Silting complexes of coherent sheaves and the Humphreys conjecture. (English) Zbl 07928013

Achar, Pramod N.; Hardesty, William

Duke Math. J. 173, No. 12, 2397-2445 (2024).

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.
Let \( \mathbf{X} \) 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 \( \lambda \).
Let \(W\) 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 [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 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, arXiv:1812.09960].

Reviewer: Evgenii L. Bashkirov (Minsk)

MSC:

20G10	Cohomology theory for linear algebraic groups
14F08	Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
20G15	Linear algebraic groups over arbitrary fields

Keywords:

Frobenius kernels; Humphreys conjecture; support varieties

Citations:

Zbl 1475.20073; Zbl 0919.17013

References:

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