Representation homology of simply connected spaces (2007.10844v1)

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 $G$.