Silting complexes of coherent sheaves and the Humphreys conjecture (Q6620662)

Let $ \mathbf{G} $ be a connected reductive group over an algebraically closed field $ \Bbbk $ of characteristic $p$. Assume that $p$ is greater than the Coxeter number for $ \mathbf{G} $. Let $ \mathbf{G}_{1} $ be its first Frobenius kernel and let $ G = \mathbf{G} / \mathbf{G}_{1} $ be its Frobenius twist. Let $ \mathcal{N} $ be the nilpotent variety in the Lie algebra of $G$. Then the algebra $ \mathrm{Ext}_{ \mathbf{G}_{1} }^{\bullet} ( \Bbbk , \Bbbk ) $ is ($G$-equivariantly) isomorphic to the coordinate ring $ \Bbbk [ \mathcal{N} ] $ and for any $ \mathbf{G} $-module $M$, the $ \mathbf{G}_{1} $-cohomology $ H^{\bullet} ( \mathbf{G}_{1}, M ) = \mathrm{Ext}_{ \mathbf{G}_{1} }^{\bullet} ( \Bbbk , M ) $ has the structure of a $G$-equivariant graded $ \Bbbk [ \mathcal{N} ] $-module, or equivalently, a $ G\times \mathbb{G}_{m} $-equivariant (quasi-)coherent sheaf on $ \mathcal{N} $. The main goal of the paper under review is to give a new description of this cohomology in the case where $M$ is an indecomposable tilting $ \mathbf{G} $-module.\N\NLet $ \mathbf{X} $ be the weight lattice of $G$ and let $ \mathbf{X}^{+}\subset \mathbf{X} $ be the set of dominant weights. For $ \lambda\in \mathbf{X}^{+} $, let $ \mathsf{T} ( \lambda ) $ be the indecomposable tilting $ \mathbf{G} $-module of highest weight $ \lambda $.\N\NLet $W$ be the Weyl group of $ \mathbf{G} $ and let $ W_{\mathrm{ext}} = W\ltimes \mathbf{X} $ be its extended affine Weyl group. For $ \lambda\in \mathbf{X} $, let $ t_{\lambda} $ denote the corresponding element of $ W_{\mathrm{ext}} $. For $ \lambda\in \mathbf{X}^{+} $, let $ w_{\lambda} $ be the unique element of minimal length in the double coset $ Wt_{\lambda}W\subset W_{\mathrm{ext}} $. The following action of the group $ W_{\mathrm{ext}} $ on $ \mathbf{X} $ is called the $p$-dilated dot action of $ W_{\mathrm{ext}} $ on $ \mathbf{X} $: for $ w = v\ltimes t_{\lambda}\in W_{\mathrm{ext}} $ and $ \mu\in \mathbf{X} $, $ w\cdot\mu $ is defined to be $ v( \mu + p\lambda + \rho ) - \rho $, where $ \rho $ is one-half the sum of the positive roots. According to Lemma 8.7 in [\textit{P. Achar} et al., Transform. Groups, 24, No. 3, 597--657 (2019; Zbl 1475.20073))], $H^{\bullet} ( \mathbf{G}_{1}, \mathsf{T} ( \mu ) ) = 0 $ unless $ \mu = w_{\lambda}\cdot 0 $ for some $ \lambda\in\mathbf{X}^{+} $. The main theorem of the article (Theorem 8.1) describes $ H^{\bullet} ( \mathbf{G}_{1}, \mathsf{T} ( \mu ) ) $ in the case where $ \mu = w_{\lambda}\cdot 0 $ for some $ \lambda\in\mathbf{X}^{+} $. It confirms a relative version of a conjecture due to \textit{J. E. Humphreys} [AMS/IP Stud. Adv. Math. 4, 69--80 (1997; Zbl 0919.17013)] as well as part of a refinement of this conjecture proposed by the first authot et al. [``Conjectures on tilting modules and antispherical $p$-cells'', Preprint, \url{arXiv:1812.09960}].