On Harish-Chandra modules over quantizations of nilpotent orbits

2023/09/20 by Losev, Ivan, Yu, Shilin
#17B10 #17B35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2309.11191

Abstract

Let G be a semisimple algebraic group over the complex numbers and K be a connected reductive group mapping to G so that the Lie algebra of K gets identified with a symmetric subalgebra of \mathfrakg. So we can talk about Harish-Chandra (\mathfrakg,K)-modules, where \mathfrakg is the Lie algebra of G. The goal of this paper is to give a geometric classification of irreducible Harish-Chandra modules with full support over the filtered quantizations of the algebras of the form ℂ[\mathbbO], where \mathbbO is a nilpotent orbit in \mathfrakg with codimension of the boundary at least 4. Namely, we embed the set of isomorphism classes of irreducible Harish-Chandra modules into the set of isomorphism classes of irreducible K-equivariant suitably twisted local systems on \mathbbO∩ \mathfrakk^⊥. We show that under certain conditions, for example when K⊂ G or when \mathfrakg≅ \mathfrakso_n,\mathfraksp_2n, this embedding is in fact a bijection. On the other hand, for \mathfrakg=\mathfraksl_n and K=Spin_n, the embedding is not bijective and we give a description of the image. Finally, we perform a partial classification for exceptional Lie algebras.

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