# Motivic homotopy theory of the classifying stack of finite groups of Lie
type

Can Yaylali Mail<EMAIL_ADDRESS>
Department of Mathematics, TU Darmstadt, Schlossgartenstraße 7, 64289
Darmstadt, Germany
ORCID: 0000-0003-0150-3622

###### Abstract

Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite
group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel
$B$ of $G$. We relate the (rational) Tate motives of $\textup{B}G^{F}$ with
the $T$-equivariant Tate motives of the flag variety $G/B$.

On the way, we show that for a reductive group $G$ over a field $k$, with
maximal Torus $T$ and absolute Weyl group $W$, acting on a smooth finite type
$k$-scheme $X$, we have an isomorphism $A^{n}_{G}(X,m)_{\mathbb{Q}}\cong
A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}$ extending the classical result of Edidin-
Graham to higher equivariant Chow groups in the non-split case.

We also extend our main result to reductive group schemes over a regular base
that admit maximal tori. Further, we apply our methods to more general
quotient stacks. In this way, we are able to compute the motive of the stack
of $G$-zips introduced by Pink-Wedhorn-Ziegler for reductive groups over
fields of positive characteristic.

###### Contents

  1. 1 Introduction
  2. 2 Rational equivariant motivic homotopy theory
  3. 3 Equivariant motives under reductive groups
    1. 3.1 Preliminaries on reductive group schemes
    2. 3.2 Torsors under finite groups
    3. 3.3 The relation between the motives of $\left[X/T\right]$ and $\left[X/G\right]$
  4. 4 Torsors under tori
    1. 4.1 The motive of $T$-torsors
    2. 4.2 Motivic cohomology of $T$-torsors
  5. 5 Motivic cohomology of quotients up to isogeny
  6. 6 Generalizations
  7. A Existence of maximal tori over Prüfer domains

##  1 Introduction

Let $G$ be a reductive group over $\mathbb{F}_{q}$, a finite field of
characteristic $p>0$, and $\varphi\colon G\rightarrow G$ the $q$-Frobenius.
Then $G$ acts on itself via $\varphi$-conjugation, i.e. $(g,h)\mapsto
gh\varphi(g)^{-1}$. The stabilizer of the neutral element is denoted by
$G^{F}$. If $\overline{\mathbb{F}}_{q}$ denotes an algebraic closure of
$\mathbb{F}_{q}$, then $G^{F}(\overline{\mathbb{F}}_{q})=G(\mathbb{F}_{q})$ is
a finite group of Lie-type. The representation of finite groups of Lie type
over fields of characteristic $0$ was studied by Deligne and Lusztig (cf.
[DL76]). In their article they construct representations of
$G(\mathbb{F}_{q})$ by an action on the $\ell$-adic cohomology of certain
varieties, for $\ell\neq p$. Roughly, the varieties in question are
constructed by intersection of Bruhat strata and graph of Frobenius.

Let us fix a Borel $B$ of $G$.111Any reductive group over a finite field is
quasi-split and thus admits a Borel. The Bruhat strata of $\left[G/B\right]$
are induced by the Bruhat decomposition of $G$ via pullback along
$G/B\rightarrow G/B\times^{G}G/B\cong\left[B\backslash G/B\right]$. In this
article, we want to analyze the cohomological connection between
$\textup{B}G^{F}$ and $\left[B\backslash G/B\right]$, i.e. study how their
motivic categories are related.

The derived category of $\ell$-adic sheaves
$D(\textup{B}G^{F},\mathbb{Q}_{\ell})$, for $\ell\neq p$, encodes information
about the action of $G^{F}$ on $\ell$-adic cohomology. One can show that
$\textup{B}G^{F}\cong\left[G/_{\varphi}G\right]$, where $G$ acts on itself via
$\varphi$-conjugation. The restriction of the $\varphi$-conjugation to $B$
yields an adjunction

${D(\left[G/_{\varphi}G\right],\mathbb{Q}_{\ell})}$${D(\left[G/_{\varphi}B\right],\mathbb{Q}_{\ell}).}$

On the other hand there is an adjunction

${D(\left[G/_{\varphi}B\right],\mathbb{Q}_{\ell})}$${D_{B}(G/B,\mathbb{Q}_{\ell}),}$

which is induced via the graph of $\varphi$ (here the right hand side denotes
the derived category of $B$-equivariant $\mathbb{Q}_{\ell}$-modules on the
flag variety). Thus, it seems natural that the study of these two adjunctions
should lead to information about the geometric representation theory of
$G^{F}$ and connection to the classical theory of Deligne-Lusztig.

Instead of rewriting the theory of Deligne-Lusztig in the derived setting, we
want to understand the adjunctions above in the motivic setting with rational
coefficients. The idea is that first after $\ell$-adic realization, we get the
classical situation back and further this could lead to information about the
$\mathbb{Q}$-representations of $G^{F}$, as we are naturally working with
rational coefficients.

### Motives and connection to representation theory

Motives were famously envisioned by Grothendieck to capture similar behavior
of cohomology theories in an abelian category. The construction of such a
category is not an easy task and has been studied for many years. The main
approach is to define a derived category of motives with the hope to find a
$t$-structure on it, so that the heart of this $t$-structure defines the
abelian category of motives. To capture functorial behavior on cohomology
theories one demands a full six functor formalism for the derived category of
motives. There are several versions of the derived category of motives which
agree under certain assumptions. One version was constructed by Cisinki and
Déglise in the case of rational coefficients, which we denote by
$\operatorname{DM}$ (cf. [CD19]). They show that the assignment
$X\mapsto\operatorname{DM}(X)$ from smooth $k$-schemes indeed admits a six
functor formalism $(\otimes\dashv\underline{\operatorname{Hom}},f^{*}\dashv
f_{*},f_{!}\dashv f^{!})$ and agrees with the classical construction of Morel.
In particular, they show that motivic cohomology, i.e.
$\operatorname{Hom}_{\operatorname{DM}(X)}(1_{X},1_{X}(n)[m])$ agrees with
Bloch’s higher Chow groups $A^{n}(X,2n-m)_{\mathbb{Q}}$. With the help of the
$6$-functor formalism, we can define the motive of an $k$-scheme $\pi\colon
X\rightarrow\operatorname{Spec}(k)$ resp. the global sections via

$M_{k}(X)\coloneqq\pi_{!}\pi^{!}1_{Y}\quad\textup{resp.}\quad
R\Gamma_{k}(X,\mathbb{Q})\coloneqq\pi_{*}\pi^{*}1_{k}$

computing motivic cohomology resp. cohomology. The existence of a
$t$-structure is a more delicate problem and in general not known. Levine
shows that for a particular class of schemes $X$, e.g. finite fields or affine
spaces over finite fields, a $t$-structure exists on the full triangulated
subcategory of Tate motives
$\operatorname{DTM}(X)\subseteq\operatorname{DM}(X)$ generated by the
$1_{X}(n)$ for $n\in\mathbb{Z}$ (cf. [Lev93]). Further, using weight
structures one can see that $\operatorname{DTM}(\mathbb{F}_{p})$ is equivalent
to the bounded derived category of $\mathbb{Q}$-vector spaces. We also have
realization functors
$\textup{real}_{\ell}\colon\operatorname{DTM}(X)\rightarrow
D_{\operatorname{\acute{e}t}}(X,\mathbb{Q}_{\ell})$ for $\ell\neq p$ that is
conservative and $t$-exact for the perverse $t$-structure on
$D_{\operatorname{\acute{e}t}}(X,\mathbb{Q}_{\ell})$. Let us remark that if
for a morphism $f\colon X\rightarrow Y$ we have
$f_{*}1_{X}\in\operatorname{DTM}(Y)$, then this automatically induces an
adjunction of $\operatorname{DTM}(X)$ and $\operatorname{DTM}(Y)$. In
particular, the adjunction $f^{*}\dashv f_{*}$ restricts to Tate motives. We
call morphisms with such a property Tate.

Let us now explain the relation between Tate motives and geometric
representation theory. For simplicity, let us assume that $G/\mathbb{F}_{q}$
is a split reductive group. Let $T$ be a split maximal torus inside a Borel
$B$ of $G$. In [SW18], Soergel and Wendt show that for schemes stratified by
affine spaces, such as the flag variety $G/B$, one can define a subcategory of
$\operatorname{DM}$ called stratified Tate motives that admits a
$t$-structure. This $t$-structure is glued from the $t$-structure on the
strata. For $G/B$ with the Bruhat stratification, we will denote the category
of stratified Tate motives with $\operatorname{DTM}_{(B)}(G/B)$. Soergel and
Wendt show that $\operatorname{DTM}_{(B)}(G/B)$ is equivalent to the bounded
derived category $D^{b}({\mathcal{O}}^{\mathbb{Z},ev})$ of graded
${\mathcal{O}}\coloneqq H^{*}(G/B)$-modules concentrated in even degrees. To
connect this to representations, we have to go further and endow motivic
cohomology with group actions.

For this, we need to define equivariant motives. To make sense of the
following construction, we need to work in the setting of $\infty$-categories.
The idea is to define $\operatorname{DM}({\mathcal{X}})$ for an Artin stack
${\mathcal{X}}$ via gluing along the atlas. As we essentially have to glue the
derived category this only makes sense in the $\infty$-categorical framework.
Then this gluing can be defined via right Kan-extension from schemes to Artin
stacks (cf. [RS20]). As one would expect motivic cohomology of a quotient
stack $[X/G]$, where $X$ is a smooth $k$-scheme and $G$ is a linear algebraic
group, yields the equivariant Chow groups $A^{n}_{G}(X,2n-m)$ of Edidin and
Graham (cf. [RS20]). In this way, one can also extend the stratified Tate
motives to Artin stacks. In the case of the flag variety, Soergel, Virk and
Wendt show that $\operatorname{DTM}(\left[B\backslash G/B\right])$ is
equivalent to the bounded derived category of bi-Soergel-modules (cf.
[SVW18]). Further, they show that applying $K_{0}$ yields an isomorphism to
the Iwahori-Hecke algebra and Verdier duality yields the Kazhdan-Lusztig
involution. In particular, the stratified Tate motives of $\left[B\backslash
G/B\right]$ with the weight structure and $6$-functor formalism carries
information about the $\ell$-adic geometric representations of $G$.

### Connection between finite groups of Lie type and their associated flag
variety

As we have seen above the geometric representation theory of the flag variety
is linked to stratified Tate motives. This particular connection uses that the
flag variety is stratified by affine spaces and that Tate motives behave
nicely under this stratification. We expect that the geometric representation
theory of $G^{F}$ is linked to Tate motives on $\textup{B}G^{F}$, the
classifying space of $G^{F}$. In this way, we want to relate representation
theory of $G^{F}$ with the representation theory of the associated flag
variety.

We will do this by linking the Tate motives of $\textup{B}G^{F}$ to Tate
motives on $\left[B\backslash G/B\right]$ in the following way. The stack
$\textup{B}G^{F}$ is equivalent to $\left[G/_{\varphi}G\right]$, where $G$
acts on itself via $\varphi$-conjugation. Let us fix a maximal torus
$T\subseteq B$. Let $T$ act on $G$ also via $\varphi$-conjugation. We can
embed $T$ into $T\times T$ via $t\mapsto(t,\varphi(t))$. If we now let
$T\times T$ act on $G$ via $(t,t^{\prime},g)\mapsto tgt^{\prime-1}$, we get a
zigzag of Artin stacks

(1)
${\left[G/_{\varphi}G\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$

It is a fact that $\operatorname{DM}(\left[G/T\times
T\right])\simeq\operatorname{DM}(\left[B\backslash G/B\right])$. Thus, on the
level of motivic categories, this zigzag yields adjunctions between
$\operatorname{DM}(\textup{B}G^{F})$ and $\operatorname{DM}(\left[B\backslash
G/B\right])$. Now we can formulate the leading question of this article:

($\ast$) Do these adjunctions preserve Tate motives?

The answer to this question is positive and yields a first point to access
motivic representation theory of $G^{F}$ via the motivic geometric
representation theory of $G$.

### Equivariant motivic homotopy theory of reductive groups

From now on let $\tilde{B}$ be a an excellent noetherian scheme of dimension
$\leq 2$ and $S$ a regular connected $\tilde{B}$-scheme of finite type. We
will first work with reductive $S$-group schemes $H$ that admit split maximal
tori $Z$. This is not a classical notion and any such reductive group scheme
may not be split or even quasi-split (in sense of [SGA3] or [Con14]). But in
this case, one has that the Weyl-group scheme $N_{H}(Z)/Z$ is represented by a
constant group scheme associated to a finite constant group $W_{H}$, which we
will call the Weyl group of $Z$ in $H$ (cf. Section 3.1).

As $S$ is regular noetherian, any maximal torus splits after passage to a
Galois cover and we deduce the answer to our leading question ($\ast$ ‣ 1)
from the split case.

From now on let $G$ be a reductive $S$-group scheme that admits a maximal
torus $T$. Our main question is about the behavior of Tate motives under the
induced maps on $\operatorname{DM}$ corresponding to the zigzag (1). We will
work in a more general setting and look at $a$ and $b$ separately.

#### Equivariant motives and passage to Tori

The morphism $a$ resembles the motivic version of a more classical problem on
classical problem on Chow groups. Let $X$ be an $S$-scheme locally of finite
type with $G$-action, what is the relation between $A^{\bullet}_{G}(X)$ and
$A^{\bullet}_{T}(X)$? In [EG98] Edidin and Graham answer this question for
rational Chow groups in the case where $S=\operatorname{Spec}(k)$ is the
spectrum of a field and $T$ is split, i.e.
$A^{\bullet}_{G}(X)_{\mathbb{Q}}\cong A^{\bullet}_{T}(X)^{W}_{\mathbb{Q}}$,
where $W$ denotes the Weyl group of $T$ in $G$.

This isomorphism is just a shadow of an equivalence that can be seen
motivically.

######  Theorem 1 (3.14).

Let $G$ be a reductive $S$-group scheme with split maximal torus $T$ and Weyl
group $W$. Assume $G$ acts on an $S$-scheme $X$ locally of finite type. Then
the natural map

$\left[X/T\right]\rightarrow\left[X/G\right]$

is Tate.

Further, we have

$R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}.$

The idea of the proof of Theorem 1 is to factorize
$\left[X/T\right]\rightarrow\left[X/G\right]$ into

$\left[X/T\right]\rightarrow\left[X/N_{G}(T)\right]\rightarrow\left[X/G\right].$

Then the first map of the factorization is naturally a $W$-torsor and the
second map a $G/N_{G}(T)$-bundle. For torsors under finite groups étale
descent relates motives via $W$-invariants. For $G/N_{G}(T)$-bundles it
suffices to see that $R\Gamma_{S}(G/N_{G}(T),\mathbb{Q})$ is trivial. We will
prove this by reducing the triviality to the equivalence of the map
$K_{0}(S)_{\mathbb{Q}}\rightarrow K_{0}(G/B)^{W}_{\mathbb{Q}}$, which after
comparison with rational Chow theory is classical.

Applying Theorem 1 to motivic cohomology in the case, where
$S=\operatorname{Spec}(k)$, we can extend the classical result to higher Chow
groups even in the non-split case, generalizing the analogous result from
[Kri13] from split reductive groups to non-split groups. The idea is that any
maximal torus splits, after passage to a finite Galois extension. Then we
reduce to the split case, using that motives of torsors under finite groups
are related by taking invariants.

######  Corollary 2 (3.15).

Let $k$ be a field an let $G$ be a reductive $k$-group scheme with maximal
torus $T$ and absolute Weyl group $W$. Assume $G$ acts on a smooth finite type
$k$-scheme $X$. Then for all $n,m\in\mathbb{Z}$, we have

$A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}.$

The same argumentation as before works also over general base and arbitrary
reductive group schemes, to show that
$\left[X/T\right]\rightarrow\left[X/G\right]$ is Tate. This is because over a
connected noetherian normal base any torus splits after passage to a finite
Galois extension.

######  Corollary 3 (3.16).

Let $G$ be a reductive $S$-group scheme that admits a maximal torus $T$.
Assume $G$ acts on an $S$-scheme $X$ locally of finite type. Then the natural
map

$\left[X/T\right]\rightarrow\left[X/G\right]$

is Tate.

#### Motives of $T$-torsors

We will now analyze morphism $b$. For this let $\varphi\colon G\rightarrow G$
be an isogeny that fixes $T$. Consider the embedding $T\hookrightarrow T\times
T$ given by the graph of $\varphi$. The quotient $T\times T/T$ under this
embedding is isomorphic to $T$. In particular, this isomorphism gives the map

$b\colon\left[G/_{\varphi}T\right]\rightarrow\left[G/T\times T\right]$

the structure of a $T$-torsor. So, to understand the motivic behavior of $b$
it is enough to understand motives of $T$-torsors.

Thus, let $X\rightarrow Y$ be a morphism of Artin stacks that is a $T$-torsor.
Classically, Chow groups in this setting can be computed rather easily. For
each character $\chi\in T\rightarrow\mathbb{G}_{\textup{m}}$ we get a
$1$-dimensional representation $\kappa(\chi)$ of $T$. This yields a line
bundle $L_{\chi}\coloneqq X\times^{T}\kappa(\chi)$ on $Y$. Multiplication with
the first Chern class of $L_{\chi}$ yields an action of the character group
$\hat{T}$ of $T$ on $A^{\bullet}(Y)$. If $T$ is split, the morphism $b$ yields

(2) $A^{\bullet}_{T}(G)\cong A^{\bullet}_{T\times
T}(G)/\hat{T}A^{\bullet}_{T\times T}(G)$

(cf. [Tot14]).

Again, this is just a shadow of computations for oriented cohomology theories.

The idea is the following. If $T$ is split, we have
$T\cong\mathbb{G}_{\textup{m}}^{r}$ for some $r\in\mathbb{N}$. By applying
successive $\mathbb{G}_{\textup{m}}$-quotients, we can write

$X\rightarrow X_{1}\rightarrow X_{2}\rightarrow\dots\rightarrow X_{r}\cong Y,$

where $X_{i}\coloneqq\left[X/\mathbb{G}_{\textup{m}}^{i}\right]$. Each of the
maps $X_{i-1}\rightarrow X_{i}$ is a $\mathbb{G}_{\textup{m}}$-torsor. So we
may reduce to the case, where $T=\mathbb{G}_{\textup{m}}$. In this case, we
can follow [HPL21] and assign the line bundle ${\mathcal{L}}\coloneqq
X\times^{\mathbb{G}_{\textup{m}}}\mathbb{A}^{1}$ over $Y$. Multiplication with
the first Chern class of ${\mathcal{L}}$ yields a fibre sequence

$M_{k}(X)\rightarrow M_{k}(Y)\rightarrow M_{k}(Y)(1)[2].$

Applying this construction in the split case to motivic cohomology yields a
generalization of (2).

######  Corollary 4 (4.6).

Assume that $S=\operatorname{Spec}(k)$ and $T$ is a split torus. Let
$X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks over $S$. Then

$A^{\bullet}(X)_{\mathbb{Q}}\cong
A^{\bullet}(Y)_{\mathbb{Q}}/\hat{T}A^{\bullet}(Y)_{\mathbb{Q}}.$

If $X$ and $Y$ are represented by quotients of qcqs smooth schemes by
diagonalizable group schemes, for example for $b$ as above, we can actually
replace the Chow ring with equivariant $K_{0}$. More generally, the analogous
statement holds for any oriented cohomology theory that is $m$-connective (cf.
Remark 4.7).

Further, combining the above with the fact that any torus over $S$ splits
after passage to a finite Galois extension, we also see the following.

######  Proposition 5 (4.3).

Let $f\colon X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks over $S$.
Then $f$ is a Tate map.

#### Applications to quotients up to conjugation by isogeny

We will summarize our results above to see that the maps relating
$\textup{B}G^{F}$ and $\left[G/T\times T\right]$ are Tate. But we haven’t
particularly used the fact that we are interested in Frobenius-conjugation but
rather conjugation up to isogeny. Thus, we can work in a more general setting,
that we describe in the following.

Let $P$ resp. $Q$ be parabolics inside $G$ with Levi-components $L$ resp. $M$.
Let $\varphi\colon L\rightarrow M$ be an isogeny. Then $L$ acts on $G$ via
$(l,g)\mapsto lg\varphi(l)^{-1}$. Let $T$ be a maximal torus of $G$ contained
in $L$. Fix a $g_{0}\in G(S)$ such that $g_{0}\varphi(T)g_{0}^{-1}=T$ and
denote by $\widetilde{\varphi}$ the composition of $\varphi$ and
$g_{0}$-conjugation. We can embed $T$ into $T\times T$ via
$t\mapsto(t,\widetilde{\varphi}(t))$.

######  Theorem 6 (5.1).

In the setting above, we have the following zigzag of Tate maps

${\left[G/_{\varphi}L\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$

Further, if $S=\operatorname{Spec}(k)$ and $T$ is split, we can compute the
Chow ring of $\left[G/_{\varphi}L\right]$ as

$A^{\bullet}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong\left(A^{\bullet}_{T}(G/T)_{\mathbb{Q}}/\hat{T}A^{\bullet}_{T}(G/T)_{\mathbb{Q}}\right)^{W_{L}},$

where $W_{L}$ denotes the Weyl group of $T$ in $L$.

Using results about equivariant $K$-theory by Uma and Krishna and our results
on the cohomology theory of $T$-torsors, we can extend the situation above to
equivariant $K$-theory and generalize a result by Brokemper.

######  Corollary 7 (5.4).

In the setting above assume that $S=\operatorname{Spec}(k)$, $G$ is split with
respect to $T$ and that the derived group of $G$ is simply connected. Then we
have

$K_{0}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
R(T)_{\mathbb{Q}}^{W_{L}}/(f-\tilde{\varphi}f\mid f\in
R(T)_{\mathbb{Q}}^{W}),$

where $W_{L}$ denotes the Weyl group of $T$ in $L$ and $W$ as before denotes
the Weyl group of $T$ in $G$.

######  Example 8.

Let us give two interesting examples, where Theorem 6 can be used.

  1. 1.

Let $k=\mathbb{F}_{q}$ be a finite field with $q$-elements and assume
$S=\operatorname{Spec}(k)$. If we set $L=G$ and $\varphi$ the $q$-Frobenius,
then we precisely get the situation of the beginning back. In particular, we
see that there is an adjunction between $\operatorname{DTM}(\textup{B}G^{F})$
and $\operatorname{DTM}(\left[B\backslash G/B\right])$.

  2. 2.

Let $k$ be a finite field of characteristic $p>0$ and assume
$S=\operatorname{Spec}(k)$. Another interesting example is the stack of
$G$-zips of Pink-Wedhorn-Ziegler (cf. Example 5.6). In particular, we can
recover the computations of Brokemper [Bro18] for Chow groups (up to some
computations of loc.cit.).

This answers our main question ($\ast$ ‣ 1) positively.

######  Remark 9.

If we work with reductive $S$-group schemes that admit split maximal tori,
then we will see in our main text, that we can drop the connectedness
hypothesis on the results above.

#### Structure of this article

We start this article by recalling properties of the $\infty$-category of
motives and how to extend this to arbitrary locally of finite type Artin
stacks. After defining the necessary notions for this article, we quickly
recollect some computational aspects.

Afterwards, we start to focus on motives on schemes with group action. First,
we explain how to achieve a group action on motives and how torsors under
finite groups of Artin stacks have a particular behavior. Then we concentrate
on the case $T\subseteq G$, a split maximal torus inside a reductive group.
Namely, we show that the relation between $T$-equivariant Chow groups and
$G$-equivariant Chow groups extend to the motivic case.

Next, we show that $T$-torsors of Artin stacks are Tate and explicitly compute
motivic cohomology implying the classical case of Chow groups.

In the end, we focus on reductive groups with conjugation up to isogeny. We
use our results from before to get the desired adjunction of Tate motives with
the $T$-equivariant flag variety. We end the paper with ideas for
generalization, that we want to address in the future.

#### Setup

  1. $\bullet$

Throughout, we fix a noetherian excellent scheme $\tilde{B}$ of dimension at
most $2$ and a regular scheme $S$ of finite type over $\tilde{B}$.

  2. $\bullet$

An Artin stack is an algebraic stack in the sense of [Sta22]. When we write
“stack”, we always mean “Artin stack”. Every Artin stack (and hence scheme)
will be locally of finite type over $S$ and any morphism of Artin stacks will
be an $S$-morphism.

  3. $\bullet$

Throughout, we will work in the setting of $\infty$-categories and freely use
the language of $\infty$-categories. In particular, presheaves will be always
considered as presheaves with values in $\infty$-groupoids.

  4. $\bullet$

Throughout $\operatorname{DM}$ denotes the Beilinson motives with coefficients
in $\mathbb{Q}$.

  5. $\bullet$

We will work with reductive group schemes over $S$ (cf. [SGA3]), i.e.
$S$-affine smooth group schemes such that the geometric fibers are connected
reductive groups.

  6. $\bullet$

We say that a reductive $S$-group scheme $G$ is quasi-split if it so in the
sense of [SGA3, Exp. 24 §3.9].

  7. $\bullet$

Let $X$ be an $S$-scheme and $G$ an $S$-group scheme acting on $X$. Then we
denote by $\left[X/G\right]$ the associated quotient stack in the étale
topology.

  8. $\bullet$

For a $S$-group scheme $G$, we denote its (étale) classifying stack by
$\textup{B}G\coloneqq\left[S/G\right]$, where $G$ acts trivially on $S$.

### Acknowledgement

I would like to thank Torsten Wedhorn for multiple discussions and comments on
the earlier version, as well as his communication of Appendix A and the idea
to use reductive group schemes with split maximal tori, generalizing the
earlier versions. Further, I would like to thank Paul Ziegler, who
communicated this project and shared his thoughts with me. Finally, I would
like to thank Rizacan Ciloguli, Arnaud Éteve, Tim Holzschuh, Marc Hoyois,
Adeel Khan, Timo Richarz, Jakob Scholbach, Fabio Tanania, Thibaud van den Hove
for fruitful discussions and feedback.

This project was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - project number 524431573, the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and
Arithmetic of Uniformized Structures, project number 444845124 and by the
LOEWE grant ‘Uniformized Structures in Algebra and Geometry’.

##  2 Rational equivariant motivic homotopy theory

In this section, we want to recall some properties of the category of
(rational) motives and how to extend this to Artin stacks locally of finite
type. We expect that most readers are familiar with the notion of motives and
the $6$-functor formalism and refer to [RS20, Syn. 2.1.1] for an overview of
the properties of the $6$-functor formalism.

Nevertheless, to prevent confusion, let us quickly recall some common notation
and remarks.

######  Remark 2.1 ([RS20], [CD19]).

In the following any scheme and any morphism will be considered in the
category of finite type $S$-schemes, $\textup{Sch}_{S}^{\textup{ft}}$.

  1. (i)

For any $S$-scheme $X$, $\operatorname{DM}(X)$ is a stable, presentable,
closed symmetric monoidal $\infty$-category. The $\otimes$-unit will be
denoted by $1_{X}$. It has all limits and colimits.

  2. (ii)

The assignment $X\mapsto\operatorname{DM}(X)$ can be upgraded to a presheaf of
symmetric monoidal $\infty$-categories

$\operatorname{DM}^{*}\colon\textup{Sch}_{S}^{\textup{ft}}\rightarrow\textup{Cat}_{\infty}^{\otimes},\
X\mapsto\operatorname{DM}(X),\ f\mapsto f^{*}.$

For any morphism of schemes $f\colon X\rightarrow Y$, there is an adjunction

${f^{*}\colon\operatorname{DM}(Y)}$${\operatorname{DM}(X)\colon f_{*}.}$

  3. (iii)

If $f$ is smooth, then $f^{*}$ has a left adjoint, denoted $f_{\sharp}$.

  4. (iv)

The assignment $X\mapsto\operatorname{DM}(X)$ can be upgraded to a presheaf of
$\infty$-categories

$\operatorname{DM}^{!}:(\textup{Sch}_{S}^{\textup{ft}})^{\operatorname{op}}\rightarrow\textup{Cat}_{\infty},\
X\mapsto\operatorname{DM}(X),f\mapsto f^{!}.$

For each $f$, there is an adjunction

$f_{!}:\operatorname{DM}(X)\rightleftarrows\operatorname{DM}(Y):f^{!}.$

For any factorization $f=p\circ j$ with $j$ an open immersion and $p$ a proper
map, there is a natural equivalence $f_{!}\cong p_{*}j_{\sharp}$.

  5. (v)

Both functors $\operatorname{DM}^{*}$ and $\operatorname{DM}^{!}$ are sheaves
for the $h$-topology (cf. [RS20, Thm. 2.1.13])

  6. (vi)

For the projection $p:\mathbb{G}_{\textup{m},S}\times_{S}X\rightarrow X$, and
any $M\in\operatorname{DM}(X)$, the map induced by the counit
$p_{\sharp}p^{*}M[-1]\rightarrow M[-1]$ in $\operatorname{DM}(X)$ is a split
monomorphism. The complementary summand is denoted by $M(1)$. The functor
$M\mapsto M(1)$ is an equivalence with inverse denoted by $M\mapsto M(-1)$.
For any integer $n$ the $n$-fold composition is denoted by $M\mapsto M(n)$ and
in the future, we will abbreviate $\langle n\rangle\coloneqq(n)[2n]$.

Let $\underline{X}$ be a prestack222This notion is from [RS20]., i.e. presheaf
of anima on the category of rings. There are several approaches to the
$\infty$-category $\operatorname{DM}(\underline{X})$. If $\underline{X}$ is an
Artin stack over a field $k$ Hoskins-Lehalleur use a construction similar to
equivariant Chow groups. One resolves $\underline{X}$ by open substacks
$(\underline{X}_{i})$ such that on each $\underline{X}_{i}$ there is a vector
bundle $V_{i}$ together with an open $U_{i}$ with a free $G$-action such that
the codimension of $V_{i}\setminus U_{i}$ tends towards infinity (cf.
[HPL21]). This idea was further generalized to arbitrary cohomology theories
by Khan-Ravi by a construction they call lisse extension (cf. [KR21]). This
construction was already used for the motive of classifying stacks
$\textup{B}G$ by Morel-Voevodsky (cf. [MV99, §4]). Totaro then gave an
explicit computation of the motive of $\textup{B}\mathbb{G}_{\textup{m}}$ over
a field (cf. [Tot16] and Example 2.8).

Alternatively to the construction of Hoskins-Lehalleur resp. Khan-Ravi,
Richarz-Scholbach give a construction via certain left and right Kan extension
(cf. [RS20]). Their approach is based on gluing the motivic structure on
Beilinsion motives to arbitrary prestacks. Indeed, $\operatorname{DM}(-)$
satisfies $h$-descent, so it is rather formal to extend the six functor
formalism to Artin stacks locally of finite type, we will use this approach.
One should note that this was also discussed in [Kha19], to extend the
$6$-functor formalism to higher Artin stacks.

For computations of the underlying motives it seems to be better to work with
the definition of Hoskins-Lehalleur resp. Morel-Voevodsky. Let
$f\colon{\mathfrak{X}}\rightarrow\operatorname{Spec}(k)$ be a smooth Artin
stack and let $M({\mathfrak{X}})$ denote the $k$-linear motive of
${\mathfrak{X}}$ defined in [HPL21]. This defines an object in
$\operatorname{DM}^{*}(\operatorname{Spec}(k))$. Let $1_{k}$ denote the unit
in $\operatorname{DM}^{*}(\operatorname{Spec}(k))$. We will see in Corollary
2.7 that if $f$ is smooth, we have $M({\mathfrak{X}})\simeq
f_{\sharp}f^{*}1_{k}$. In particular, if we use the approach of [RS20] and
define a motive of a prestack as the $\sharp$-push/$*$-pull of the unit, we
see that the notion of motives on Artin stacks as in loc.cit. agrees with the
classical ones.

To define $\operatorname{DM}$ for prestacks, we will follow [RS20]. For the
following definition let us fix a regular cardinal $\kappa$. Let
$\textup{Aff}_{S}^{\textup{ft}}$ denote the (Nerve of the) category of affine
schemes of finite type over $S$. We let $\textup{Aff}_{S}^{\kappa}$ denote the
$\kappa$-pro-completion of $\textup{Aff}_{S}^{\textup{ft}}$. Further,
$\textup{DGCat}_{\textup{cont}}$ denotes the $\infty$-category of presentable
stable $\mathbb{Q}$-linear dg-$\infty$-categories with colimit preserving
functors.

######  Definition 2.2 ([RS20]).

Let $y\colon\textup{Aff}_{S}^{\kappa}\hookrightarrow
P(\textup{Aff}_{S}^{\kappa})$ be the Yoneda embedding. We define the functor

$\operatorname{DM}_{S}\colon
P(\textup{Aff}_{S})^{\operatorname{op}}\rightarrow\textup{DGCat}_{\textup{cont}}$

via left Kan extension along the inclusion
$\textup{Aff}_{S}^{\textup{ft}}\subseteq\textup{Aff}_{S}^{\kappa}$ and right
Kan extension along $\textup{Aff}_{S}^{\kappa}P(\textup{Aff}_{S}^{\kappa})$,
where all the functors are given via $!$-pullback. For a prestack
$\underline{X}\in P(\textup{Aff}_{S}^{\kappa})$, we define the
$\infty$-category of $S$-linear motives (with rational coefficients) of
$\underline{X}$ as $\operatorname{DM}_{S}(\underline{X})$.

As noted in [RS20], for applications we can take $\kappa$ to be large enough,
so that all affine schemes of interest are inside $\textup{Aff}_{S}^{\kappa}$.
Thus, for once and for all we may fix a regular cardinal $\kappa$ and drop it
from the notation.

Khan showed in [Kha19, Thm. A.5] that this method of extending the theory of
motives to (derived) Artin stacks, does not loose the $6$-functor formalism.
One way to see this, is that we can use the DESCENT program in [LZ12], since
the Beilinsion motives satisfy étale descent, in our context. As mentioned in
[Kha19], this is equivalent to the construction of [RS20].

######  Theorem 2.3.

Let $\widetilde{\operatorname{DM}}$ be the restriction of $\operatorname{DM}$
to Artin stacks locally of finite type over $S$. Then
$\widetilde{\operatorname{DM}}$ is compatible with the $6$-functor formalism
in the sense of [RS20, Syn. 2.1.1].

###### Proof.

The proof is the same as [Kha19, Thm A.5]333As mentioned in op.cit., the
method of extending the $6$-functor formalism works with any motivic category
that satisfies étale descent. and follows from the results of [RS20]. ∎

######  Definition 2.4 ([RS20]).

Let ${\mathfrak{X}}$ be an Artin $S$-stack locally of finite type with
structure morphism $f\colon{\mathfrak{X}}\rightarrow S$. Then we define the
(rational) $S$-linear motive of ${\mathfrak{X}}$ as
$M_{S}({\mathfrak{X}})\coloneqq f_{!}f^{!}1_{S}$. If
$S=\operatorname{Spec}(A)$ is affine, we write $M_{A}({\mathfrak{X}})$.

We further define the global sections of ${\mathfrak{X}}$ over $S$ to be
$R\Gamma_{S}({\mathfrak{X}},\mathbb{Q})\coloneqq f_{*}f^{*}1_{S}$.

######  Remark 2.5.

Let $f\colon{\mathfrak{X}}\rightarrow S$ be an Artin stack locally of finite
type over $S$. If $f$ is smooth, then relative purity implies that
$f_{\sharp}f^{*}\simeq f_{!}f^{!}$ and in particular, we see with

$\operatorname{Hom}_{\operatorname{DM}(S)}(M_{S}({\mathfrak{X}}),1(n)[m])\simeq\operatorname{Hom}_{\operatorname{DM}(S)}(1(-n)[-m],R\Gamma_{S}({\mathfrak{X}},\mathbb{Q}))$

that $M_{S}({\mathfrak{X}})$ computes motivic homology and
$R\Gamma_{S}({\mathfrak{X}},\mathbb{Q})$ motivic cohomology.

######  Notation 2.6.

Let $G$ be an $S$-group scheme locally of finite type acting on an $S$-scheme
$X$ locally of finite type, via a morphism $a$. For the quotient stack
$\left[X/G\right]$, we can define a simplicial object in $S$-schemes locally
of finite type, via its Bar-resolution

${\dots}$${G\times_{S}X}$${X}$$\scriptstyle{p}$$\scriptstyle{a}$

We denote the corresponding simplicial functor with
$\textup{Bar}^{\bullet}(X,G)$.

There is also an alternative way to define motives of algebraic stacks via the
Bar-resolution. For each $n\geq 0$ let $\mathbb{Q}(\textup{Bar}^{n}(X,G))$ be
the constant étale sheaf with coefficients in $\mathbb{Q}$ associated to
$\textup{Bar}^{n}(X,G)$. This yields a simplicial object in étale sheaves of
$S$-schemes locally of finite type with rational coefficients. The complex
associated to this simplicial object induces a motive
$M_{S}(\textup{Bar}^{\bullet}(X,G))$ in $\operatorname{DM}(S)$. For
$S=\operatorname{Spec}(k)$ the spectrum of a field, Hoskins-Lehalleur explain
in [HPL21] that this definition is equivalent to their definition of a motive
of an Artin stack. The naturally arising question is if
$M_{S}(\textup{Bar}^{\bullet}(X,G))$ is equivalent to
$M_{S}(\left[X/G\right])$ as defined in Definition 2.4. If $\left[X/G\right]$
is representable by a smooth scheme, then the answer is positive and follows
by cohomological descent for $\operatorname{DM}$ with respect to the
$h$-topology (cf. [CD19, Thm. 14.3.4, Prop. 5.2.10]). Thus, the answer stays
positive for smooth Artin stacks, as $\operatorname{DM}$ satisfies $h$-descent
by gluing (cf. [RS20, Thm. 2.2.16]).

######  Corollary 2.7.

Let $k$ be a field and $S=\operatorname{Spec}(k)$. Let $X$ be a smooth
$S$-scheme and $G$ be a smooth $S$-group scheme acting on $X$ with structure
map $f\colon\left[X/G\right]\rightarrow S$. Then $M_{S}(\left[X/G\right])$ is
equivalent to $M_{S}(\textup{Bar}^{\bullet}(X,G))$.

###### Proof.

This follows from [HPL21, Prop. A.7] and the discussion above. ∎

We can use Corollary 2.7 to compute the motive of $B\mathbb{G}_{\textup{m}}$
as in [Tot16].

######  Example 2.8.

Let $k$ be a field. Further, let $\mathbb{G}_{\textup{m},k}$ act trivially on
$\operatorname{Spec}(k)$. Then

$M_{k}(\textup{B}\mathbb{G}_{\textup{m},k})\simeq\mathop{\textup{colim}}_{i\in\mathbb{N}}M_{k}(\mathbb{P}^{i}_{k})\simeq\bigoplus_{i\geq
0}1_{k}\langle i\rangle.$

######  Remark 2.9.

In the following we want to understand the Gysin sequence for algebraic
stacks. Let us quickly recall it in the scheme case.

Let $i\colon Z\hookrightarrow X$ be a closed immersion of $S$-schemes of pure
codimension $n$ with open complement $U$. Let us assume that $Z$ and $X$ are
smooth over $S$. In particular, we see that $i$ is equivalently a regular
closed immersion of codimension $n$. Then there exists a fibre sequence of the
form

$M_{S}(U)\rightarrow M_{S}(X)\rightarrow M_{S}(Z)\langle n\rangle$

(cf. [CD19, 11.3.4]).

We are going to replace $S$ by a smooth Artin stack ${\mathfrak{Y}}$ over $S$
and $X$ by a smooth Artin stack ${\mathfrak{X}}$ over ${\mathfrak{Y}}$. For
this let us also recall the notion of a (regular) closed immersion of a
certain codimension for Artin stacks.

Let $\iota\colon{\mathfrak{Z}}\hookrightarrow{\mathfrak{X}}$ be a closed
immersion of locally noetherian Artin stacks. Let $X\rightarrow{\mathfrak{X}}$
be a smooth atlas. Then $\iota$ is representable and we define the codimension
of ${\mathfrak{Z}}$ as the codimension of
${\mathfrak{Z}}\times_{{\mathfrak{X}}}X$ in $X$ (cf. [Oss15, §6]). We can also
define the notion of a regular immersion in that way (cf. [Sta22, 06FM]) and
the notion of its codimension. In particular, a closed immersion of smooth
Artin $S$-stacks ${\mathfrak{Z}}\hookrightarrow{\mathfrak{X}}$ is
automatically regularly immersed and the codimension of the regular immersion
agrees with the codimension as a closed substack.

######  Lemma 2.10 (The Gysin sequence).

Let $f\colon{\mathfrak{X}}\rightarrow{\mathfrak{Y}}$ be a smooth schematic
morphism of smooth Artin $S$-stacks. Further let
$i\colon{\mathfrak{Z}}\hookrightarrow{\mathfrak{X}}$ be a closed immersion of
(pure) codimension $n$ such that ${\mathfrak{Z}}$ is smooth over
${\mathfrak{Y}}$ with open complement
$j\colon{\mathfrak{U}}\rightarrow{\mathfrak{X}}$. Further, let us denote
$f_{0}\coloneqq f\circ j$ and $\bar{f}\coloneqq f\circ i$. Then there exists
the following fibre sequence

$f_{0!}f_{0}^{!}1_{{\mathfrak{Y}}}\rightarrow
f_{!}f^{!}1_{{\mathfrak{Y}}}\rightarrow\bar{f}_{!}\bar{f}^{!}1_{{\mathfrak{Y}}}\langle
n\rangle.$

###### Proof.

Let $Y\rightarrow{\mathfrak{Y}}$ be a smooth atlas. Let us define $X\coloneqq
Y\times_{{\mathfrak{Y}}}{\mathfrak{X}}$ and let $\check{C}(Y)_{\bullet}$ resp.
$\check{C}(X)_{\bullet}$ denote the corresponding Čech nerves. By construction
$\check{C}(X)_{\bullet}$ is obtained by $\check{C}(Y)_{\bullet}\times_{Y}X$.
So, by functoriality we get maps

${f_{\bullet!}\colon\operatorname{DM}(\check{C}(Y)_{\bullet})}$${\operatorname{DM}(\check{C}(X)_{\bullet})\colon
f_{\bullet}^{!}}$

that induce the maps $f_{!}$ and $f^{!}$ after passing to the limit. By
construction we have a pullback diagram

${\check{C}(X)_{\bullet}}$${\check{C}(Y)_{\bullet}}$${{\mathfrak{X}}}$${{\mathfrak{Y}}.}$$\scriptstyle{f_{\bullet}}$$\scriptstyle{j_{X,\bullet}}$$\scriptstyle{j_{Y,\bullet}}$

In particular, by smoothness of the atlas and the exchange equivalence, we
have

$j^{*}_{Y,\bullet}f_{!}f^{!}1_{Y}\simeq
f_{\bullet!}f_{\bullet}^{!}1_{\check{C}(Y)_{\bullet}}.$

Thus, by smoothness we can use that
$\operatorname{DM}^{*}({\mathfrak{Y}})\simeq\operatorname{DM}^{!}({\mathfrak{Y}})$
and descent to see that

$\mathop{\textup{colim}}_{\Delta}f_{\bullet!}f_{\bullet}^{!}1_{\check{C}(Y)_{\bullet}}\simeq
f_{!}f^{!}1_{Y}.$

Analogously, we can write

$\mathop{\textup{colim}}_{\Delta}f_{0\bullet!}f_{0\bullet}^{!}1_{\check{C}(Y)_{\bullet}}\simeq
f_{0!}f^{!}_{0}1_{Y},\quad\mathop{\textup{colim}}_{\Delta}\bar{f}_{\bullet!}\bar{f}_{\bullet}^{!}1_{\check{C}(Y)_{\bullet}}\simeq\bar{f}_{!}\bar{f}^{!}1_{Y}.$

Therefore, we may assume that ${\mathfrak{Y}}$ is representable by a scheme
and by representability of $f$ also ${\mathfrak{X}},{\mathfrak{U}}$ and
${\mathfrak{Z}}$ are representable by schemes. Hence, the result now follows
from the classical Gysin sequence (cf. [CD19, 11.3.4]). ∎

Lastly, let us define Tate motives. As we mentioned in the introduction, the
existence of a motivic $t$-structure is still an open problem. For a field $k$
Levine proved that under certain vanishing assumptions on motivic cohomology
in $\operatorname{DM}(k)$, the so called Beilinson-Soulé vanishing conjecture,
such a $t$-structure exists on the full stable subcategory generated by Tate
twists $1_{k}(n)$ (cf. [Lev93]). The Beilinson-Soulé vanishing conjecture
holds for example for finite fields.

######  Definition 2.11 ([RS20]).

Let $X$ be an Artin $S$-stack locally of finite type. We define the category
of Tate motives $\operatorname{DTM}(X)$ to be the full stable subcategory of
$\operatorname{DM}(X)$ generated by $1_{X}(n)$, for $n\in\mathbb{Z}$. An
element $M\in\operatorname{DM}(X)$ is Tate, if $M\in\operatorname{DTM}(X)$.

A map $f\colon X\rightarrow Y$ of Artin stacks locally of finite type over $S$
is called Tate if $f_{*}1_{X}$ is Tate.

Levine further shows, that existence of a weight structure on Tate motives for
a field imply that the heart of $\operatorname{DTM}(\mathbb{F}_{p})$ under the
motivic $t$-structure is equivalent to the category of graded finite
dimensional $\mathbb{Q}$-vector spaces
$(\mathbb{Q}\textup{-VS}^{\mathbb{Z}})$. In particular, using a result of
Wildeshaus, we see for example that
$\operatorname{DTM}(\mathbb{F}_{p})\simeq{\mathcal{D}}^{b}(\mathbb{Q}\textup{-VS}^{\mathbb{Z}})$,
where ${\mathcal{D}}^{b}$ denotes the bounded derived category (cf. [Wil09]).

Let us give a particularly interesting example of a Tate map that will be used
later on.

######  Example 2.12.

Let $G$ be a split reductive $S$-group scheme and $B\subseteq G$ a Borel. Then
we claim that the structure map of the flag variety $G/B\rightarrow S$ is
Tate.

Indeed, the Bruhat decomposition of $G/B$ yields a stratification by affine
spaces indexed by the Weyl group. The length of each Weyl element yields a
partial order on the associated Schubert varieties. With this order, one can
show using standard arguments that
$R\Gamma_{S}(G/B,\mathbb{Q})\simeq\bigoplus_{w\in W}1_{S}\langle
l(w)-n\rangle$, where $n$ denotes the relative dimension of $G/B$ over $S$
(cf. [Yay23] or [Bac19] for more details on analogous problems).

######  Remark 2.13 ([RS20]).

If $f\colon X\rightarrow Y$ is a smooth morphism of Artin $S$-stacks locally
of finite type and $X$ is smooth over $S$, then $D_{Y}(f_{*}1_{X})\simeq
f_{!}D_{X}(1_{X})\simeq f_{!}1_{X}\langle\Omega_{X/S}\rangle$ and
$D_{Y}(f_{!}1_{X})\simeq f_{*}1_{X}\langle\Omega_{X/S}\rangle$. Thus,
$f_{*}1_{X}$ is Tate if and only if $f_{!}1_{X}$ is Tate.

In the literature one also considers the stable cocomplete $\infty$-category
generated by Tate twists (cf. [RS20]). This is usually also referred to as
“Tate motives” but we will differentiate these from our definition. An example
of such motives is given by the motive of
$\textup{B}\mathbb{G}_{\textup{m},k}\rightarrow\operatorname{Spec}(k)$, the
classifying stack of $\mathbb{G}_{\textup{m},k}$ over a field $k$. Indeed, in
Example 2.8 we have seen
$M_{k}(\textup{B}\mathbb{G}_{\textup{m}})\simeq\bigoplus_{n\in\mathbb{Z}}1_{k}\langle
n\rangle$ and thus this lies in the ind-completion of the category of Tate
motives.

######  Definition 2.14.

Let $X$ be an Artin $S$-stack locally of finite type. We will call an
$M\in\operatorname{DM}(X)$ completed Tate if it is already in the full stable
cocomplete subcategory generated by $1_{X}(n)$.

##  3 Equivariant motives under reductive groups

Let us give an outlook of the next subsections in the split case. The non-
split case will then be a corollary.

Assume that $S$ is connected and let $G$ be a reductive $S$-group scheme and
$T$ a split maximal torus in $G$ with Weyl group scheme $W_{S}$. We will see
in Section 3.1 that $W_{S}$ is a constant group scheme associated to the
finite group $W\coloneqq W_{S}(S)$, which we call Weyl group of $T$ in $G$.

Now let $X$ be an $S$-scheme locally of finite type with $G$ action. In this
section, we want to show that the natural map
$f\colon\left[X/T\right]\rightarrow\left[X/G\right]$ is Tate, i.e.
$f_{*}1_{\left[X/T\right]}$ is a Tate motive in
$\operatorname{DM}(\left[X/G\right])$.

The key idea is to use the factorization
$\left[X/T\right]\xrightarrow{g}\left[X/N\right]\xrightarrow{h}\left[X/G\right]$,
where $N$ is the normalizer of $T$ in $G$. We note that per definition the map
$g\colon\left[X/T\right]\rightarrow\left[X/N\right]$ is a $W$-torsor. As
$W_{S}$ is finite étale, we see that after passage to an étale cover,
$\left[X/T\right]$ is isomorphic to the disjoint union of $\left[X/N\right]$
indexed by $W$. Thus, automatically $g$ is Tate.

The map $h\colon\left[X/N\right]\rightarrow\left[X/G\right]$ is a
$G/N$-torsor. Up to taking $W$-invariant, we can identify $G/N$ with $G/B$. On
$G/B$, we have stratification by Schubert cells, which are affine spaces. In
this way, we can decompose $p_{*}1_{G/B}$ as a direct sum of twists and shifts
indexed by $W$, where $p\colon G/B\rightarrow S$ is the structure map. This
will enable us to reduce the question if $h$ is Tate to ordinary $K$-theory.
More precisely, it will be enough to show that
$K_{0}(S)_{\mathbb{Q}}\rightarrow K(G/T)^{W}_{\mathbb{Q}}$ is an isomorphism,
which after reduction to the case of Chow groups is classically known.

Before coming to our main result of this section, we will first introduce
group actions on motives. In the end, we will see that our computations before
showed that, we have $R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\cong
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}$, analogous to the case of Chow
groups.

######  Remark 3.1.

A key argument in this section, is that torsors under finite étale group
schemes have related motives by taking invariants of the action. This is a
major obstruction for the generalization to integral coefficients, as we
expect that this is only satisfied if we have étale descent (c.f. [CD19, Thm.
3.3.32]). Nevertheless, using the theory of étale motives it should not be to
difficult to have analogous results after inverting only the residue
characteristics of our base scheme.

###  3.1 Preliminaries on reductive group schemes

Let $G$ be a reductive $S$-group scheme. In the following article, we want to
work with reductive group schemes that admit maximal tori. Even though this is
automatically satisfied if $S$ is the spectrum of a field, this is not true
for general base schemes (cf. [Con15]). We will first proof our results under
the assumption that the maximal torus of $G$ is split. This is not equivalent
to splitness of $G$ or even quasi-splitness.

In general one can find two main notions of quasi-split in the literature. In
SGA 3 a quasi-split reductive group scheme admits a Borel pair as well as a
global section of the associated Dynkin diagram (cf. [SGA3, Exp. XXIV]).
Another definition is given by Conrad (cf. [Con14]), where he only enforces
existence of a Borel subgroup. If $S$ is affine, this automatically implies
the existence of a maximal torus contained in $B$.

As mentioned in the introduction, for applications it is enough to consider
only the existence of a maximal torus. If the maximal torus is split, we can
also talk about the associated Weyl group, as the next lemma shows.

We want to thank Torsten Wedhorn for explaining this and elaborating that it
is enough to consider reductive group schemes with split maximal tori instead
of split reductive group schemes.

######  Lemma 3.2.

Assume that $S$ is connected. Let $T$ be a split maximal torus inside $G$ with
normalizer $N_{G}(T)$, then $W_{S}\coloneqq N_{G}(T)/T$ is represented by a
constant finite group scheme.

###### Proof.

Let us first recall some well known fact (cf. [SGA3, Exp. XXII §3]).

The quotient $W_{S}$ is represented by a finite étale $S$-scheme and that
there is an open immersion $\iota\colon
W_{S}\hookrightarrow\underline{\operatorname{Aut}}_{S}(T)$. As $T$ is split,
$\underline{\operatorname{Aut}}_{S}(T)$ is the constant group associated to
the automorphism group $\operatorname{Aut}(X^{\bullet}(T))$. In particular,
$\underline{\operatorname{Aut}}_{S}(T)\cong\coprod_{\operatorname{Aut}(X^{\bullet}(T))}S$.

Note that $\underline{\operatorname{Aut}}_{S}(T)$ is separated and thus
$\iota$ is finite. As any finite monomorphism is a closed immersion (cf.
[Sta22, 03BB]), the morphism $\iota$ is an open and closed immersion. By
assumption $S$ is connected and therefore the image of $\iota$ has to be a
disjoint union of connected components, i.e. $W_{S}\cong\coprod_{H}S$, for
some subset $H\subseteq\operatorname{Aut}(X^{\bullet}(T))$. Finally, since
$W_{S}\rightarrow S$ is finite, the set $H$ has to be finite. ∎

######  Definition 3.3.

Assume that $S$ is connected and that $G$ admits a split maximal torus. Then
we call the finite group $W\coloneqq(N_{G}(T)/T)(S)$ the Weyl group of
$G$444If $G$ is split reductive, then this agrees with the classical notion..

Let us give some examples of reductive groups that admit maximal tori.

######  Example 3.4.

Let us give examples of schemes $S$ and $G$ as above.

  1. $\bullet$

If $S=\operatorname{Spec}(k)$ is the spectrum of a field, then any reductive
group admits a maximal torus (cf. [SGA3, Exp. XIV Thm 1.1]).

  2. $\bullet$

If $S$ is any connected regular $\tilde{B}$-scheme of finite type, then any
quasi-split reductive $S$-group scheme $G$ suffices.

  3. $\bullet$

Assume, that $A$ is a Dedekind domain and $S=\operatorname{Spec}(A)$. Then $G$
admits a maximal torus if its generic fibre quasi-split (cf. Appendix A).

  4. $\bullet$

Brian Conrad constructed examples of reductive groups over
$\operatorname{Spec}(\mathbb{Z})$ that do not admit maximal tori (cf.
[Con15]). All of these examples concern non-classical reductive groups and the
author is not aware of any other counterexamples.

###  3.2 Torsors under finite groups

To warm up, we first show that torsors under finite groups are Tate. In
particular, it will be clear that the canonical map
$\left[X/T\right]\rightarrow\left[X/N\right]$, considered in the beginning of
Section 3, is Tate. We will not need any connectedness hypothesis on $S$ for
the results of this section.

######  Lemma 3.5.

Let $\Gamma$ be a finite étale $S$-group scheme. Let $f\colon X\rightarrow Y$
be a $\Gamma$-torsor of Artin $S$-stacks locally of finite type. Then $f$ is
Tate, i.e. $f_{*}1_{X}$ is a Tate motive in $\operatorname{DM}(Y)$.

###### Proof.

Let $n$ denote the degree of $\Gamma$ over $S$. Then we claim that the natural
map $\coprod_{i=1}^{n}1_{Y}\rightarrow f_{*}1_{X}$ induced via the unit of
$f_{*}f^{*}$ is an equivalence.

Indeed, by $h$-descent we may assume that $f$ is given by the trivial
$\Gamma$-torsor $\Gamma\times_{S}Y\rightarrow Y$ and $Y$ is represented by a
scheme. In particular, $f$ is a finite étale cover of degree $n$. After
passage to an étale cover, we may assume that
$\Gamma\times_{S}Y\cong\coprod_{i=1}^{n}Y$ implying the claim. ∎

Next, let us analyze the structure of the motives with respect to the base.
For this, we need to understand group actions on motives and taking fixed
points under these actions.

An action of a group $\Gamma$ on a motive $M$ in $\operatorname{DM}(X)$ is map
$\Gamma\rightarrow\operatorname{Aut}_{\operatorname{DM}(S)}(M)$ that is a
group homomorphism on $\pi_{0}$. Or equivalently, it is a map
$\Sigma_{\Gamma}\rightarrow\operatorname{DM}(X)$ (here $\Sigma_{\Gamma}$
denotes the deloop of the group $\Gamma$ seen as a discrete category - usually
this is denoted with $\textup{B}\Gamma$ but to avoid confusion, we changed the
notation).

######  Definition 3.6.

Let $M$ be a motive in $\operatorname{DM}(X)$ with an action by a finite group
$\Gamma$. Then we define the homotopy $\Gamma$-fixed points of $M$, denoted by
$M^{h\Gamma}$, as the limit of the action map
$\Sigma_{\Gamma}\rightarrow\operatorname{DM}(X)$.

######  Definition and Remark 3.7.

Let $X$ be an Artin $S$-stack locally of finite type and with an action by a
finite group $\Gamma$. For each $\gamma$, we have an action map
$a_{\gamma}\colon X\rightarrow X$. By construction of $\operatorname{DM}(X)$
this defines a map $\gamma.\colon 1_{X}\rightarrow 1_{X}$ by lax-monoidality
of the $*$-pushforward. This endows any $M\in\operatorname{DM}(X)$ with an
action via $\gamma.$. We define the $\Gamma$-fixed points of $M$, denoted by
$M^{\Gamma}$ as the image555Note that (the homotopy category of)
$\operatorname{DM}(X)$ is pseudo-abelian and hence for any idempotent operator
we can define its image. of the map

$p=\frac{1}{\\#\Gamma}\sum_{\gamma\in\Gamma}\gamma.$

The canonical map $M^{\Gamma}\rightarrow M$ defines an equivalence
$M^{\Gamma}\xrightarrow{\sim}M^{h\Gamma}$ (cf. [CD19, 3.3.21]).

Let $X\rightarrow Y$ be a map of $\Gamma$-torsor of Artin stacks locally of
finite type over $S$ (here we see $\Gamma$ as a constant group scheme on $S$).
Then the $\Gamma$-torsor $Y$-automorphisms of $X$ is isomorphic to $\Gamma$
and thus we get a $\Gamma$-action on $f_{*}f^{*}M$ for any
$E\in\operatorname{DM}(Y)$ (via $*$-pushforward of a $\Gamma$-torsor
$Y$-automorphism). Note that $f_{*}f^{*}E$ can be used to compute the motivic
cohomology of $X$ with coefficients in $E$ for smooth $f$ as

$\operatorname{Hom}_{\operatorname{DM}(Y)}(1_{Y},f_{*}f^{*}E)=\operatorname{Hom}_{\operatorname{DM}(Y)}(f^{*}1_{Y},f^{*}E)=\operatorname{Hom}_{\operatorname{DM}(Y)}(M_{Y}(X),E).$

In particular, we see that666Kernels in triangulated categories are
monomorphisms and thus on the level of homotopy groups induce short exact
sequences, dualy cokernels induce epimorphisms. Thus, applying the homotopy
invariance functor to the Hom-spectrum
$\underline{\operatorname{Hom}}_{\operatorname{DM}(Y)}(1_{Y},f_{*}f^{*}E)$, we
see that $\pi_{0}$ precisely gives us the invariants of the induced group
action on $\pi_{0}$.

$\operatorname{Hom}_{\operatorname{DM}(Y)}(1_{Y},(f_{*}f^{*}E)^{\Gamma})=\operatorname{Hom}_{\operatorname{DM}(Y)}(M_{Y}(X),E)^{\Gamma}.$

Note that as $\Gamma$ is finite, the $\Gamma$-torsor $f$ is étale and proper,
hence $f_{*}f^{*}\simeq f_{!}f^{!}$.

Let $k$ be a field and $X$ a scheme locally of finite type over
$\operatorname{Spec}(k)$. Further, let $K/k$ be a finite Galois extension and
let us denote the base change of $X$ to $\operatorname{Spec}(K)$ by $X_{K}$.
The Chow groups of $X$ and $X_{K}$ are related by the fixed points under the
Galois group, i.e.
$\textup{CH}_{n}(X)=\textup{CH}_{n}(X_{K})^{\operatorname{Gal}(K/k)}$. As one
expects, this also holds motivically. This is due to Ayoub and Cisinki-
Déglise. As noted in Remark 3.1, we do not expect this to hold, when we do not
impose étale descent. Thus, we do not expect the next lemma to hold with
integral coefficients.

######  Lemma 3.8.

Let $\Gamma$ be a finite group. Let $f\colon X\rightarrow Y$ be a
$\Gamma$-torsor of Artin $S$-stacks locally of finite type. Then the unit
factors as $\operatorname{id}\rightarrow(f_{*}f^{*})^{\Gamma}\rightarrow
f_{*}f^{*}$ and the map $\operatorname{id}\rightarrow(f_{*}f^{*})^{\Gamma}$ is
an equivalence.

###### Proof.

The factorization of the unit follows from the description of
$(f_{*}f^{*})^{\Gamma}$ as a limit. We claim that
$\operatorname{id}\rightarrow(f_{*}f^{*})^{\Gamma}$ is an equivalence. It
suffices to check this after base change to a smooth atlas of $Y$. In
particular, we may assume that $Y$ is represented by a scheme. Since
$\operatorname{DM}_{\mathbb{Q}}$ satisfies $h$-descent, we may assume777Using
[CD19, Prop. 3.3.31], we see that
$M_{Y}(X)^{\Gamma}\simeq\varphi_{*}\varphi^{*}1_{Y}$, where
$\varphi\colon({\mathscr{X}},\Gamma)\rightarrow Y$ is the induced morphism of
the diagram $\Sigma_{\Gamma}\rightarrow\textup{Sch}_{S}$ that maps the single
point of the category $\Sigma_{\Gamma}$ to $X$ and the morphisms to the
actions. Then we can base change using [CD19, Prop. 3.1.17]. that $f$ is a
trivial $\Gamma$-torsor, where it follows from [Ayo07, Prop. 2.1.166]. ∎

######  Example 3.9.

Let us consider the $W$-torsor
$f\colon\left[X/T\right]\rightarrow\left[X/N\right]$ in the beginning of
Section 3. As $W$ is finite, Lemma 3.8 implies that
$f_{*}f^{*}1_{\left[X/T\right]}^{W}\simeq
f_{!}f^{!}1_{\left[X/T\right]}^{W}\simeq 1_{\left[X/N\right]}$. After
$\sharp$\- resp. $*$-pushforward to the base $S$ along the structure map
$\left[X/N\right]\rightarrow S$, we see that

$M_{S}(\left[X/N\right])\simeq M_{S}(\left[X/T\right])^{W}\textup{ resp.
}R\Gamma(S,\left[X/N\right])\simeq R\Gamma(S,\left[X/T\right])^{W}.$

Now assume that $S=\operatorname{Spec}(k)$ is the spectrum of a field.
Applying the latter equivalence to motivic cohomology yields

$A^{n}_{N}(X,m)\cong A^{n}_{T}(X,m)^{W}$

for the equivariant intersection theory of $X$.

###  3.3 The relation between the motives of $\left[X/T\right]$ and
$\left[X/G\right]$

We have seen, that the map $\left[X/T\right]\rightarrow\left[X/N\right]$ is
Tate and how to use the action of the Weyl group to compute the motive of
$\left[X/N\right]$ with respect to the motive of $\left[X/T\right]$. Now our
goal is to show that the map $\left[X/N\right]\rightarrow\left[X/G\right]$ is
Tate and how to compute the $!$-push/pull of this map. This will be achieved
by analyzing the motive of $G/N$. Using that $G/T\rightarrow G/N$ is again a
$W$-torsor, we will reduce to the case of the flag variety $G/B$. For this, we
will need that equivariant motives do not see the action of split unipotent
subgroups.

Now let us recall the definition of a split unipotent subgroup. These are
extensions of vector bundles, e.g. a Borel $B$ containing a maximal split
torus $T$ is an extension of $T$ by a split unipotent subgroup.

######  Definition 3.10.

An algebraic $S$-group scheme $U$ is called split-unipotent if there exists a
normal series, i.e. a filtration $U=U_{n}\supseteq
U_{n-1}\supseteq\dots\supseteq U_{0}=0$ such that $U_{i}$ is normal in
$U_{i+1}$, with each successive quotient is isomorphic a vector bundle
$\mathbb{V}({\mathcal{E}})$, where ${\mathcal{E}}$ is a finite locally free
${\mathcal{O}}_{S}$-module.

######  Example 3.11.

The $S$-subgroup scheme of unipotent matricies $\mathbb{U}_{n,S}$ in
$\operatorname{GL}_{n,S}$ is split unipotent. More generally, let $G$ be a
reductive $S$-group scheme and $P$ be a parabolic in $G$, then the unipotent
radical $R_{u}(P)$ of $P$ is split unipotent (cf. [SGA3, Exp. XXVI Prop.
2.1]).

######  Lemma 3.12.

Let $G$ be a linear algebraic $S$-group scheme. Consider a split exact
sequence of $S$-group schemes

$1\rightarrow U\rightarrow G\rightarrow H\rightarrow 1$

where $U$ is split unipotent. Chose a splitting $\pi\colon H\hookrightarrow
G$. Let $X$ be an $S$-scheme locally of finite type with an $F$-action. Then
the $!$-pullback induces an equivalence

$\pi^{!}\colon\operatorname{DM}(X/G)\xrightarrow{\sim}\operatorname{DM}(X/H).$

###### Proof.

This is analogous to the proof of [RS20, Prop. 2.2.11] but for completion, we
give a proof.

The morphism $\pi^{!}$ induces a morphism
$\pi_{n}^{!}\colon\operatorname{DM}(\textup{Bar}^{n}(X,G))\rightarrow\operatorname{DM}(\textup{Bar}^{n}(X,H))$.
Using [BN19, Lem. B.6], it suffices to show that
$\pi^{!}\colon\operatorname{DM}(G)\rightarrow\operatorname{DM}(H)$ is fully
faithful. By assumption $\pi$ is a $U$-torsor and using étale descent, we may
assume that $\pi$ is given by the trivial $U$-torsor, i.e. $\pi\colon U\times
H\rightarrow H$ is given by the projection. Replacing $H$ by $S$, it suffices
to show that
$\pi^{!}\colon\operatorname{DM}(S)\rightarrow\operatorname{DM}(U)$ is fully
faithful. As $U$ is split unipotent, it has a filtration by subgroup $U_{i}$
with successive quotients isomorphic to a vector bundle. Using the same
argumentation as above, we may assume that $U$ is a vector bundle, in which
case the assertion is clear by homotopy invariance. ∎

Let $G$ be a split reductive $S$-group scheme and let $B$ be a Borel contain
$T$ inside $G$. Let $X$ be an $S$-scheme locally of finite type with
$B$-action. The above lemma shows that
$\operatorname{DM}(\left[X/T\right])\simeq\operatorname{DM}(\left[X/B\right])$.
We also have that $G/T\rightarrow G/N$ is a $W$-torsor. The results on torsors
under finite groups combined with the above lemma then yields that
$R\Gamma_{S}(G/N,\mathbb{Q})\simeq R\Gamma_{S}(G/B,\mathbb{Q})^{W}$. The next
result will use this fact.

######  Proposition 3.13.

Let $G$ be a reductive $S$-group scheme with maximal torus $T$. Let $N$ denote
the normalizer of $T$ in $G$. Further, let $X$ be an $S$-scheme locally of
finite type with $G$-action and
$f\colon\left[X/N\right]\rightarrow\left[X/G\right]$ the canonical map. Then
the unit $1_{\left[X/G\right]}\rightarrow f_{*}f^{*}1_{\left[X/G\right]}$ is
an equivalence. In particular, $f_{*}1_{\left[X/N\right]}$ is a Tate motive in
$\operatorname{DM}(\left[X/G\right])$.

Let us summarize the idea of the proof. We will show that the unit
$1_{\left[X/G\right]}\rightarrow f_{*}f^{*}1_{\left[X/G\right]}$ is an
equivalence. To see this we will use that we have the following pullback
diagram

${G/N\times_{S}X}$${X}$${\left[X/N\right]}$${\left[X/G\right]}$$\scriptstyle{\operatorname{pr}}$

(cf. [Sta22, 04Y4]). In particular, after étale-descent, we may assume that
$f$ is given by the projection $G/N\rightarrow S$. Again, by étale descent, we
may assume that $G$ is split with respect to $T$. Let $B$ be a Borel of $G$
containing $T$. Using our calculations about torsors under finite groups, it
is enough to show that the induced map $1_{S}\rightarrow
R\Gamma_{S}(G/T,\mathbb{Q})^{W}$ is an equivalence. But as
$R\Gamma_{S}(G/T,\mathbb{Q})^{W}\simeq R\Gamma_{S}(G/B,\mathbb{Q})^{W}$ will
reduce this question to a classical question about Chow rings, at least when
$S$ is the spectrum of a field, namely if the pullback map

$\textup{CH}_{n}(S,m)_{\mathbb{Q}}\rightarrow\textup{CH}_{n}(G/B,m)_{\mathbb{Q}}^{W}$

is an isomorphism for all $n,m\in\mathbb{Z}$. But as we can identify
$\textup{CH}_{n}(G/B,m)_{\mathbb{Q}}^{W}$ with
$\textup{CH}_{n}(G/G,m)_{\mathbb{Q}}$, this is clear (cf. [Kri13, Cor. 8.7]).
In the case when $S$ is not the spectrum of a field, we have to use $K$-theory
and as the flag variety admits an integral model, we can reduce to
$S=\operatorname{Spec}(\mathbb{Z})$. Up to $K_{1}$ the rational $K$-theory of
$\mathbb{Z}$ and $\mathbb{Q}$ are isomorphic. Using that the flag variety
admits a stratification by affine spaces, we may assume the
$S=\operatorname{Spec}(\mathbb{Q})$, as the only part we have to worry about,
$K_{1}(\mathbb{Z})_{\mathbb{Q}}$, vanishes.

###### Proof of Proposition 3.13.

Throughout this proof, we will denote for readability the structure map of an
Artin stack ${\mathcal{X}}$ to $S$ with $p_{{\mathcal{X}}}$.

To show that the unit

$1_{\left[X/G\right]}\rightarrow f_{*}f^{*}1_{\left[X/G\right]}$

is an equivalence, we may assume by étale descent that the map $f$ is given by
the structure map $G/N\rightarrow S$. Again, by étale descent, we may assume
that $G$ is split with respect to $T$. Then it is enough to show that the unit
$1_{S}\rightarrow R\Gamma_{S}(G/N,\mathbb{Q})$ is an equivalence, since the
pullback of this equivalence along $p_{X}$ yields the desired equivalence.

Note that the natural map $g\colon G/T\rightarrow G/N$ is naturally a
$W$-torsor. Hence, by Lemma 3.8, we see that $1_{G/N}\simeq(g_{*}1_{G/T})^{W}$
and thus $R\Gamma_{S}(G/N,\mathbb{Q})\simeq p_{G/N*}(g_{*}1_{G/T})^{W}$. Since
$p_{G/N*}$ is a right adjoint it commutes with limits. Thus, we have

$p_{G/N*}(g_{*}1_{G/T})^{W}\simeq(p_{G/N*}g_{*}1_{G/T})^{W}\simeq(p_{G/T*}1_{G/T})^{W}\simeq
R\Gamma_{S}(G/T,\mathbb{Q})^{W}$

By Lemma 3.12, we see that $R\Gamma_{S}(G/T,\mathbb{Q})\simeq
R\Gamma_{S}(G/B,\mathbb{Q})$. By Example 2.12, we have that

$R\Gamma_{S}(G/B,\mathbb{Q})\simeq\bigoplus_{w\in W}1_{S}\langle
l(w)-n\rangle,$

where $n$ is the relative dimension of the flag variety $G/B$. In particular,
$R\Gamma_{S}(G/B,\mathbb{Q})$ and thus $R\Gamma_{S}(G/T,\mathbb{Q})$ is Tate.
As the $W$-invariants are defined as an image of a map, we see that
$R\Gamma_{S}(G/T,\mathbb{Q})^{W}$ is also Tate. The $\infty$-category of Tate
motives over $S$ is the stable subcategory of $\operatorname{DM}(S)$ generated
by $1(r)$, for $r\in\mathbb{Z}$. Therefore, the natural map $1_{S}\rightarrow
R\Gamma_{S}(G/T,\mathbb{Q})^{W}$ is an equivalence if and only if the induced
map

(3.1)
$\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}(r),1_{S})\rightarrow\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}(r),R\Gamma_{S}(G/T,\mathbb{Q})^{W})$

is an equivalence for all $r\in\mathbb{Z}$ (here
$\underline{\operatorname{Hom}}$ denotes the inner-$\operatorname{Hom}$). If
$r>0$, then the $m$-th homotopy group of the left hand side is isomorphic to
$K_{-2r+m}(S)^{(-r)}$ and thus vanishes as the negative Adams eigenspaces
vanish per definition (cf. [Sou85]). By Remark 3.7 and the computation of
$p_{G/B*}1_{G/B}$, we see that

$\pi_{m}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}(r),R\Gamma_{S}(G/T,\mathbb{Q})^{W})\cong(\bigoplus_{w\in
W}K_{-2r+m}(S)^{(l(w)-n-r)})^{W},$

which also vanishes for $r>0$ as $l(w)-n\leq 0$. Therefore, we may assume from
now on that $r\leq 0$. Before continuing with the proof, let us first assume
that $S=\operatorname{Spec}(k)$ is the spectrum of a field $k$. Then by the
above, we are reduced to show that the restriction map

$A^{r}(S,m-2r)\rightarrow A^{r}(G/B,m-2r)^{W}$

is an isomorphism (by homotopy invariance and identification of motivic
cohomology with higher Chow-theory). But as $A^{r}(G/B,m-2r)^{W}\cong
A^{r}(G/G,m-2r)=A^{r}(S,m-2r)$ (cf. [Kri13, Cor. 8.7]), the restriction above
yields the identity on $A^{r}(S,m-2r)$ which is trivially an isomorphism.
Hence, we are done in the case of $S=\operatorname{Spec}(k)$.

Next, let us show how to reduce to the case of $S=\operatorname{Spec}(k)$. As
$r\leq 0$, we may use commutativity of $\underline{\operatorname{Hom}}$ with
limits, to see that (3.1) is an equivalence if and only if

(3.2) $\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,1_{S})\rightarrow\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,R\Gamma_{S}(G/T,\mathbb{Q})^{W})$

is an equivalence. Further, since $S$ is finite dimensional (cf. [GW10, Prop.
14.109]), it is a fact that for every $n\in\mathbb{Z}$, the groups
$K_{n}(S)^{(i)}$ vanish for all but finitely many $i\in\mathbb{Z}$ (cf.
[Sou85, §2]). Therefore, the morphism (3.2) is an equivalence for all $r\leq
0$ if and only if

$\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,1_{S})\rightarrow\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,R\Gamma_{S}(G/T,\mathbb{Q})^{W})$

is an isomorphism. Equivalently, we can write this morphism as

$\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S},1_{S}\langle
r\rangle)\rightarrow\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(M_{S}(G/T),1_{S}\langle
r\rangle)^{W}$

The motive $M_{S}(G/T)$ is a direct sum of shifts and twists of the unit of
$S$ and therefore compact (cf. [CD19, Thm. 11.1.13]). By compactness of
$1_{S}$ and $M_{S}(G/T)$, the above morphism is an equivalence if and only if
the morphism

$\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S},\bigoplus_{r\in\mathbb{Z}}1_{S}\langle
r\rangle)\rightarrow\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(M_{S}(G/T),\bigoplus_{r\in\mathbb{Z}}1_{S}\langle
r\rangle)^{W}$

is an equivalence. By construction of the rational K-theory spectrum in
$\operatorname{DM}(S)$, denoted by $\textup{KGL}_{S,\mathbb{Q}}$, we see that
$\bigoplus_{r\in\mathbb{Z}}1_{S}\langle
r\rangle\simeq\textup{KGL}_{S,\mathbb{Q}}$ (cf. [CD19, Lem. 14.1.4]). Thus, as
$G/T$ is representable by a scheme (cf. [SGA3, Exp. IX Thm. 5.1]), the
$n$-homotopy group of the right hand side is isomorphic to
$K_{n}(G/T)_{\mathbb{Q}}^{W}$, the rational higher $K$-theory of $G/T$.
Further, the properties of the $K$-theory spectrum yield that the induced
morphism $K_{n}(S)_{\mathbb{Q}}\rightarrow K_{n}(G/T)_{\mathbb{Q}}$ is given
by the pullback along $G/T\rightarrow S$ (cf. [CD19, §13.1]). Taking
$W$-invariants yields the map

(3.3) $K_{n}(S)_{\mathbb{Q}}\rightarrow K_{n}(G/T)_{\mathbb{Q}}^{W}$

on the $n$-th homotopy groups. We have to show that this map is an isomorphism
for all $n\geq 0$. In the following we will freely identify
$K_{n}(G/T)_{\mathbb{Q}}$ with $K_{n}(G/B)_{\mathbb{Q}}$ and $K$-theory with
$G$-theory as we are only working with smooth schemes. As $G$ is split
reductive, there exists a model
$(G_{\mathbb{Z}},B_{\mathbb{Z}},T_{\mathbb{Z}})$ over
$\operatorname{Spec}(\mathbb{Z})$ such that $G/B$ is the pullback of the
associated flag variety $G_{\mathbb{Z}}/B_{\mathbb{Z}}$. As $G/B$ (resp.
$G_{\mathbb{Z}}/B_{\mathbb{Z}}$) is stratified by affine spaces, we have an
equivalence of $K$-theory spectra $K(G/B)_{\mathbb{Q}}\simeq
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}}$
(cf. [Jos01, Cor. 4.3]), here $K(-)_{\mathbb{Q}}$ denotes the underlying
$E_{\infty}$-ring spectrum in $\operatorname{DM}(\mathbb{Z})$, in particular
$K(\mathbb{Z})_{\mathbb{Q}}=\textup{KGL}_{\mathbb{Z},\mathbb{Q}}$ and
$K(S)_{\mathbb{Q}}$ (resp. $K(G/B)_{\mathbb{Q}}$) is just the $*$-pushforward
of $\textup{KGL}_{S,\mathbb{Q}}$ (resp. $\textup{KGL}_{G/B,\mathbb{Q}}$) to
$\operatorname{DM}(\mathbb{Z})$. We will show that the induced morphism

$K(S)_{\mathbb{Q}}\rightarrow K(G/B)_{\mathbb{Q}}^{W}$

is an equivalence if and only if

$K(\mathbb{Z})_{\mathbb{Q}}\rightarrow
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}^{W}.$

This follows from the following equivalence,

$(K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}})^{W}\simeq
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}^{W}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}},$

which we will show holds true. This is shown similarly to the analogous facts
on representations of finite groups, but we will give an argument.

For a scheme $X$ let $\textup{DK}(X)$ of $X$ denote the $\infty$-category of
$KGL_{X,\mathbb{Q}}$-modules in $\operatorname{DM}(X)$. This can also be
extended to stacks by étale descent.888Note that the étale localized
$K$-theory spectrum of stacks does not agree with the genuine rational
$K$-theory spectrum. As we will only need the existence of a six functor
formalism, we will just ignore the technicalities. By abuse of notation will
denote the tensor unit of $DK(X)$ with $1_{X}$ and just for the following will
even drop the $X$ in the notation, as it is clear in which categories the
units are. Also, we will again denote the inner-$\operatorname{Hom}$ by
$\underline{\operatorname{Hom}}$. Let us fix the following diagram with
pullback squares

${G_{\mathbb{Z}}/B_{\mathbb{Z}}}$${\mathbb{Z}}$${S}$${\left[W\backslash
G_{\mathbb{Z}}/B_{\mathbb{Z}}\right]}$${\left[\mathbb{Z}/W\right]}$${\left[S/W\right]}$${\mathbb{Z}}$${S,}$$\scriptstyle{g}$$\scriptstyle{b}$$\scriptstyle{f}$$\scriptstyle{a^{\prime}}$$\scriptstyle{a}$

here $W$ acts on $\mathbb{Z}$ resp. $S$ trivially and $a,b$ are the natural
projection. Note that by Lemma 3.8, we have that
${g_{*}g^{*}}^{W}\simeq\operatorname{id}$. Using this, smooth base change,
projection formula999Note that $f$ is proper, as $W$ is a finite constant
group scheme. and adjunctions, we get the following chain of equivalences

$\displaystyle(K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}})^{W}$
$\displaystyle\simeq\underline{\operatorname{Hom}}_{\textup{DK}(\left[W/\mathbb{Z}\right])}(1,g_{*}g^{*}(b_{*}1\otimes
a^{\prime}_{*}1))^{W}$
$\displaystyle\simeq\underline{\operatorname{Hom}}_{\textup{DK}(\left[W/\mathbb{Z}\right])}(1,b_{*}1\otimes
a^{\prime}_{*}1)$
$\displaystyle\simeq\underline{\operatorname{Hom}}_{\textup{DK}(\mathbb{Z})}(1,f_{*}(b_{*}1\otimes
a^{\prime}_{*}1))$ $\displaystyle\simeq
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}^{W}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}}.$

Note that we also have to use that
$K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}$ is a direct sum of the unit,
to see that the $\underline{\operatorname{Hom}}$ agrees with the tensor
product of spectra (see below for an explicit description).

Thus, as the structure map $G/B\rightarrow S$ is induced via pullback, we see
that $K(S)_{\mathbb{Q}}\rightarrow K(G/B)^{W}_{\mathbb{Q}}$ is induced via

$K(\mathbb{Z})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}}\xrightarrow{\alpha^{W}\otimes\operatorname{id}_{S}}K(G_{\mathbb{Z}}/B_{\mathbb{Z}})^{W}_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}},$

where $\alpha\colon
G_{\mathbb{Z}}/B_{\mathbb{Z}}\rightarrow\operatorname{Spec}(\mathbb{Z})$
denotes the structure map. In particular, we may assume that
$S=\operatorname{Spec}(\mathbb{Z})$.

As $G/B$ admits an affine cell decomposition, we see that

$K(G/B)_{\mathbb{Q}}\simeq\bigoplus_{w\in
W}K(\mathbb{Z})_{\mathbb{Q}},\textup{ and
}K(G_{\mathbb{Q}}/B_{\mathbb{Q}})_{\mathbb{Q}}\simeq\bigoplus_{w\in
W}K(\mathbb{Q})_{\mathbb{Q}},$

where $G_{\mathbb{Q}}/B_{\mathbb{Q}}$ denotes the generic fiber of $G/B$ (cf.
[Yay23]). It is well known that for any $n\neq 1$ there is an isomorphism
$K_{n}(\mathbb{Z})_{\mathbb{Q}}\cong K_{n}(\mathbb{Q})_{\mathbb{Q}}$ and
$K_{1}(\mathbb{Z})_{\mathbb{Q}}=0$. Thus, it is clear that
$K_{1}(\mathbb{Z})_{\mathbb{Q}}\rightarrow K_{1}(G/B)_{\mathbb{Q}}^{W}$ is an
isomorphism. For the other $K$-groups, we can use the identification of
$K_{n}(\mathbb{Z})_{\mathbb{Q}}$ with $K_{n}(\mathbb{Q})_{\mathbb{Q}}$ and may
assume that $S=\operatorname{Spec}(\mathbb{Q})$. This concludes the proof via
the argument in the Chow group case. ∎

Combining Proposition 3.13 and the results of Section 3.2 now yields the main
result of this section.

######  Theorem 3.14.

Let $G$ be a reductive $S$-group scheme that admits a split maximal torus $T$
with Weyl group $W$. Assume $G$ acts on an $S$-scheme $X$ locally of finite
type. Then the natural map

$\left[X/T\right]\rightarrow\left[X/G\right]$

is Tate.

Further, if $S$ is connected, we have

$R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}.$

###### Proof.

Let $N$ be the normalizer of $T$ in $G$. We can factor the map in the theorem
as
$\left[X/T\right]\xrightarrow{f}\left[X/N\right]\xrightarrow{g}\left[X/G\right]$.
The first part of the theorem follows immediately with Lemma 3.5 and
Proposition 3.13.

For the proof of the second claim, we apply Lemma 3.8 and Proposition 3.13 and
get

$\displaystyle(p_{\left[X/T\right]*}1_{\left[X/T\right]})^{W}\simeq(p_{\left[X/G\right]*}g_{*}f_{*}1_{\left[X/T\right]})^{W}$
$\displaystyle\simeq
p_{\left[X/G\right]*}g_{*}(f_{*}1_{\left[X/T\right]})^{W}$
$\displaystyle\simeq p_{\left[X/G\right]}g_{*}1_{\left[X/N\right]}\simeq
p_{\left[X/G\right]*}1_{\left[X/G\right]}.$

∎

Let $S=\operatorname{Spec}(k)$ be the spectrum of a field and $G$ a split
reductive group over $k$ with split maximal torus $T$ and Weyl group $W$.
Edidin and Graham have shown in [EG98] that for any smooth scheme $X$ with
$G$-action, we have $A^{\bullet}_{G}(X)\cong A^{\bullet}_{T}(X)^{W}$. Later on
Krishna has shown in [Kri13, Cor. 8.7] that this result can be generalized to
higher Chow groups, at least when $X$ is quasi-projective. Our Theorem 3.14 is
the motivic analog of both of these results. In fact, we will see in the next
corollary, that applying Theorem 3.14 to motivic cohomology recovers both
Edidin-Graham’s and Krishna’s result. Moreover, as any reductive group over a
field splits after passage to a finite Galois extension, we can use the
results of Section 3.2 to extend this comparison of (higher) equivariant Chow
groups to arbitrary reductive groups over fields. We also note that Levine
showed that Bloch’s higher Chow groups also satisfy a localization sequence,
even in the non quasi-projective case. The following corollary summarizes this
discussion.

######  Corollary 3.15.

Assume that $S=\operatorname{Spec}(k)$ is the spectrum of a field. Let $G$ be
a reductive $S$-group scheme with maximal torus $T$. Let $W$ denote the
absolute Weyl101010Let $\bar{k}$ be an algebraic closure of $k$, then
${\mathcal{W}}\coloneqq N_{G}(T)/T$ is a finite étale group scheme of $S$ and
we set the absolute Weyl group to be $W\coloneqq{\mathcal{W}}(\bar{k})$ group
of $T$ in $G$. Assume $G$ acts on a smooth finite type $S$-scheme $X$. Then we
have

$R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}.$

In particular, applying this result to motivic cohomology yields for all
$n,m\in\mathbb{Z}$

$A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}.$

###### Proof.

Any reductive group over $k$ becomes split after passing to a finite Galois
extension $K/k$ (cf. [SGA3, Exp. XXII Cor. 2.4]). Thus, let $K$ be such an
extension, so that $T_{K}$ is a split maximal torus. Then we have the
following diagram with pullback squares

${\left[X_{K}/T_{K}\right]}$${\left[X_{K}/G_{K}\right]}$${\operatorname{Spec}(K)}$${\left[X/T\right]}$${\left[X/G\right]}$${\operatorname{Spec}(k).}$$\scriptstyle{f}$$\scriptstyle{p_{T_{K}}}$$\scriptstyle{g}$$\scriptstyle{p_{G_{K}}}$$\scriptstyle{p_{K}}$$\scriptstyle{p_{T}}$$\scriptstyle{p_{G}}$

As $K/k$ is a finite Galois extension, the morphisms $p_{K}$ and thus also $f$
are $\Gamma\coloneqq\operatorname{Gal}(K/k)$-torsors. Then Lemma 3.8 yields
the equivalence

$p_{G*}1_{\left[X/G\right]}\simeq(p_{K*}p_{K}^{*}p_{G*}1_{\left[X/G\right]})^{\Gamma}.$

As the diagram above has cartesian squares, we can use smooth base change, to
see that $p_{K}^{*}p_{G*}\simeq p_{G_{K}*}g^{*}$. But by Theorem 3.14, we have

$p_{G_{K}*}g^{*}1_{\left[X/G\right]}\simeq
R\Gamma_{K}(\left[X_{K}/G_{K}\right],\mathbb{Q})\simeq
R\Gamma_{K}(\left[X_{K}/T_{K}\right],\mathbb{Q})^{W}.$

Commutativity of the above diagram yields
$p_{K*}R\Gamma_{K}(\left[X_{K}/T_{K}\right],\mathbb{Q})\simeq
p_{T*}f_{*}1_{\left[X_{K}/T_{K}\right]}.$ Thus, we have

$p_{G*}1_{\left[X/G\right]}\simeq((p_{T*}f_{*}1_{\left[X_{K}/T_{K}\right]})^{W})^{\Gamma}.$

As limits commute with limits, we may write the right hand side as
$(p_{T*}(f_{*}f^{*}1_{\left[X/T\right]})^{\Gamma})^{W}$ and again by Lemma
3.8, we have $(f_{*}f^{*}1_{\left[X/T\right]})^{\Gamma}\simeq
1_{\left[X/T\right]}$ concluding the proof.

The result about motivic cohomology follows from Remark 3.7 by using [RS20,
Thm. 2.2.10] which proves that motivic cohomology for smooth quotient stacks
is computed by the higher equivariant Chow groups of Edidin-Graham.111111In
loc.cit. they assume some properties on the groups and on $X$, as Edidin-
Graham need these assumptions to compare higher Chow theory of stacks and
equivariant higher Chow theory [EG98, Prop. 13 (b)]. The assumptions in
loc.cit. are needed as Bloch only shows the existence of a long exact sequence
for higher Chow groups in the case where $X$ is quasi-projective. This result
was extended by Levine to all finite type $k$-schemes (cf. [Lev01]). Thus, the
comparison of Edidin-Graham and hence also of Richarz-Scholbach go through in
the case of the corollary. ∎

We can apply the same reasoning as above to see in the next corollary that the
map $\left[X/T\right]\rightarrow\left[X/G\right]$ is Tate even if $T$ is not
split. The main point is that groups of multiplicative type over locally
noetherian normal schemes are isotrivial, i.e. become diagonalizable after
passing to a finite étale cover (cf. [SGA3, Exp. X Thm. 5.16]). In particular,
after passing to a Galois cover of our base, we can reduce to Theorem 3.14
using the results of Section 3.2.

######  Corollary 3.16.

Let $S$ be a connected scheme. Let $G$ be a reductive $S$-group scheme that
admits a maximal torus $T$. Assume $G$ acts on an $S$-scheme $X$ locally of
finite type. Then the natural map

$\left[X/T\right]\rightarrow\left[X/G\right]$

is Tate.

###### Proof.

As any étale cover of a connected scheme can be refined by a Galois cover (cf.
[Sta22, 03SF]), there exists a Galois cover $\tilde{S}\rightarrow S$ such that
$\tilde{T}\coloneqq T\times_{S}\tilde{S}$ splits (cf. [SGA3, Exp. X Thm.
5.16]). Let us denote the base change of $X$ under $\tilde{S}\rightarrow S$
with $\tilde{X}$. We get an induced action of $\tilde{G}$ on $\tilde{X}$. In
particular, we get a pullback square diagram

${\left[\tilde{X}/\tilde{T}\right]}$${\left[\tilde{X}/\tilde{G}\right]}$${\left[X/T\right]}$${\left[X/G\right].}$$\scriptstyle{\tilde{f}}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$

Let $\Gamma$ denote the finite étale $S$-automorphism group of $\tilde{S}$.
Then $g$ and $h$ are $\Gamma$-torsors. Thus, we can compute

$f_{*}1_{\left[X/T\right]}\simeq(f_{*}h_{*}1_{\left[\tilde{X}/\tilde{T}\right]})^{\Gamma}\simeq(g_{*}\tilde{f}_{*}1_{\left[\tilde{X}/\tilde{T}\right]})^{\Gamma}.$

The morphism $\tilde{f}$ is Tate by Theorem 3.14 and $g$ is also Tate by Lemma
3.5. Further, taking invariants under finite group actions is given by taking
the image of the fixed point operator (cf. Remark 3.7), i.e. by extensions,
and thus preserves Tate motives. In particular, $f_{*}1_{\left[X/T\right]}$ is
a Tate motive. ∎

##  4 Torsors under tori

Let us fix a torus $T$ over $S$ (cf. [SGA3, Exp. IX]).

In this subsection, we want to understand the motivic homotopy theory of
torsors under $T$. More precisely, let us consider the following situation.
Let $X\rightarrow Y$ be a $T$-torsors of Artin $S$-stacks locally of finite
type. We want to understand the relation between $M_{S}(X)$ and $M_{S}(Y)$.

Let us recall the classical case of Chow theory. For this paragraph let us
assume that $X$ and $Y$ are smooth Artin stacks over $\operatorname{Spec}(k)$,
where $k$ is a field. Then the group $\hat{T}$ of characters of $T$ acts on
$A^{\bullet}(Y)$ in the following way. Let $\chi\in\hat{T}$ be a character and
consider its associated $1$-dimensional representation $\kappa(\chi)$. The
quotient $L_{\chi}\coloneqq X\times^{T}\kappa(\chi)$ is representable by a
line bundle over $Y$. Multiplication with the first Chern class of $L_{\chi}$
yields an action of $\hat{T}$ on $A^{\bullet}(Y)$. If $T$ is split, then for
Chow rings it is known that

$A^{\bullet}(X)\cong A^{\bullet}(Y)/\hat{T}A^{\bullet}(Y).$

Our goal is to extend this result to motivic homotopy theory, so that it
generalizes the result about Chow theory and yields a similar statement for
$K$-theory. Even though we work with Beilinson motives, we will remark in the
end how to extend this to integral $K$-theory under some assumptions on $X$
and $Y$.

###  4.1 The motive of $T$-torsors

Let us denote the character group of $T$ with $\hat{T}$. Let $X\rightarrow Y$
be a $T$-torsor of Artin stacks locally of finite type over $S$ and
$\chi\in\hat{T}$ a character. Let $\mathbb{G}_{\textup{m},S}$ act via left
multiplication in $\mathbb{A}^{1}_{S}$. Then $T$ acts via $\chi$ on
$\mathbb{A}^{1}_{S}$ and thus, $X\times^{T}\mathbb{A}^{1}_{S}\rightarrow
X/T\cong Y$ yields a line bundle over $Y$. The action of the first Chern class
of $X\times^{T}\mathbb{A}^{1}_{S}$ on the motivic cohomology will be described
by a Gysin sequence (cf. Proposition 4.2).

######  Notation 4.1.

Assume that $T$ is split. In the following we want to split up the $T$-torsor
$f\colon X\rightarrow Y$ into a sequence of
$\mathbb{G}_{\textup{m},S}$-torsors. Note that by splitness
$T\cong\mathbb{G}_{\textup{m},S}^{r}$ for some $r\in\mathbb{N}$. Fixing a
numbering of the $\mathbb{G}_{\textup{m},S}$-components of $T$, we can embed
for any $1\leq k\leq r$ the product $\mathbb{G}_{\textup{m},S}^{k}$ into $T$
by $\operatorname{id}_{\mathbb{G}_{\textup{m},S}^{k}}\times 1^{r-k}$. Then
$\mathbb{G}_{\textup{m},S}^{k}$ acts on $X$ via this embedding. We get a
sequence

$X\rightarrow X/\mathbb{G}_{\textup{m},S}\rightarrow
X/\mathbb{G}_{\textup{m},S}^{2}\rightarrow\dots\rightarrow X/T\cong Y$

of $\mathbb{G}_{\textup{m},S}$-torsors. We denote the induced maps
$X/\mathbb{G}_{\textup{m},S}^{i}\rightarrow Y$ with $f_{i}$.

######  Proposition 4.2.

Assume that $T\cong\mathbb{G}_{\textup{m},S}^{r}$ for some $r\in\mathbb{N}$.
Let $X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks over $S$. Then,
there exists a filtration

$M_{Y}(X)=M_{0}\rightarrow M_{1}\rightarrow\dots\rightarrow M_{r}=1_{Y}$

in $\operatorname{DM}(S)$, where $M_{i}\coloneqq
M_{Y}(X/\mathbb{G}_{\textup{m},S}^{i})$ such that the cofiber of
$M_{Y}(X_{i-1})\rightarrow M_{Y}(X_{i})$ is given by $M_{Y}(X_{i})\langle
1\rangle$ and the map $M_{Y}(X_{i})\rightarrow M_{Y}(X_{i})\langle 1\rangle$
is induced by multiplication with
$c_{1}(X_{i-1}\times^{\mathbb{G}_{\textup{m}}}\mathbb{A}^{1}_{S})$.

###### Proof.

This follows by successively using [HPL21, Prop. 2.32]. But for completion let
us give a proof by recalling the argument.

The morphism $X_{i-1}\rightarrow X_{i}$ is a $\mathbb{G}_{m}$-torsor. In
particular, the scheme ${\mathcal{L}}_{i}\coloneqq
X_{i-1}\times_{S}^{\mathbb{G}_{\textup{m},S}}\mathbb{A}^{1}_{S}$ is a line
bundle over $X_{i}$. Let $s\colon X_{i}\hookrightarrow V_{i}$ denote the zero
section. Certainly, the complement of the closed immersion $s$ is isomorphic
to $X_{i-1}$. Then the Gysin sequence of Lemma 2.10 yields a fibre sequence

$M_{Y}(X_{i-1})\rightarrow M_{Y}(V_{i})\simeq
M_{Y}(X_{i})\xrightarrow{\varphi}M_{Y}(X_{i})\langle 1\rangle.$

By construction of the Gysin sequence $\varphi$ is given by multiplication
with $c_{1}({\mathcal{L}}_{i})$. This concludes the proof. ∎

######  Corollary 4.3.

Assume $S$ is connected and let $f\colon X\rightarrow Y$ be a $T$-torsor of
smooth Artin stacks over $S$. Then $f$ is a Tate map.

If $T$ is split, we may drop the connectedness hypothesis on $S$.

###### Proof.

If $T$ is split, then Proposition 4.2 implies that $M_{Y}(X)$ is a successive
extension of $1_{Y}$. Thus, the result follows from Remark 2.13.

In the non-split case, this is completely analogous to the proof of Corollary
3.16. ∎

###  4.2 Motivic cohomology of $T$-torsors

Now we will assume that $T$ is split and fix an $r\in\mathbb{N}$ such that
$T\cong\mathbb{G}_{\textup{m},S}^{r}$.

For any Artin stack $X$ locally of finite type over $S$, we will denote its
motivic cohomology with

$H^{p}(X,\mathbb{Q}(n))\coloneqq\operatorname{Hom}_{\operatorname{DM}(S)}(M_{S}(X),\mathbb{Q}(n)[p]).$

If $X$ is representable by a smooth $S$-scheme, then we have
$H^{p}(X,\mathbb{Q}(n))\cong K_{2n-p}(X)^{(n)}$. If $S=\operatorname{Spec}(k)$
is the spectrum of a field and $X$ is smooth and over $S$, we have
$H^{p}(X,\mathbb{Q}(n))\cong A^{n}(X,2n-p)$. Note that for a smooth Artin
$S$-stack $X$ the motivic cohomology vanishes automatically in certain degrees
by descent and vanishing of negative $K$-theory for regular schemes, i.e. we
have that $H^{p}(X,\mathbb{Q}(n))\cong 0$ for $p>2n$.

######  Notation 4.4.

Consider a $\chi\in\hat{T}$. The associated line bundle $L_{\chi}\coloneqq
X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi})$ yields a map
$H^{p+2}(Y,\mathbb{Q}(n+1))\rightarrow H^{p}(Y,\mathbb{Q}(n))$, by
multiplication with the Chern class of $L_{\chi}$. We denote the image of this
map with $c_{1}(L_{\chi})H^{p}(Y,\mathbb{Q}(n))$.

######  Proposition 4.5.

In the setting of Proposition 4.2 let us further fix an $n\in\mathbb{Z}$.
Then, we have

$H^{2n}(X,\mathbb{Q}(n))\cong
H^{2n}(Y,\mathbb{Q}(n))/\hat{T}H^{2n}(Y,\mathbb{Q}(n)).$

###### Proof.

First let us note that it is enough121212Any character is generated by
primitive characters and the corresponding $1$-dimensional representation is
given by the associated tensor product. to show that

$H^{2n}(X,\mathbb{Q}(n))=H^{2n}(Y,\mathbb{Q}(n))/\langle
c_{1}(X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi_{i}}))H^{2n}(Y,\mathbb{Q}(n))\rangle,$

where $\chi_{i}$ is a primitive character in $\hat{T}$. Let
$X=X_{0}\rightarrow X_{1}\rightarrow\dots\rightarrow X_{r}=Y$ be the sequence
of Proposition 4.2. For each $0\leq i\leq r$ this yields a long exact sequence
on motivic cohomology

$\displaystyle\dots\rightarrow H^{2n+2}(X_{i},\mathbb{Q}(n+1))$
$\displaystyle\rightarrow H^{2n}(X_{i},\mathbb{Q}(n))$
$\displaystyle\rightarrow H^{2n}(X_{i-1},\mathbb{Q}(n))\rightarrow
H^{2n+3}(X_{i},\mathbb{Q}(n+1))\rightarrow\dots.$

We have $H^{2n+3}(X_{i},\mathbb{Q}(n+1))=0$ and thus get an exact sequence of
the form

$H^{2n+2}(X_{i},\mathbb{Q}(n+1))\xrightarrow{a}H^{2n}(X_{i},\mathbb{Q}(n))\xrightarrow{b}H^{2n}(X_{i-1},\mathbb{Q}(n))\rightarrow
0.$

The map $b$ is the usual pullback on motivic cohomology. The map $a$ is
induced by multiplication with the Chern class of the line bundle
${\mathcal{L}}_{i}=X_{i-1}\times^{\mathbb{G}_{\textup{m}}}_{S}V({\mathcal{E}}_{\chi_{i}})$.
As $X_{r}=Y$, we have $H^{2n}(X_{r-1},\mathbb{Q}(n))\cong
H^{2n}(Y,\mathbb{Q}(n))/c_{1}({\mathcal{L}}_{r})H^{2n}(Y,\mathbb{Q}(n))$.
Hence, inductively we see that

$H^{2n}(X,\mathbb{Q}(n))\cong H^{2n}(Y,\mathbb{Q}(n))/\langle
c_{1}({\mathcal{L}}_{i})H^{2n}(Y,\mathbb{Q}(n))\rangle_{1\leq i\leq r}.$

We are left to show that
$c_{1}({\mathcal{L}}_{i})H^{2n}(Y,\mathbb{Q}(n))=c_{1}(X\times_{S}^{T}V({\mathcal{E}}_{i}))H^{2n}(X,\mathbb{Q}(n))$.
For this let us start with $i=r$. Then by construction
$X\times_{S}^{T}V({\mathcal{E}}_{\chi_{r}})\cong
X_{r-1}\times_{S}^{\mathbb{G}_{\textup{m}}}V({\mathcal{E}}_{\chi_{r}})$.
Inductively, we may replace $Y$ by $X_{i}$, where the claim again follows by
construction.

∎

######  Corollary 4.6.

Let $S=\operatorname{Spec}(k)$ be the spectrum of a field and Let
$X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks of over $S$. Then

$A^{\bullet}(X)_{\mathbb{Q}}\cong
A^{\bullet}(Y)_{\mathbb{Q}}/\hat{T}A^{\bullet}(Y)_{\mathbb{Q}}.$

###### Proof.

This follows immediately from Proposition 4.5. ∎

######  Remark 4.7.

Proposition 4.2 and Proposition 4.5 can be extended to other cohomology
theories in the following way. Let us fix a $T$-torsor $X\rightarrow Y$ of
smooth Artin $S$-stacks.

  1. (1)

(Rational étale localized cohomology theories) Let us fix a $T$-torsor
$X\rightarrow Y$ of smooth Artin $S$-stacks. Let
$M\in\textup{SH}(S)_{\mathbb{Q},\operatorname{\acute{e}t}}$ be an oriented
$E_{\infty}$-ring spectrum and let us denote its pullback to any smooth Artin
$S$-stack $Z$ with $M_{Z}$. The orientation of $M$ yields a Chern class map

$c_{1}\colon\operatorname{Pic}(Z)\rightarrow
H^{2}(Z,M(1))\coloneqq\operatorname{Hom}_{\textup{SH}(Z)}(1_{Z},M_{Z}(1)[2]).$

In the same fashion as before, for any character $\chi\in\hat{T}$, we can
define $c_{1}(L_{\chi})H^{n}(Y,M)$.

Assume there exists a $p\in\mathbb{Z}$ such that $M$ satisfies

$H^{m}(Z,M)\coloneqq\operatorname{Hom}_{\textup{SH}(Z)}(1_{Z},M_{Z}[m])=0$

for all $m>p$. Then

$H^{p}(X,M)\cong H^{p}(Y,M)/\langle
c_{1}(X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi_{i}}))H^{p}(Y,M)\rangle.$

If we take for example $M=\textup{KGL}_{\mathbb{Q},S}$, the rational
$K$-theory spectrum in $\textup{SH}(S)$, then we get

$K_{0}(X)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}\cong
K_{0}(Y)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}/\langle
c_{1}(X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi_{i}}))K_{0}(Y)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}\rangle,$

where $K_{0}(-)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}$ denotes the étale
localized rational $K$-theory.

  2. (2)

(Integral $K$-theory) The extension to integral $K$-theory is more subtle, as
we have to restrict ourselves to certain algebraic stacks131313These are
called scalloped stacks (cf. [KR21]). to make sense of the stable homotopy
category. For simplicity, we may assume that $X=\left[X^{\prime}/H\right]$ and
$Y=\left[Y^{\prime}/F\right]$ are represented by quotients of quasi-projective
schemes by diagonalizable group schemes. Then there is a well defined notion
of a stable homotopy category $\textup{SH}(X)$ resp. $\textup{SH}(Y)$ together
with a functorial $E_{\infty}$-ring object KGL that represents equivariant
$K$-theory (cf. [KR21]). Further, Bott-periodicity yields an orientation on
KGL. In particular, using that $\textup{KH}(X)$ and $\textup{KH}(Y)$ are
connective141414This holds Nisnevich locally by loc.cit. and by descent for
KH. (cf. [HK19, Thm. 5.7]), we see that Corollary 4.6 holds for integral
$K_{0}$ in this case, i.e.

$K_{0}(X)\cong K_{0}(Y)/\hat{T}K_{0}(Y).$

This result can be glued to the class of so called scalloped stacks (cf.
[KR21] for the notion of scalloped stacks and the construction of SH).

##  5 Motivic cohomology of quotients up to isogeny

We are now ready to answer the main question of this article ($\ast$ ‣ 1). Let
us assume that $S$ is connected and let us fix a reductive $S$-group scheme
$G$ and assume that it admits a maximal torus $T$.

For completeness, we will work in a more general setting than in the beginning
of the introduction.

Let $P$ resp. $Q$ be parabolics inside $G$ with Levi-components $L$ resp. $M$.
Assume that $T$ is contained in $L$. Let $\varphi\colon L\rightarrow M$ be an
isogeny. Then $L$ acts on $G$ via $(l,g)\mapsto lg\varphi(l)^{-1}$. We are
interested in the quotient of this action, which we denote by
$\left[G/_{\varphi}L\right]$ or rather its motive. To do so, we follow the
idea of Brokemper in the proof of [Bro16, Prop. 1.2].

As $\varphi$ is an isogeny the image of $T$ is again a maximal torus. In
particular, up to conjugation by an element $g_{0}\in G(S)$, we may identify
$T$ with $\varphi(T)$. The $g_{0}$-conjugation of $G$ induces an isomorphism
$G\rightarrow G$ that is $L$-equivariant, where $L$ acts on the right hand
side via $(l,g)\mapsto lgg_{0}^{-1}\varphi(l)^{-1}g_{0}$. In particular, after
replacing $M$ resp. $Q$ by their $g_{0}^{-1}$-conjugation, we may assume that
$\varphi(T)=T$. Then we have the following embedding $T\hookrightarrow T\times
T$, via $t\mapsto(t,\varphi(t))$. The quotient under this embedding is
$T\times T/T\cong T$. Thus, the naturally induced morphism
$\left[G/_{\varphi}T\right]\rightarrow\left[G/T\times T\right]$, where
$T\times T$ acts on $G$ via $(t,t^{\prime},g)\mapsto tgt^{\prime-1}$, is a
$T$-torsor. This leaves us with the following picture

${\left[G/_{\varphi}L\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$

For morphism $a$ we can apply Corollary 3.16 and see that it is Tate. The
morphism $b$ is by the above a $T$-torsor. Therefore, we can use Corollary 4.3
to see that $b$ is also Tate.

If $T$ is split, we can say more. In the split case, we have
$R\Gamma_{S}(\left[G/_{\varphi}L\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[G/_{\varphi}T\right],\mathbb{Q})^{W_{L}},$ where $W_{L}$
denotes the Weyl group of $T$ in $L$ (cf. Theorem 3.14). Further, we can
compute the motive resp. the motivic cohomology of
$\left[G/_{\varphi}T\right]$ via the motive resp. the motivic cohomology of
$\left[G/T\times T\right]$ using Proposition 4.2 and Proposition 4.5.

If $G$ admits a Borel $B$ containing $T$, then by invariance under extensions
of unipotent groups, we can identify $\operatorname{DM}(\left[G/T\times
T\right])$ with $\operatorname{DM}(\left[T\backslash G/B\right])$ (cf. Lemma
3.12). Therefore, with all of the above we see that the $T$-equivariant
motivic cohomology resp. motive of the flag variety $G/B$ yields results about
the motivic cohomology resp. the motive of $\left[G_{\varphi}/L\right]$. But
the author has shown151515Even though in the cited article, we assume that $S$
is affine, all the arguments go through in $\operatorname{DM}$ without this
assumption. in [Yay23] that the motive of $\left[T\backslash G/B\right]$ is
computed by $M_{S}(G/B)\otimes M_{S}(BT)$.

If further $S=\operatorname{Spec}(k)$, we have seen that

$M_{S}(\textup{B}T)\cong\bigotimes_{i=1}^{r}M_{S}(\textup{B}\mathbb{G}_{\textup{m}})=\bigotimes_{i=1}^{r}\bigoplus_{j\geq
0}1_{k}\langle i\rangle,$

which is completed Tate (cf. Example 2.8). As the motive of the flag variety
$G/B$ is also Tate (cf. Example 2.12), we see that $M_{S}(T\backslash G/B)$ is
completed Tate. Summarizing all of the above yields the following theorem.

######  Theorem 5.1.

We have the following diagram of Tate maps

${\left[G/_{\varphi}L\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$

If $T$ is split, we may drop the connectedness hypothesis on $S$.

Further, if $S=\operatorname{Spec}(k)$ and $T$ is split, the motives
$R\Gamma_{S}(\left[G/_{\varphi}L\right],\mathbb{Q})$ and
$M_{S}(\left[G/_{\varphi}L\right])$ are completed Tate motives in
$\operatorname{DM}(S)$. And, we can compute the Chow ring of
$\left[G/_{\varphi}L\right]$ as

$A^{\bullet}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong\left(A^{\bullet}_{T}(G/T)_{\mathbb{Q}}/\hat{T}A^{\bullet}_{T}(G/T)_{\mathbb{Q}}\right)^{W_{L}}$

###### Proof.

The first assertion is the discussion above (for the non-connected case use
Theorem 3.14). Together with the discussion above, the second resp. third
assertion follows again from the discussion above and Remark 2.13 resp.
Corollary 4.6. ∎

######  Remark 5.2.

The last isomorphism in Theorem 5.1 is also valid in the case, where $S$ is
not the spectrum of a field, after replacing the Chow groups with the $(2n,n)$
motivic cohomology group for $n\in\mathbb{Z}$ as in Proposition 4.5.

######  Remark 5.3.

If $G$ is a split reductive group with split maximal torus $T$, Brokemper has
shown that one can give a more explicit computation of the Chow ring of
$\left[G_{\varphi}/L\right]$ using the computations of Brion [Bri97] (cf.
[Bro16, Prop. 1.2]). To be more precise, we can write the last isomorphism of
Theorem 5.1 as

$A^{\bullet}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
S^{W_{L}}/(f-\varphi f\mid f\in S_{+}^{W_{G}}),$

where $S=\operatorname{Sym}_{\mathbb{Q}}(\hat{T})\cong
A^{\bullet}_{T}(\ast)_{\mathbb{Q}}$, $S_{+}$ are the elements of positive
degree and $W_{G}$ is the Weyl group of $T$ in $G$. A more detailed
computation can be found in the proof of [Bro16, Prop. 1.2].

Remark 4.7 enables us to get an analogous result of Remark 5.3 for rational
$K_{0}$.

######  Proposition 5.4.

In the setting above, let us assume that $S=\operatorname{Spec}(k)$ is the
spectrum of a field, the derived group of $G$ is simply connected and $T$ is a
split maximal torus. Then we have

$K_{0}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
R(T)_{\mathbb{Q}}^{W_{L}}/(f-\tilde{\varphi}f\mid f\in
R(T)_{\mathbb{Q}}^{W_{G}}).$

###### Proof.

For $K_{0}$, we could not produce an analogue of Corollary 3.15 using Theorem
3.14 and thus have to use a result of Krishna on equivariant $G$-theory (cf.
[Kri14, Lem. 9.2]). For completion we recall the main argument.

First, we may replace $Q$ and $M$ by $\prescript{g_{0}}{}{Q}$ and
$\prescript{g_{0}}{}{M}$ and assume that $\varphi(T)=T$. In particular,
$\tilde{\varphi}$-conjugation of $G$ by $T$ is just $\varphi$-conjugation.

Now we embed $T$ into $T\times_{S}T$ by $t\mapsto(t,\varphi(t))$. Let
$T\times_{S}T$ act on $G$ by $(t,t^{\prime}).g\coloneqq tgt^{\prime-1}$. This
yields a morphism
$\left[G/_{\varphi}T\right]\rightarrow\left[G/T\times_{S}T\right]$. This is
$T\cong T\times_{S}T/T$-torsor. Thus, by Remark 4.7, we have

$\displaystyle K_{0}(\left[G/_{\varphi}T\right])_{\mathbb{Q}}$
$\displaystyle\cong
K_{0}(\left[G/T\times_{S}T\right])_{\mathbb{Q}}/\hat{T}K_{0}(\left[G/T\times_{S}T\right])_{\mathbb{Q}}.$

By homotopy invariance, we have
$K_{0}(\left[G/T\times_{S}T\right])_{\mathbb{Q}}\cong
K_{0}^{T}(G/B)_{\mathbb{Q}}$. Therefore, we are reduced to classical
statements about $T$-equivariant $K$-theory of flag varieties (cf. [Uma13])
and get

$K_{0}(\left[G/_{\varphi}T\right])_{\mathbb{Q}}\cong
R(T)_{\mathbb{Q}}/(f-\tilde{\varphi}f\mid f\in R(T)_{\mathbb{Q}}^{W_{G}}).$

It follows161616Even though Krishna works with $G=\operatorname{GL}_{n}$ his
proof of [Kri14, Lem. 9.2] goes through in our case. from [Kri14, Lem. 9.2]
that

$K_{0}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
K_{0}(\left[G/_{\varphi}T\right])_{\mathbb{Q}}^{W_{L}}.$

Thus, it suffices to show that
$IR(T)_{\mathbb{Q}}^{W_{L}}=IR(T)_{\mathbb{Q}}\cap R(T)^{W_{L}}_{\mathbb{Q}}$,
where $I=(f-\tilde{\varphi}f\mid f\in R(T)_{\mathbb{Q}}^{W_{G}})$, but this
follows from faithfully flatness of $R(T)^{W_{L}}\hookrightarrow R(T)$ (cf.
[Ste75, Thm. 1.2] \- see also the proof of [Bro16, Prop. 1.2] resp. [Bro18,
Prop. 2.3.2] for a detailed argument in the Chow group case). ∎

We want to give two motivating examples of quotients
$\left[G/_{\varphi}L\right]$ as above, that appear naturally and apply Theorem
5.1 to see that their motives are Tate. The classifying stack of finite groups
of Lie-type and the stack of $G$-zips. Both are examples in characteristic
$p>0$. In the last section, we want to use Theorem 5.1 to give an idea how we
want to approach geometric representation theory of finite groups of Lie-type
$G^{F}$ by relating it motivically to geometric representation of the
Langlands dual of $G$ (see below for the notation).

######  Example 5.5.

Let us assume that $S=\operatorname{Spec}(\mathbb{F}_{q})$ be a finite field
of characteristic $p>0$. We set $\varphi\colon G\rightarrow G$ to be the
$q$-Frobenius. This is an isogeny and thus, we can apply Theorem 5.1 in this
setting. Further, the stack $\left[G/_{\varphi}G\right]$ is isomorphic to
$\textup{B}G^{F}$, where $G^{F}$ is the stabilizer group scheme of the neutral
element (cf. [Bro16, Lem. 2.1]). It is well known that
$G^{F}(\overline{\mathbb{F}}_{q})\cong G(\mathbb{F}_{q})$, where
$\overline{\mathbb{F}}_{q}$ denotes an algebraic closure of $\mathbb{F}_{q}$.
Thus, we see that the motive of the classifying stack of a finite group of
Lie-type is Tate. Further, we are able to relate Tate motives of
$\left[T\backslash G/B\right]$, where $B$ is a Borel containing $T$ with Tate
motives of $\textup{B}G^{F}$ via the diagram in Theorem 5.1. This answers our
main question ($\ast$ ‣ 1) of the article.

One of Brokemper’s applications of his computations is the computation of the
Chow ring of $G$-zips. In a similar fashion we will apply the above results
and show that the motive of the stack of $G$-zips over a field is completed
Tate.

######  Example 5.6.

Let $k$ be a field of characteristic $p>0$ and let $S=\operatorname{Spec}(k)$.
Let $G,P,Q$ be as above. Let us denote the unipotent radical of $P$ resp. $Q$
with $R_{u}(P)$ resp. $R_{u}(Q)$. Further, let $\varphi\colon
P/R_{u}(P)\rightarrow Q/R_{u}(Q)$ be an isogeny. The datum
${\mathcal{Z}}\coloneqq(G,P,Q,\varphi)$ is called algebraic zip-datum. To
every algebraic zip-datum like above, we can associate the group

$E_{{\mathcal{Z}}}\coloneqq\\{(p,q)\in P\times
Q\mid\varphi(\bar{p})=\bar{q}\\}.$

The group $E_{{\mathcal{Z}}}$ acts on $G$ via conjugation $(p,q).g\coloneqq
pgq^{-1}$. The quotient stack
$G\textup{-Zip}\coloneqq\left[G/E_{{\mathcal{Z}}}\right]$ is called the stack
of G-zips. There are also alternative constructions using a Tannakian
formalism on the stack of $F$-zips (cf. [PWZ15]). In op.cit. there is also an
explicit description of the points of $G\textup{-Zip}$. Let $L\subseteq P$ be
a Levi-component of $P$. Then as seen in the proof of [Bro18, Thm. 2.4.4]
there is a split exact sequence

$1\rightarrow R_{u}(P)\times R_{u}(Q)\rightarrow E_{{\mathcal{Z}}}\rightarrow
L\rightarrow 1,$

where the splitting is induced by $L\hookrightarrow E_{{\mathcal{Z}}}$,
$l\mapsto(l,\varphi(l))$. Therefore, by homotopy invariance, we have
$M_{S}(G\textup{-Zip})\simeq M_{S}(\left[G/_{\varphi}L\right])$ which is
completed Tate by Theorem 5.1 and the discussion before the theorem.

##  6 Generalizations

In this section, we want to give an overview on the integral version of
Theorem 3.14 and Theorem 5.1. We want to mention three questions that came up
naturally during the work on this article that we want to address in the
future.

  1. (1)

Under what assumptions can we transport all of these results to motives
defined via Spitzweck’s motivic cohomology ring spectrum?

  2. (2)

Is it enough to invert the residue characteristics of the base for all of our
results?

  3. (3)

Can the results of this article be extended to other cohomology theories?

Let us go a bit more into detail. So, let $S=\operatorname{Spec}(k)$ be the
spectrum of a field and $X$ be an $S$-scheme locally of finite type. further,
let $G$ be a split reductive group over $k$ with split maximal torus $T$ and
associated Weyl group $W$. Assume $G$ acts on $X$.

Question (1) is rather straightforward. For Chow groups one can see that if
$A^{\bullet}_{G}(X)\cong A^{\bullet}_{T}(X)^{W}$ holds integrally if any only
if $G$ is special. But if we assume that $G$ is special, then any $G$-torsor
is trivial Zariski-locally. If we define integral motives
$\operatorname{DM}_{\mathbb{Z}}$ on Prestacks via right Kan extension of
Spitzweck motives (cf. [Spi18]), we see that $\operatorname{DM}_{\mathbb{Z}}$
satisfies Nisnevich and in particular Zariski descent. Thus, up to
technicalities, we expect that all the arguments after Section 3.3 go through.
Crucially, Section 3.2 has to be worked out in this context, as both Ayoub and
Cisinski-Déglise use rational coefficients. This is not surprising, as one can
see that the particular motivic behavior of torsors under finite groups should
yield étale descent. We hope that in the case of special group there is a work
around.

Question (2) is addressed in a similar fashion, instead of Spitzweck motives,
we can use étale motive (cf. [CD16]). As these still satisfy étale descent
again all the arguments after Section 3.3 should go through. We expect that
inverting the residue characteristics should be enough to recover the
statements of Section 3.2 in this case. But still one needs to prove the
necessary results, which we expect to be rather straightforward.

Question (3) needs more careful treatment. The results about $T$-torsors of
Section 4 can be extended to other oriented cohomology theories, as we have
seen. Section 3.3 is more difficult as it boils down to vanishing results on
cohomology theories. The key part is Proposition 3.13. We expect that this
proposition still holds for modules over the étale $K$-theory ring spectrum
but we did not check this thoroughly. If one wants to work with genuine
$K$-theory, then this statement can not be proven via étale descent and thus
needs a more careful treatment. For other cohomology theories one can possibly
give a precise vanishing assumption to extend the results of this article.

For completion let us note, that the computation of the global sections in
Theorem 3.14 will not work for arbitrary cohomology theories. If one takes for
example Chow-Witt cohomology and $G=\operatorname{SL}_{2}$, then the Chow-Witt
ring of $\textup{B}\mathbb{G}_{\textup{m}}$ is trivial, while the Chow-Witt
ring of $\textup{B}G$ is not (cf. [Lev22]).

##  Appendix A Existence of maximal tori over Prüfer domains

The following is essentially from the authors notes of a seminar talk from
Torsten Wedhorn and we do not claim originality (cf. [Wed23]). I want to thank
him for allowing me to use his ideas.

For our main result (Theorem 5.1), we need to work with reductive group
schemes that admit maximal tori. As we mentioned in Section 6 not every
reductive group scheme admits such a maximal tori. In this short appendix, we
want to prove that a reductive group scheme over sufficiently nice base, e.g.
Dedekind domains, with quasi-split generic fibre admits a maximal torus. In
fact, to prove this, we only need to assumptions on the base. First, we will
need that the base is affine and secondly the stalks of the base are valuation
rings. The following idea was communicated to us by Torsten Wedhorn.

Rings with the property that the stalks are valuation rings were studied
before and we will give them a name according to literature.

######  Definition A.1 ([FS01]).

An integral domain $A$ is called Prüfer domain, if for any prime
${\mathfrak{p}}\subseteq A$, the localization $A_{{\mathfrak{p}}}$ is a
valuation ring.

######  Example A.2.

Any Dedekind domain is a Prüfer domain conversely, any noetherian Prüfer
domain is in fact a Dedekind domain (cf. [Sta22, 00II]). So, Prüfer domains
can be seen as the non-noetherian analog of Dedekind domains.

Another example of Prüfer domains, are so called Bézout domains, i.e. integral
domains in which every finitely generated ideal is principal. In particular,
the Picard group of a Bézout domain is trivial. This includes all PID’s and
valuation rings, but also rings such as the algebraic integers
$\overline{\mathbb{Z}}$ (cf. [DB72, pp. 243-245] for more examples).

In the following let us fix a Prüfer domain $A$ and set
$S\coloneqq\operatorname{Spec}(A)$.

######  Lemma A.3.

Let us denote $K\coloneqq\operatorname{Quot}(A)$. Further, let $X\rightarrow
S$ be a proper morphism of finite presentation. Then the map $X(A)\rightarrow
X(K)$ is bijective.

###### Proof.

Let $\sigma\in X(K)$. By the valuative criterion for properness, we can find
for any $s\in S$ a unique lift $\sigma_{s}\in X({\mathcal{O}}_{S,s})$ of
$\sigma$. As $X$ is of finite presentation, we can find an open neighborhood
$U_{s}$ of $s$ and a lift $\sigma_{U_{s}}\in X(U_{s})$ of $\sigma_{s}$. By
uniqueness of the lifts $\sigma_{s}$, the $\sigma_{U_{s}}$ glue to a unique
lift $\sigma_{A}\in X(A)$ of $\sigma$. ∎

Applying this lemma to the scheme representing Borels in a reductive $S$-group
scheme $G$, we see that if the generic fiber of $G$ is quasi-split, then $G$
is quasi-split. The existence of a maximal torus in $G$ follow now from the
fact maximal tori contained in $B$ are a $R_{u}(B)$-torsor. Writing everything
out we get the following proposition.

######  Proposition A.4.

Let $G$ be a reductive $S$-group scheme. If the generic fibre of $G$ is quasi-
split, then $G$ admits a Borel pair. Further, if $\operatorname{Pic}(A)=0$ and
the generic fibre of $G$ is split, then $G$ is split.

###### Proof.

Let $K\coloneqq\operatorname{Quot}(A)$ and $\operatorname{Bor}$ be the scheme
classifying the Borels of $G$ over $A$. As $G_{K}$ is quasi-split, there
exists an element $B_{K}\in\operatorname{Bor}(K)$. The scheme
$\operatorname{Bor}$ is proper and of finite presentation (cf. [SGA3, Exp.
XXII Cor. 5.8.3]) and thus we can apply Lemma A.3 to $\operatorname{Bor}$ and
see that $B_{K}$ lifts uniquely to a Borel $B$ of $G$. Let $U$ denote the
unipotent radical of $B$. The functor of maximal tori in $B$ is representable
by a smooth affine $S$-scheme $X$ and is further a $U$-torsor (cf. [SGA3, Exp.
XXII Cor. 5.6.13]). But $H^{1}(S,U)=0$, as $S$ is affine (cf. [SGA3, Exp. XXII
Cor. 5.9.6]), and therefore $X\rightarrow S$ admits a section.

If further $T_{K}$ is split and $\operatorname{Pic}(A)=0$, then we claim that
$G$ is split. For this let ${\bar{\eta}}$ be the geometric point corresponding
to $K$. As $S$ is integral and normal (cf. [Sta22, 00IC]), $T$ splits after
passage to a finite Galois cover of $S$ (cf. [Con15, Cor. B.3.6]). This
induces an action of $\pi_{1}^{\operatorname{\acute{e}t}}(A,{\bar{\eta}})$ on
the character group $X^{\bullet}(T)$, which is trivial if and only if $T$ is
split. As $T_{K}$ is split, the action of
$\pi_{1}^{\operatorname{\acute{e}t}}(K,{\bar{\eta}})$ on $X^{\bullet}(T_{K})$
is trivial. Since $\pi_{1}^{\operatorname{\acute{e}t}}(A,{\bar{\eta}})$ is a
quotient of $\pi_{1}^{\operatorname{\acute{e}t}}(K,{\bar{\eta}})$ (cf. [Sta22,
0BQM]), we deduce that $\pi_{1}^{\operatorname{\acute{e}t}}(A,{\bar{\eta}})$
acts trivially on $X^{\bullet}(T)$. Therefore, $T$ is split and since
$\operatorname{Pic}(A)=0$, we see that $G$ is split (cf. [SGA3, Exp. XXII
Prop. 2.2]). ∎

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