# Minimal-dimensional representations of reduced enveloping algebras for gl$$_n$$

Goodwin, Simon M., Topley, Lewis (2019) Minimal-dimensional representations of reduced enveloping algebras for gl$$_n$$. Compositio Mathematica, 155 (8). pp. 1594-1617. ISSN 0010-437X. (doi:10.1112/S0010437X19007474) (KAR id:83305)

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## Abstract

Let g=gl$$_N$$(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$$^d$$χ , 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$$^d$$χ. As a consequence, we deduce that they are all parabolically induced from a one-dimensional module for U$$_0$$(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(gl$$_N$$(C)) . To obtain these results, we reduce to the case where χ is nilpotent, and then classify the one-dimensional modules for the corresponding restricted W -algebra.

Item Type: Article 10.1112/S0010437X19007474 general linear Lie algebras, reduced enveloping algebras, finite W-algebras, Yangians Q Science > QA Mathematics (inc Computing science) Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Lewis Topley 07 Oct 2020 12:16 UTC 16 Feb 2021 14:15 UTC https://kar.kent.ac.uk/id/eprint/83305 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-4701-4384