Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for jet algebras

2013/08/13 by Masoud Kamgarpour, Kamgarpour, Masoud
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.AG #math.QA #math.RT

paper · pdf · doi:10.48550/arxiv.1308.2995

First draft

arxiv created 2013/08/13 · arxiv updated 2013/08/15

Abstract

Let \fg be a simple finite-dimensional complex Lie algebra with a Cartan subalgebra \fh and Weyl group W. Let \fg_n denote the Lie algebra of n-jets on \fg. A theorem of Rais and Tauvel and Geoffriau identifies the centre of the category of \fg_n-modules with the algebra of functions on the variety of n-jets on the affine space \fh^*/W. On the other hand, a theorem of Feigin and Frenkel identifies the centre of the category of critical level smooth modules of the corresponding affine Kac-Moody algebra with the algebra of functions on the ind-scheme of opers for the Langlands dual group. We prove that these two isomorphisms are compatible by defining the higher residue of opers with irregular singularities. We also define generalized Verma and Wakimoto modules and relate them by a nontrivial morphism.

Related

• A Geometric Proof of the Feigin-Frenkel Theorem
• Jet modules
• Self-extensions of Verma modules and differential forms on opers
• Vertex algebras and Hodge structures
• A Feigin-Frenkel theorem with n singularities