Section 4. (0N03) · ScalingStacks

4. Motivic Springer theory

In this section, we introduce the general setup and definitions of Springer motives and the motivic extension algebra. Moreover, we introduce the local purity and finiteness conditions (PT) and (FO). We then combine the results of Sections 2 and 3 to obtain our main formality results for Springer motives.

4.1. The setup

Recall that all varieties are over $k=\overline{\mathbb{F}}_{p}.$ Let $G$ be a linear algebraic group. Let $\mu_{i}:\widetilde{\mathcal{N}}_{i}\to\mathcal{N}$ be a collection of $G$ -equivariant proper maps where each $\widetilde{\mathcal{N}}_{i}$ is smooth and connected. Consider the collections of objects

\displaystyle\mathcal{T}^{Spr}=\{\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}})\in\operatorname{DM}_{G}(\mathcal{N})\}\text{ and }\widehat{\mathcal{T}}^{Spr}=\bigcup_{n\in\mathbb{Z}}\mathcal{T}^{Spr}(n)[2n].

0MYP

Definition 4.1. The triangulated category of Springer motives $\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})$ is the full subcategory of $\operatorname{DM}_{G}(\mathcal{N})$ generated by $\widehat{\mathcal{T}}^{Spr}$ with respect to isomorphism, direct summands and triangles,

\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})=\langle\widehat{\mathcal{T}}^{Spr}\rangle_{\cong,\inplus,\Delta}\subset\operatorname{DM}_{G}(\mathcal{N}).

We define two conditions, that are assumed in some of the later results.

(PT)

Pure Tate. For each $x\in\mathcal{N},$ the motive $\operatorname{M}(\mu_{i}^{-1}(\{x\})$ is pure Tate.
(FO)

Finite Number of Orbits. The images $\mu_{i}(\widetilde{\mathcal{N}}_{i})\subset\mathcal{N}$ have finitely many $G$ -orbits.

0MYQ

Remark 4.2. Condition (FO) allows for simple induction arguments. In many settings it can be weakened such that all arguments still work. For instance, there is a quite straightforward adaption to ind-varieties with possibly infinitely many orbits.

4.2. Motivic extension algebras

We define now the motivic and $\ell$ -adic extension algebras.

0MYR

Definition 4.3. The motivic extension algebra $E$ is the $\mathbb{Z}$ -graded locally unital algebra

E=\operatorname{End}^{\bullet}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mathcal{T}^{Spr})

which has a grading induced by the autoequivalence $\langle 1\rangle=(1)[2],$ see (2.3). More explicitly, for $n\in\mathbb{Z}$ the $n$ -th graded part of $E$ is

E^{n}=\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]).

For every prime $\ell\neq 0$ we define the $\ell$ -adic extension algebra $E^{\acute{e}t}_{\ell}$ in the same way, replacing $\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})$ by $\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).$

The realisation functor induces a morphism of algebras $\operatorname{Real}_{\ell}:E\to E^{\acute{e}t}_{\ell}$ . The motivic extension algebra can be understood in terms of $G$ -equivariant Chow groups of the Steinberg varieties $Z_{i,j}=\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}$ equipped with a convolution product.

0MYS

Proposition 4.4. Let $d_{j}=\dim\widetilde{\mathcal{N}}_{j}.$ There is a natural isomorphism

\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])=\operatorname{CH}_{d_{j}-n}^{G}(Z_{i,j})_{\mathbb{Q}}.

0MYT

Proof. Let $Z=Z_{i,j}$ with projections $\pi_{i,j}:Z\to\widetilde{\mathcal{N}}_{i,j}.$ For a variety $X$ denote by $\operatorname{fin}_{X}:X\to\operatorname{pt}$ the structure map. Then by using various adjunctions, base change and $\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{*}=\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{!}(-d_{j})[-2d_{j}]$ since $\widetilde{\mathcal{N}}_{j}$ is smooth, we get

	$\displaystyle\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}($	$\displaystyle\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(\widetilde{\mathcal{N}}_{i},\mathbb{Q})}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}},\mu_{i}^{!}\mu_{j,*}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(\widetilde{\mathcal{N}}_{i},\mathbb{Q})}(\operatorname{fin}_{\widetilde{\mathcal{N}}_{i}}^{}\mathbb{Q},\pi_{i,}\pi_{j}^{!}\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{*}\mathbb{Q}(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{fin}_{\widetilde{\mathcal{N}}_{i},}\pi_{i,}\pi_{j}^{!}\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{!}\mathbb{Q}(n-d_{j})[2(n-d_{j})])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{fin}_{Z,*}\operatorname{fin}_{Z}^{!}\mathbb{Q}(n-d_{j})[2(n-d_{j})])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{M}^{c}(Z)(n-d_{j})[2(n-d_{j})]).$

Now the last term is isomorphic to $\operatorname{CH}^{G}_{d_{j}-n}(Z)_{\mathbb{Q}}$ by [Kel17, Theorem 5.3.14]. ∎

0MYU

Remark 4.5. Since $Z_{i,j}$ is not necessarily equidimensional we need to work with Chow groups indexed by the dimension of cycles here.

The convolution of two cycles $\alpha\in\operatorname{CH}^{G}_{d_{j}-n}(Z_{i,j})_{\mathbb{Q}}$ and $\beta\in\operatorname{CH}^{G}_{d_{k}-m}(Z_{j,k})_{\mathbb{Q}}$ is given by the formula

(4.1)

\displaystyle\alpha\star\beta=p_{*}\delta^{!}(\alpha\times\beta)\in\operatorname{CH}^{G}_{d_{k}-n-m}(Z_{i,k})_{\mathbb{Q}}

where $\alpha\times\beta$ is the exterior product and

	$\displaystyle\delta$	$\displaystyle:\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}\to\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}=Z_{i,j}\times Z_{j,k}\text{ and}$
	$\displaystyle p$	$\displaystyle:\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}\to\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}=Z_{i,k}$

are the diagonal and projection maps.

Convolution of cycles and composition of morphisms are compatible in the obvious way. In the non-equivariant case this is proven and discussed in detail in [Fan16]. The proof can be adapted to the equivariant case by using that both equivariant motivic sheaves and equivariant Chow groups are defined in terms of their non-equivariant versions of approximations of the Borel construction. We will not present the details here.

0MYV

Corollary 4.6. There is an isomorphism of graded algebras

(4.2)

\displaystyle E^{\bullet}\cong\bigoplus_{i,j}\operatorname{CH}_{d_{j}-\bullet}^{G}(Z_{i,j})_{\mathbb{Q}}

Similarly, the $\ell$ -adic extension algebra can be described in terms of $\ell$ -adic Borel–Moore homology, see [CG10, Section 8.6].

0MYW

Proposition 4.7. There is an isomorphism of graded algebras

(4.3)

\displaystyle(E^{\acute{e}t}_{\ell})^{\bullet}\cong\bigoplus_{i,j}H^{BM,G}_{2(d_{j}-\bullet)}(Z_{i,j},\mathbb{Q}_{\ell}(d_{j}-\bullet)).

0MYX

Remark 4.8. The above discussion should be a shadow of the following conjectural general theory. There should be a Chow weight structure on the category $\operatorname{DM}_{G}^{gm}(\mathcal{N})$ similarly to the non-equivariant case, see Example 2.2(3). The heart of this Chow weight structure should be equivalent to a category of equivariant relative Chow motives $\operatorname{Chow}_{G}(\mathcal{N},\mathbb{Q})$ in which the composition of morphisms is defined via convolution as in (4.1). Since by assumption $\widetilde{\mathcal{N}}_{i}$ is smooth and $\mu_{i}$ is projective the motive $\mu_{!}(\mathbb{Q}_{M_{i}})$ should be in the heart and correspond to the relative Borel–Moore motive $M^{BM}(\widetilde{\mathcal{N}}_{i}/\mathcal{N})$ in the category $\operatorname{Chow}_{G}(N,\mathbb{Q}).$

In the non-equivariant case this is shown to be true by Fangzhou [Fan16]. To define a weight structure in the equivariant case, one would need appropriate $G$ -equivariant resolution of singularities or alterations, see [SVW18, Remark II.4.15].

In this article we get around this problem by defining a weight structure on the subcategory $\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})$ by brute force using the conditions (PT) and (FO).

4.3. Main results

We will now combine the formalism of weight structures and weight complex functors from Section 2 with our results on pointwise pure Tate motives from Section 3.6 to obtain the following main result.

0MYY

Theorem 4.9. Assume that (PT) and (FO) hold. Then there is an equivalence of categories

Moreover, for all primes $\ell\neq p$ the $\ell$ -adic realisation functor $\operatorname{Real}_{\ell}$ gives an isomorphism $E\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\cong E^{\acute{e}t}_{\ell}$ and acts as a degrading functor with respect to the Tate-twist $(1)$ in the sense of [BGS96]

The following is the most important ingredient in order to prove Theorem 4.9.

0MYZ

Proposition 4.10. Assume that the conditions (PT) and (FO) are fulfilled. Then the collection of objects $\widehat{\mathcal{T}}^{Spr}$ in $\operatorname{DM}_{G}(\mathcal{N})$ is tilting.

0MZ0

Proof. The condition (PT) implies that all objects in $\widehat{\mathcal{T}}^{Spr}$ are pointwise pure Tate by Proposition 3.21. Now, let $M=\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}})(k)[2k]$ and $N=\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(l)[2l].$ The subvariety $k_{i,j}:\mathcal{N}_{i,j}=\mu_{i}(\widetilde{\mathcal{N}}_{i})\cup\mu_{j}(\widetilde{\mathcal{N}}_{j})\hookrightarrow\mathcal{N}$ is closed since $\mu_{i}$ and $\mu_{j}$ are proper. Since $M$ and $N$ are supported on $\mathcal{N}_{i,j}$ there is an equality

(4.4)

\operatorname{Hom}_{\operatorname{DM}_{G}(N,\mathbb{Q})}(M,N[n])=\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N}_{i,j},\mathbb{Q})}(k_{i,j}^{*}M,k_{i,j}^{!}N[n]).

The condition (FO) ensures that there are only finitely many $G$ -orbits in $\mathcal{N}_{i,j}.$ Moreover $k_{i,j}^{*}M$ and $k_{i,j}^{!}N$ are $*$ - and $!$ -pointwise pure Tate. Hence by Proposition 3.19 the right hand side of (4.4) vanishes for $n\neq 0$ and the statement follows. ∎

By combining Propositions 4.10 and 2.6 we can define a weight structure $w$ on $\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})$ with heart $\langle\widehat{\mathcal{T}}^{Spr}\rangle_{\cong,\inplus,\oplus}.$ Now, the first statement of Theorem 4.9 follows from Corollary 2.16. The remaining statements are implied by the following:

0MZ1

Proposition 4.11. Assume that the conditions (PT) and (FO) are fulfilled. Let $M,N\in\operatorname{DTM}^{Spr}_{G}(\mathcal{N})$ and $\ell\neq p$ be a prime. The natural isomorphisms $\operatorname{Real}_{\ell}\circ(1)\to\operatorname{Real}_{\ell}$ from (3.1) induce isomorphisms

\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{DM}_{G}(N,\mathbb{Q}_{\ell})}(M,N(n))\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N))

and for $M,N\in\operatorname{DTM}^{Spr}_{G}(\mathcal{N})^{w=0}$ isomorphisms

\operatorname{Hom}_{\operatorname{DM}_{G}(N,\mathbb{Q}_{\ell})}(M,N)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N)).

0MZ2

Proof. Let $M,N\in\operatorname{DTM}^{Spr}_{G}(\mathcal{N}).$ All objects in $\widehat{\mathcal{T}}^{Spr}$ are supported on a closed subset of $\mathcal{N}$ consisting of finitely many $G$ -orbits by condition (FO) and are pointwise pure Tate using condition (PT) and Proposition 3.21. Now $M$ and $N$ are constructed from the objects in $\widehat{\mathcal{T}}^{Spr}$ by a finite combination of taking direct summands, finite direct sums and triangles. Hence $M$ and $N$ are also supported on a closed subset of $\mathcal{N}$ consisting finitely many $G$ -orbits and are pointwise mixed Tate. Now the statement follows from Proposition 3.20. ∎

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2