Minimal dimensional representations of reduced enveloping algebras for gln
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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.
|Number of pages||24|
|Early online date||11 Jul 2019|
|Publication status||Published - 1 Aug 2019|
- general linear Lie algebras, reduced enveloping algebras, Finite W-algebras