Derived character maps of Lie representations and Chern--Simons forms

Abstract

We study the derived representation scheme $ \drep_{\g}(\fra) $ parametrizing the representations of a Lie algebra $ \fra $ in a reductive Lie algebra $ \g $. We define two canonical maps $\, \Tr_{\g}(\fra):\, \hc_{\bullet}^{(r)}(\fra) \to \h_{\bullet}[\drep_{\g}(\fra)]^G $ and $ \Phi_{\g}(\fra):\,\h_{\bullet}[\drep_{\g}(\fra)]^G \to \h_{\bullet}[\drep_{\frh}(\fra)]^{\bW} $, called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. The Drinfeld trace is defined on the $r$-th Hodge component of the cyclic homology of the universal enveloping algebra $\cU\fra $ of the Lie algebra $ \fra $ and depends on the choice of a $G$-invariant polynomial $ P \in \sym^r(\g^*)^G $ on the Lie algebra $ \g$. The Harish Chandra homomorphism $ \Phi_{\g}(\fra) $ is a graded algebra homomorphism extending to representation homology the natural restriction map $\, k[\rep_{\g}(\fra)]^G \to k[\rep_{\frh}(\fra)]^{\bW} $, where $ \frh \subset \g $ is a Cartan subalgebra of $ \g $ and $ {\bW} $ is the associated Weyl group. We give general formulas for these maps in terms of Chern--Simons forms. As a consequence, we show that, if $ \fra $ is an abelian Lie algebra, the composite map $ \Phi_{\g}(\fra) \circ \Tr_{\g}(\fra) $ is given by a canonical differential operator defined on differential forms on $ A = \sym(\fra) $ and depending only on the Cartan data $ (\frh, {\bW}, P) $, where $ P \in \sym(\frh^*)^{\bW} $. We derive a combinatorial formula for this operator that plays a key role in the study of derived commuting schemes in \cite{BFPRW17a}.