Representation homology of simply connected spaces

Abstract

Let G $G$ be an affine algebraic group defined over a field k $k$ of characteristic 0. We study the derived moduli space of G $G$ -local systems on a pointed connected CW complex X $X$ trivialized at the basepoint of X $X$ . This derived moduli space is represented by an affine DG scheme R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ : we call the (co)homology of the structure sheaf of R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ the representation homology of X $X$ in G $G$ and denote it by HR * ( X , G ) $ \mathrm{HR}_\ast (X,G)$ . The 0-dimensional homology, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ , is isomorphic to the coordinate ring of the G $G$ -representation variety Rep G [ π 1 ( X ) ] $ {\rm {Rep}}_G[\pi _1(X)]$ of the fundamental group of X $X$ — a well-known algebro-geometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when X $X$ is simply connected, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ is trivial but HR ∗ ( X , G ) $ \mathrm{HR}_*(X,G)$ is still an interesting rational invariant of X $X$ that depends on the Lie algebra of G $G$ . In this paper, we use Quillen's rational homotopy theory to compute the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Lie and Sullivan algebraic models. When G $G$ is reductive, we also compute HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ , the G $G$ -invariant part of representation homology, and study the question when HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the 1980s and proved by Fishel, Grojnowski and Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces X $X$ for which HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is a graded symmetric algebra for any complex reductive group G $G$ .