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 pdχ, where dχ is half the dimension of the coadjoint orbit of χ. Our main theorem gives a classification of Uχ(g)-modules of dimension pdχ. 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.
Original language | English |
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Pages (from-to) | 1594-1617 |
Number of pages | 24 |
Journal | Compositio Mathematica |
Volume | 155 |
Issue number | 8 |
Early online date | 11 Jul 2019 |
DOIs | |
Publication status | Published - 1 Aug 2019 |
Keywords
- general linear Lie algebras
- reduced enveloping algebras
- Finite W-algebras