Representation homology of simply connected spaces

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