ScalingStacks

Setup and main results

Let ΞΌi:𝒩~i→𝒩\mu_{i}:\widetilde{\mathcal{N}}_{i}\to\mathcal{N} be a collection of GG-equivariant proper maps of varieties such that each 𝒩~i\widetilde{\mathcal{N}}_{i} is smooth. We consider the motivic extension algebra

E=⨁nβˆˆβ„€β¨i,jHomDMG⁑(𝒩,β„š)(ΞΌi,!(β„šπ’©~i),ΞΌj,!(β„šπ’©~j)(n)[2n]),E=\bigoplus_{n\in\mathbb{Z}}\bigoplus_{i,j}\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n]),

which is a motivic version of the β„“\ell-adic extension algebra Eβ„“e´​tE^{\acute{e}t}_{\ell} defined via the category DG⁑(𝒩,β„šβ„“)\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell}) of equivariant β„“\ell-adic sheaves.

The algebra EE can be described in classical terms as the GG-equivariant Chow groups of the Steinberg varieties Zi,j=𝒩~i×𝒩𝒩~jZ_{i,j}=\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j} equipped with a convolution product. Namely, by Corollary Β 4.6 there is an isomorphism

E≅⨁i,jCHβˆ™G​(Zi,j)β„š.E\cong\bigoplus_{i,j}\operatorname{CH}_{\bullet}^{G}(Z_{i,j})_{\mathbb{Q}}.

The motivic extension algebra is defined using the category DMG⁑(𝒩,β„š)\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q}) of GG-equivariant motivic sheaves on 𝒩,\mathcal{N}, which was introduced in [SVW18] as a motivic version of DG⁑(𝒩,β„šβ„“).\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell}). Similarly to their β„“\ell-adic counterpart, equivariant motivic sheaves are equipped with a six-functor-formalism. Their hom-spaces are governed by equivariant higher Chow groups and they admit an autoequivalence (1)(1) called Tate twist, which will serve as a shift of grading functor for us.

We define the full subcategory of Springer motives

DMGS​p​r(𝒩,β„š)=⟨μi,!(β„šπ’©~i)βŸ©β‰…,β¨­,Ξ”,(Β±1)βŠ‚DMG(𝒩,β„š),\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})=\langle\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}})\rangle_{\cong,\inplus,\Delta,(\pm 1)}\subset\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q}),

see DefinitionΒ 4.1, and introduce local purity and finiteness conditions (PT) and (FO), see SectionΒ 4.1. We will prove the following formality result showing that Springer motives can be described solely in terms of the algebra EE carrying no higher structure.

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Theorem (TheoremΒ 4.9). Assuming (PT) and (FO) there is an equivalence of categories between the category of Springer motives and the perfect derived category of graded modules of the motivic extension algebra

DMGS​p​r⁑(𝒩,β„š){\lx@inpgf@ignorespaces\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})}Dperf℀⁑(E).{\lx@inpgf@ignorespaces{\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(E)}.}∼\scriptstyle{\lx@inpgf@ignorespaces\sim}

Moreover, for all primes β„“β‰ p\ell\neq p the β„“\ell-adic realisation functor Realβ„“\operatorname{Real}_{\ell} gives an isomorphism EβŠ—β„šβ„šβ„“β‰…Eβ„“e´​tE\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\cong E^{\acute{e}t}_{\ell} and acts as a degrading functor with respect to the Tate-twist (1)(1) in the sense of [BGS96]

DMGS​p​r⁑(𝒩,β„šβ„“){\lx@inpgf@ignorespaces\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}DGS​p​r⁑(𝒩,β„šβ„“).{\lx@inpgf@ignorespaces{\operatorname{D}^{Spr}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).}}Realβ„“\scriptstyle{\lx@inpgf@ignorespaces\operatorname{Real}_{\ell}}
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Remark 1.1. The analogous statement replacing motivic sheaves by mixed β„“\ell-adic sheaves (or mixed Hodge modules) fails because there are non-trivial extensions between the Tate objects β„šβ„“β€‹(n)\mathbb{Q}_{\ell}(n) in the category of mixed β„“\ell-adic sheaves. These unwanted extensions were first addressed in the context of perverse sheaves on flag varieties and category π’ͺ\mathcal{O} in [BGS96, Section 4]. There, a workaround is proposed using the derived category of mixed β„“\ell-adic perverse sheaves which have a semisimple Frobenius action on their associated graded with respect to the weight filtration. This construction has several drawbacks when compared to motivic sheaves. First, it is not clear how to extend it to the equivariant case and to settings where no perverse tt-structure exists. Secondly, it is difficult to determine if the six functors preserve the semisimplicity of the Frobenius. This and the independence of the prime β„“\ell are the main technical advantage of motivic sheaves.

We apply the result in the setting of Lusztig’s graded affine Hecke algebra ℍ¯​(G)\overline{\mathbb{H}}(G) associated to the root datum of a reductive group G,G, see also [Ebe21].

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Theorem (TheoremΒ 5.2). Under a standard assumption on p,p, there is an equivalence of categories

DMG×𝔾mS​p​r⁑(𝒩n​i​l,β„š)β‰…Dperf℀⁑(ℍ¯​(G))\operatorname{DM}^{Spr}_{G\times\mathbb{G}_{m}}(\mathcal{N}_{nil},\mathbb{Q})\cong\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(\overline{\mathbb{H}}(G))

between the category of Springer motives on the nilpotent cone 𝒩n​i​l\mathcal{N}_{nil} and the perfect derived category of graded modules of ℍ¯​(G).\overline{\mathbb{H}}(G).

We obtain a similar result for quiver Hecke algebras (KLR algebras) R𝐝R_{\mathbf{d}} for quivers QQ in type AA and A~\widetilde{A} (with cyclic orientation).

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Theorem (TheoremΒ 6.5). There is an equivalence of categories

DMGL⁑(𝐝)S​p​r⁑(Rep⁑(𝐝),β„š)β‰…Dperf℀⁑(R𝐝)\operatorname{DM}^{Spr}_{\operatorname{GL}(\mathbf{d})}(\operatorname{Rep}(\mathbf{d}),\mathbb{Q})\cong\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(R_{\mathbf{d}})

between the category of Springer motives for representations of QQ and the perfect derived category of graded modules of R𝐝.R_{\mathbf{d}}.

A similar result holds for quiver Schur algebras, see TheoremΒ 6.6. To establish the local purity condition (PT) in this setting, we prove that partial quiver flag varieties in type A~\widetilde{A} (with cyclic orientation) admit affine pavings.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2