Generic stabilizers and the first Kac-Weisfeiler conjecture for Lie algebras of algebraic groups

Abstract

In 1971, V. Kac and B. Weisfeiler conjectured that the maximum dimension of an irreducible representation of a finite-dimensional restricted Lie algebra over an algebraically closed field in characteristic p > 0 is equal to p^((dim g - ind g)/2). This conjecture is known as KW1 and it is still open. In the first part of this thesis we prove by induction that KW1 can be reduced to restricted semisimple Lie algebras and (non-split) central extensions of semisimple Lie algebras, with p > 2. This result is new, but not easy to apply, as semisimple Lie algebras are not always a direct sum of simple Lie algebras. As a consequence, we re-prove KW1 for solvable Lie algebras with p > 2. Then, we explore how to apply this result to a restricted ideal of the Lie algebra of an algebraic group in characteristic p > 3, denoted by g. The advantage of algebraic groups is that reductive groups are well known and we can prove KW1 for Lie algebras of reductive groups and (non-split) central extensions. In particular, the base cases of the induction are proved. At every step of the reduction, we pass from g to a new restricted Lie algebra g' of smaller dimension; which is realized from the stabilizer of the action of g on a generic vector v of an irreducible restricted g-module V. In particular, one needs to prove that g' carries the same inductive assumptions as g (see 0.5). Hence, the rest of the thesis is devoted to studying the structure of such generic stabilizers. We extend a result of A.M. Popov in positive characteristic and find a lot of classes of cases for which the generic stabilizer is p-nilpotent. In the last chapter we confirm in positive characteristic some of the generic stabilizers found by G. Elashvili in characteristic 0. The study of generic stabilizers is not complete, but all the known cases are compatible with our inductive assumptions. In 2018 B. Martin, D. Stewart and L. Topley proved KW1 for Lie algebras of linear algebraic groups in large characteristic. This work contains a possible approach to lighten the assumptions on the characteristic.

Date of Award	1 Aug 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Alexander Premet (Supervisor) & Yuri Bazlov (Supervisor)