Endomorphisms of Gelfand-Graev representations

Jack Shotton (Durham University)

Wednesday 25th January, 2023 16:00-17:00 Maths 311B


The Gelfand-Graev representation is an important representation of a finite group G of Lie type that contains each 'generic' representation with multiplicity one. Its endomorphism ring E is therefore commutative. The talk is about the problem of determining the lattice of integral endomorphisms (over Z_l, for l different from the defining characteristic p of G) inside E. I will report on work of Tzu-Jan Li showing that E is, under some hypotheses, isomorphic to a ring B coming from invariant theory, via the Grothendieck ring K of mod p representations of G. I will then explain joint work of myself and Li extending this to the case where l does not divide the order of the Weyl group. The ring E arises naturally when studying generalisations of the local Langlands correspondence to coefficients in Z_l-algebras, and I will explain how the ring B is related to certain local Galois deformation rings.

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