Chern–Simons Forms and Higher Character Maps of Lie Representations

Abstract

Abstract. This paper is a sequel to [4], where we study the derived representation scheme DRep g (a) parametrizing the representations of a Lie algebra a in a reductive Lie algebra g. In [4], we constructed two canonical maps Trg(a) : HC (r)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 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 prove a combinatorial formula for this operator that plays an important role in the study of derived commuting schemes in [4].

Representation homology of simply connected spaces

“…Our next goal is to describe certain natural trace maps with values in representation homology. These maps were originally constructed in [3, 4] as (derived) characters of finite‐dimensional Lie representations. We will give a topological interpretation of these characters in terms of free loop spaces.…”

“…For $$P\,\in \,I^{m+1}({\mathfrak {g}})$$, the Chern‐Simons form $$TP(\theta )\,\in \,\mathcal {B}^{2m+1}$$ satisfies $$\delta (TP(\theta ))\,=\,P(\Omega ^{m+1})$$, where $$\Omega \,\in \,\mathcal {B}^2 \otimes {\mathfrak {g}}$$ is the curvature of $$\theta$$. Since $$\Omega ^{m+1}=0$$ by [4, Proposition A.2], $$TP(\theta )\,\in \,\mathcal {B}^{2m+1}$$ is a cocycle. It follows that $$s^{2m}TP(\theta )$$ defines a map of complexes …”

“…Fixing such a set $$ \lbrace P_1,\ldots ,P_l\rbrace $$ of generators in $$I({\mathfrak {g}})$$, we assemble the associated trace maps (4.14) into a single homomorphism of graded commutative algebras $$\begin{equation} \Lambda {\rm {Tr}}_{\mathfrak {g}}(\mathfrak {a}):\, \Lambda _k{\left[\mathop{\bigoplus}\limits_{i=1}^l \mathrm{HC}^{(d_i)}_\ast (\mathfrak {a})\right]} \xrightarrow {}\mathrm{HR}_\ast (\mathfrak {a},{\mathfrak {g}})^G. \end{equation}$$Following [3, 4], we call (4.15) the Drinfeld homomorphism for $$(\mathfrak {a},{\mathfrak {g}})$$. We note that the Drinfeld homomorphism (4.15) depends on the choice of polynomials $$\lbrace P_1,\ldots ,P_l\rbrace \subset I({\mathfrak {g}})$$, but for simplicity we suppress this in our notation.…”

“…Proof We first recall from [4] a construction of the Drinfeld traces via the Chern–Simons formalism. Let $$\mathrm{DR}(\mathcal {A}):=\Lambda _{\mathcal {A}}(\Omega ^1 \mathcal {A}[-1])$$ equipped with the differential $$d+\partial$$, where $$d$$ is the de Rham differential and $$\partial$$ is the internal differential induced by the differential on $$\mathcal {A}$$.…”

“…It follows that $$s^{2m}TP(\theta )$$ defines a map of complexes $$\begin{equation} \frac{1}{(m+1)! }s^{m}TP(\theta )\,:\,\mathcal {C}_\ast ({\mathfrak {g}}(\bar{\mathcal {A}});\mathbb {C}) \xrightarrow {}\tau ^m\overline{\mathrm{DR}}(\mathcal {A})[1].\end{equation}$$An explicit formula for the map (4.23) has been given in [21] (also see [4, Proposition A.3; 54, Equation 2.2]). By [4, Proposition A.5], the composition of the canonical projection (4.22) with (4.23) coincides with $$\Phi _P$$ (see (4.20)).…”