On the Humphreys conjecture on support varieties of tilting modules

Pramod N. Achar (1) , William Hardesty (1) , Simon Riche (2, 3)

1 LSU Mathematics - Department of Mathematics [Baton Rouge]
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The "support variety" of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel of G. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G=SLn in earlier work of the second author. In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving "relative support varieties."

Domaines: Mathématiques [math]

Pramod N. Achar, William Hardesty, Simon Riche. On the Humphreys conjecture on support varieties of tilting modules. Transformation Groups, 2019, 24, pp.597-657. ⟨hal-01576578⟩