spaces using rational models.

Identified conditions under which the G-invariant part of the homology is a free graded algebra.

Connected the algebraic properties of representation homology to the Strong Macdonald Conjecture.

Abstract

Let $G$ be an affine algebraic group defined over field $k$ of characteristic zero. We study the derived moduli space of G-local systems on a pointed connected CW complex X trivialized at the basepoint of $X$. This derived moduli space is represented by an affine DG scheme RLoc$_G(X,*)$: we call the (co)homology of the structure sheaf of RLoc$_G(X,*)$ the representation homology of $X$ in $G$ and denote it by HR$_*(X,G)$. The HR$_0(X,G)$ is isomorphic to the coordinate ring of the representation variety Rep$_G[\pi_1(X)]$ of the fundamental group of $X$ in $G$ -- a well-known algebro-geometric invariant of $X$ with many applications in topology. The case when X is simply connected seems much less studied: in this case, the HR$_0(X,G)$ is trivial but the higher representation homology is still an interesting rational invariant of $X$ depending on the algebraic group $G$. In this paper, we use rational homotopy theory to compute the HR$_*(X,G)$ for an arbitrary simply connected space $X$ (of finite rational type) in terms of its Quillen and Sullivan algebraic models. When $G$ is reductive, we also compute the $G$-invariant part of representation homology, HR$_*(X,G)^G$, and study the question when HR$_*(X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be closely related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by B. Feigin and P. Hanlon in the 1980s and proved by S. Fishel, I. Grojnowski and C. Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces $X$ for which HR$_*(X,G)^G$ is a graded symmetric algebra for any complex reductive group $G$.