Chern-simons forms and higher character maps of lie representations

Abstract

We study the derived representation scheme DRepg(a) parameterizing the representations of a Lie algebra a in a reductive Lie algebra g. In our earlier work [2], we defined two canonical maps Trg(a): HC•(r)(a) → H•[DRepg(a)]G and Φg(a): H•[DRepg(a)]G → H•[DReph(a)] W, called the Drinfeld trace and the derived Harish-Chandra homomor-phism, respectively. In this paper, we give general formulas for these maps in terms of Chern-Simons forms. As a consequence, we show that, if a is an abelian Lie algebra, the composite map Φg(a) Trg(a) is given by a canonical differential operator defined on differential forms on A = Sym(a) and depending only on the Cartan data (h, W, P), where P ∈ Sym(h∗) W. We derive a combinatorial formula for this operator that plays a key role in the study of derived commuting schemes in [2].