On the Humphreys Conjecture on Support Varieties of Tilting Modules

Abstract

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 G 1 . 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."1.2. The Humphreys conjecture. We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ G, denote by X the character lattice of T , and let X + ⊂ X be the subset of dominant weights (for the choice of positive roots such that B is the P.A.

Reductive groups, the loop Grassmannian, and the Springer resolution

“…Here we are using Sweedler's notation, with ∆(d) = d (1) ⊗ d (2) . Consider now two Hopf algebras D and E over k and a k-linear morphism of Hopf algebras ϕ : D → E. Let A and B be k-dg-algebras endowed with actions of E as above, and f : A → B be a k-linear morphism of dg-algebras which commutes with the E-actions.…”

“…(1) the functor Q sends standard, costandard, simple, and indecomposable tilting objects in Perv mix (Iw) (Gr, k) to standard, costandard, simple, and indecomposable tilting objects in Rep ∅ (G) respectively; (2) there is an isomorphism ε : Q • 1 ∼ − → Q that induces, for any F , G in Perv mix (Iw) (Gr, k) and any k ∈ Z, an isomorphism…”

“…In a different direction, in a joint work with W. Hardesty [2] we use the relation between the category Rep ∅ (G) and exotic coherent sheaves on N ∅ to obtain first results towards a proof of a conjecture of Humphreys [32] on support varieties of tilting G-modules.…”

Calculations with graded perverse-coherent sheaves

“…The derived category D b Coh G (N ) of G-equivariant coherent sheaves on N admits a remarkable t-structure whose heart is known as the category of perverse-coherent sheaves, and is denoted by PCoh(N ). This category has some features in common with classical (constructible) perverse sheaves: most importantly, every object has finite length, and the simple objects are produced by an "intermediate-extension" (or "IC") construction, starting from a pair (C, V), where C ⊂ N is a nilpotent orbit, and V is an irreducible (G × G m )-equivariant vector bundle on C. For applications of perverse-coherent sheaves to representation theory, see [AcHR1,AcRd,AriB,B2,B3]. Now let the multiplicative group G m act on N by z · x = z −2 x.…”

The ABC of p-cells

“…For the sake of completeness, we will state explicitly that these elementary relations generate the p-cell-preorder (see [AHR17,Lemma 5.3] for a proof): In the remainder of the section, we will prove some elementary properties of p-cells. In most cases we will focus on right p-cells and not state the version for left p-cells explicitly.…”