Chern-Simons forms and higher character maps of Lie representations

TL;DR

This paper provides an explicit formula for the Drinfeld trace using Chern-Simons classes, connecting Lie algebra representations with differential operators and advancing the understanding of derived representation schemes.

Contribution

It introduces a new explicit formula for the Drinfeld trace in terms of Chern-Simons classes, linking classical regulator maps with derived algebraic geometry.

Findings

Explicit formula for Drinfeld trace in terms of Chern-Simons classes

Representation of the composite map as a differential operator on forms

Derivation of a combinatorial formula for the differential operator

Abstract

This paper is a sequel to our earlier work [BFPRW], where we study the derived representation scheme DRep_{g}(A) parametrizing the representations of a Lie algebra A in a finite-dimensional reductive Lie algebra g. In [BFPRW], we defined two canonical maps Tr_{g}(A): HC^{(r)}(A) \to \H[\DRep_{g}(A)]^G and \Phi_{g}(A): H[\DRep_{g}(A)]^G \to H[\DRep_{h}(A)]^W called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give an explicit formula for the Drinfeld trace in terms of Chern-Simons classes of a canonical g-torsor associated to the pair (A, g). Our construction is inspired by (and, in a sense, dual to) the classical construction of `additive regulator maps' due to Beilinson and Feigin. As a consequence, we show that, if A is an abelian Lie algebra, the composite map Phi_{g}(A) Tr_{g}(A) is represented by a canonical differential operator acting on differential forms on Sym(A) and depending only on the Cartan data (h, W, P), where P is a W-invariant polynomial on h. We derive a combinatorial formula for this operator that plays an important role in the study of derived commuting schemes in [BFPRW].