Masoud Kamgarpour

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

Let g be a simple finite-dimensional complex Lie algebra with a Cartan subalgebra h and Weyl group W. Let g denote the Lie algebra of n-jets on g. A theorem of Rais and Tauvel and Geoffriau identifies the centre of the category of g-modules with the algebra of functions on the variety of n-jets on the affine space h*/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.

Maximal representation dimension of finite p-groups

Abstract The representation dimension rdim(G) of a finite group G is the smallest positive integer m for which there exists an embedding of G in GL m (ℂ). In this paper we find the largest value of rdim(G), as G ranges over all groups of order pn , for a fixed prime p and a fixed exponent n ⩾ 1.

Compatible Intertwiners for Representations of Finite Nilpotent Groups

We sharpen the orbit method for finite groups of small nilpotence class by associating representations to functionals on the corresponding Lie rings. This amounts to describing compatible intertwiners between representations parameterized by an additional choice of polarization. Our construction is motivated by the theory of the linearized Weil representation of the symplectic group. In particular, we provide generalizations of the Maslov index and the determinant functor to the context of finite abelian groups.