Minimal dimensional representations of reduced enveloping algebras for gln

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Abstract

Let g=glN(k), where k is an algebraically closed field of characteristic p>0, and N∈ℤ≥1. Let χ∈g and denote by Uχ(g) the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional Uχ(g)-module has dimension divisible by p, where dχ is half the dimension of the coadjoint orbit of χ. Our main theorem gives a classification of Uχ(g)-modules of dimension p. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for U0(h) for a certain Levi subalgebra h of g; we view this as a modular analogue of Mœglin's theorem on completely primitive ideals in U(glN(ℂ)). To obtain these results, we reduce to the case χ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted W-algebra.

Details

Original languageEnglish
Pages (from-to)1594-1617
Number of pages24
JournalCompositio Mathematica
Volume155
Issue number8
Early online date11 Jul 2019
Publication statusPublished - 1 Aug 2019

Keywords

  • general linear Lie algebras, reduced enveloping algebras, Finite W-algebras