Representation homology of simply connected spaces

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${}_G(X,\ast )$: we call the (co)homology of the structure sheaf of RLoc${}_G(X,\ast )$ the representation homology of $X$ in $G$ and denote it by HR${}_{\ast }(X,G)$. 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${}_{\ast }(X,G)$ 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${}_{\ast }(X,G{)}^G$, and study the question when HR${}_{\ast }(X,G{)}^G$ 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${}_{\ast }(X,G{)}^G$ is a graded symmetric algebra for any complex reductive group $G$.

39 pages. arXiv admin note: substantial text overlap with arXiv:1703.03505