Representation homology of simply connected spaces

Abstract

Let 𝐺 be an affine algebraic group defined over a field 𝑘 of characteristic 0. 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 0-dimensional homology, HR 0 (𝑋, 𝐺), is isomorphic to the coordinate ring of the 𝐺-representation variety Rep 𝐺 [𝜋 1 (𝑋)] of the fundamental group of 𝑋 -a well-known algebrogeometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when 𝑋 is simply connected, HR 0 (𝑋, 𝐺) is trivial but HR * (𝑋, 𝐺) is still an interesting rational invariant of 𝑋 that depends on the Lie algebra of 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 is reductive, we also compute HR * (𝑋, 𝐺) 𝐺 , the 𝐺-invariant part of representation homology, 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 related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the

Derived Character Maps of Groups Representations

“…In this section, we define representation homology of groups with coefficients in a commutative Hopf algebra H, following the approach of [13,14]. Taking H = O(G), where G is an affine algebraic group, we then construct the derived character maps for G-representations of Γ.…”

“…In the case when G = GL n , these maps specialize to the character maps (1.6) announced in the Introduction. We will work with homotopy simplicial groups (in the sense of Badzioch [4]), which is a more general and flexible setting than that of the usual (strict) simplicial groups used in [13,14]. In Section 3.1, we define the classifying spaces for such groups, and in Section 3.3, the cyclic bar construction and cyclic homology, both of which may be of independent interest.…”

“…i ≥ 0, are (non-abelian) homological invariants of Γ (or rather its classifying space BΓ). Following [14,13], we set Apart from groups, representation homology can be also defined for various types of algebras (e.g., associative and Lie algebras, see [9,10,8,7]) as well as for topological spaces (see [14,13,12]). What is surprising perhaps is that, in the case of groups and spaces, there is a simple construction of representation homology that does not require the use of derived algebraic geometry nor a heavy machinery of homotopical algebra.…”