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

TL;DR

This paper demonstrates the compatibility between the Feigin-Frenkel and Harish-Chandra isomorphisms for jet algebras by introducing higher residues of opers with irregular singularities and relating generalized modules.

Contribution

It establishes the compatibility of two fundamental isomorphisms in representation theory for jet algebras and introduces the concept of higher residues of irregular opers.

Findings

Compatibility of the two isomorphisms proven

Higher residues of irregular opers defined

Relation between generalized Verma and Wakimoto modules established

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.