# Aliaksandr Patotski

### First Position

Data Scientist, Microsoft### Dissertation

*Derived characters of Lie representations and Chern-Simons forms*

### Advisor

### Abstract

We study the derived representation scheme $ \mathrm{DRep}_{\mathfrak{g}}(\mathfrak{a}) $ parametrizing the representations of a Lie algebra $ \mathfrak{a} $ in a reductive Lie algebra $ \mathfrak{g} $. We define two canonical maps $\, \mathrm{Tr}_{ \mathfrak{g} }( \mathfrak{a} ):\, HC_{\bullet}^{(r)}( \mathfrak{a} ) \to H_{\bullet}[\mathrm{DRep}_{ \mathfrak{g} }( \mathfrak{a} )]^G $and $ \Phi_{ \mathfrak{g} }( \mathfrak{a} ):\, H_{\bullet}[\mathrm{DRep}_{ \mathfrak{g} }( \mathfrak{a} )]^G \to H_{\bullet}[\mathrm{DRep}_{ \mathfrak{h} }( \mathfrak{a} )]^{W} $, 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 $\mathcal{U} \mathfrak{a} $ of the Lie algebra $ \mathfrak{a} $ and depends on the choice of a $G$-invariant polynomial $ P \in Sym^r( \mathfrak{g} ^*)^G $ on the Lie algebra $ \mathfrak{g} $. The Harish Chandra homomorphism $ \Phi_{ \mathfrak{g} }( \mathfrak{a} ) $ is a graded algebra homomorphism extending to representation homology the natural restriction map $\, k[Rep_{ \mathfrak{g} }( \mathfrak{a} )]^G \to k[Rep_{ \mathfrak{h} }( \mathfrak{a} )]^{W} $, where $ \mathfrak{h} \subset \mathfrak{g} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ W $ 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 $ \mathfrak{a} $ is an abelian Lie algebra, the composite map $ \Phi_{ \mathfrak{g} }( \mathfrak{a} ) \circ Tr_{ \mathfrak{g} }( \mathfrak{a} ) $ is given by a canonical differential operator defined on differential forms on $ A = Sym( \mathfrak{a} ) $ and depending only on the Cartan data $ ( \mathfrak{h} , {W}, P) $, where $ P \in Sym( \mathfrak{h} ^*)^{W} $. We derive a combinatorial formula for this operator that plays a key role in the study of derived commuting schemes in [BFPRW17]. We study the derived representation scheme $ thrm{DRep}_{\mathfrak{g}}(\mathfrak{a}) $ parametrizing the representations of a Lie algebra $ \mathfrak{a} $ in a reductive Lie algebra $ \mathfrak{g} $. We define two canonical maps $\, \mathrm{Tr}_{ \mathfrak{g} }( \mathfrak{a} ):\, HC_{\bullet}^{(r)}( \mathfrak{a} ) \to H_{\bullet}[\mathrm{DRep}_{ \mathfrak{g} }( \mathfrak{a} )]^G $ and $ \Phi_{ \mathfrak{g} }( \mathfrak{a} ):\, H_{\bullet}[\mathrm{DRep}_{ \mathfrak{g} }( \mathfrak{a} )]^G \to H_{\bullet}[\mathrm{DRep}_{ \mathfrak{h} }( \mathfrak{a} )]^{W} $, 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 $\mathcal{U} \mathfrak{a} $ of the Lie algebra $ \mathfrak{a}$ and depends on the choice of a $G$-invariant polynomial $ P \in Sym^r( \mathfrak{g} ^*)^G$ on the Lie algebra $ \mathfrak{g} $. The Harish Chandra homomorphism $ \Phi_{ \mathfrak{g} }( \mathfrak{a} ) $ is a graded algebra homomorphism extending to representation homology the natural restriction map

$\, k[Rep_{ \mathfrak{g} }( \mathfrak{a} )]^G \to k[Rep_{ \mathfrak{h} }( \mathfrak{a} )]^{W} $, where $ \mathfrak{h} \subset \mathfrak{g}$ is a Cartan subalgebra of $ \mathfrak{g} $ and $ W $ 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 $ \mathfrak{a} $ is an abelian Lie algebra, the composite map $ \Phi_{ \mathfrak{g} }( \mathfrak{a} ) \circ Tr_{ \mathfrak{g} }( \mathfrak{a} ) $ is given by a canonical differential operator defined on differential forms on $ A = Sym( \mathfrak{a} ) $ and depending only on the Cartan data $ ( \mathfrak{h} , {W}, P) $, where $ P \in Sym( \mathfrak{h} ^*)^{W} $. We derive a combinatorial formula for this operator that plays a key role in the study of derived commuting schemes in [BFPRW17].