2.1 Kac–Moody root data

Let $\mathcal {D}=(X, \{\alpha_i\}_{i\in I}, \{\alpha_i^{\vee}\}_{i\in I})$ be a Kac–Moody root datum with generalized Cartan matrix $A = (a_{i,j})_{i,j \in I}$ . This means that X is a finite-rank lattice, $\alpha_i\in X$ , $\alpha_i^\vee\in X^\vee$ and $a_{i,j} = \langle \alpha_j, \alpha_i^{\vee}\rangle$ where by $\langle -,-\rangle$ we denote the evaluation pairing and $X^\vee$ the dual space of X.

By dualizing and exchanging roots and coroots, one obtains the Langlands dual Kac–Moody root datum $\widehat{\mathcal {D}}=(X^\vee, \{\alpha_i^{\vee}\}_{i\in I}, \{\alpha_i\}_{i\in I})$ with Cartan matrix $A^{\operatorname{tr}}.$

The Weyl group $W(\mathcal {D})\subset \operatorname{GL}(X)$ associated to the Kac–Moody datum is the group generated by the simple reflections $s_i$ with

$$s_i(\lambda)=\lambda-\langle \lambda,\alpha_i^\vee \rangle \alpha_i \quad \text{for }\lambda\in X.$$

We call a Kac–Moody root datum $\mathcal {D}$ free (or cofree) if $\{\alpha_i\}_{i\in I}$ (or $\{\alpha^\vee_i\}_{i\in I}$ ) is linearly independent. We say that $\mathcal {D}$ is of simply connected type (or adjoint type) if for all $i\in I$ , $\langle -,\alpha_i^\vee\rangle: X\to \mathbb{Z}$ (or $\langle \alpha_i,-\rangle: X^\vee\to \mathbb{Z}$ ) is surjective.

2.2 Kac–Moody groups

To a Kac–Moody root datum $\mathcal {D}$ one can associate a Kac–Moody group, Borel subgroup and maximal torus $G\supset B\supset T$ over k. Here, G and B are group ind-schemes and T is the split torus with (co-)character lattice $X(T)=X$ and $Y(T)=X^\vee$ , respectively. We denote by $\mathfrak g\supset \mathfrak b \supset \mathfrak h$ the corresponding Lie algebras, by $e_i,f_i\in \mathfrak g$ the Chevalley generators and by $\Delta_+\subset\Delta$ the set of (positive) roots.

We record some important properties of the Kac–Moody group. The Weyl group of the Kac–Moody group is $W(G,T)=N_G(T)/T$ . The action of W(G,T) on X yields a surjection

$$W(G,T)\twoheadrightarrow W(\mathcal {D})\subset \operatorname{GL}(X),$$

which is an isomorphism if the root datum is free. In this case, we simply write $W=W(G,T)= W(\mathcal {D})$ . For all $i\in I$ , there is a map $\varphi_i:\operatorname{SL}_2\to G$ such that $\varphi_i(\operatorname{diag}(t,t^{-1}))=\alpha_i^\vee(t)\in T$ and

$$d\varphi_i\left(\!\!\begin{pmatrix} 0\;\;\;\; & 1 \\ 0\;\;\;\; & 0 \end{pmatrix}\!\!\right)=e_i\quad \text{and}\quad d\varphi_i\left(\!\!\begin{pmatrix} 0\;\;\;\; & 0 \\ 1\;\;\;\; & 0 \end{pmatrix}\!\!\right)=f_i.$$

Denote the image of $\varphi_i$ by $G_i$ . The map $\varphi_i$ is injective if and only if $\langle -,\alpha_i^\vee\rangle: X\to \mathbb{Z}$ is surjective. This motivates our definition of root data of simply connected type in § 2.1.

The Lie algebra $\mathfrak n_+$ , generated by the $e_i$ , has a filtration by the ideals

$$\mathfrak n_+(m)=\prod_{\substack{\alpha \in \Delta_+\\ \operatorname{ht}(\alpha)\geqslant m}} \mathfrak g_\alpha.$$

The completion with respect to the filtration is a pro-Lie algebra $\hat{\mathfrak n}_+$ . Denote the associated pro-group by U. Then U has a filtration by normal subgroups U(m) with $\operatorname{Lie}(U(m))=\hat{\mathfrak n}_+(m)$ where $U/U(m)$ is a (finite-dimensional) linear algebraic group and $U=\lim_n U/U(n).$ The Borel subgroup is a semidirect product $B=T\ltimes U$ .

For each subset $J\subset I$ , there is a standard parabolic subgroup $P_J$ generated by B and $G_j$ for $j\in J$ . There is a Levi decomposition $P_J=L_J\ltimes U^J$ such that $L_J$ is the Kac–Moody group with Kac–Moody datum $\mathcal {D}_J=(X, \{\alpha_j\}_{j\in J}, \{\alpha_j^{\vee}\}_{j\in J})$ and $U^J\subset U.$

2.3 Kac–Moody flag varieties

The Kac–Moody flag variety is the quotient $G/B$ . For $w\in W$ denote the Bruhat cell $(G/B)_w=BwB/B\cong \mathbb{A}^{\ell(w)}$ . Then $G/B$ is an ind-variety filtered by closed projective subvarieties

$$(G/B)_{\leqslant n}=\bigcup_{\substack{w\in W\\ \ell(w)\leqslant n}} (G/B)_{w}$$

for $n\geqslant 0.$ The Hecke stack associated to the Kac–Moody datum $\mathcal {D}$ is the double quotient

$$B\backslash G/B.$$

We now show that this is a linearly reductive ind-pro stack. See § 3.4 for the definition of linearly reductive (ind-)pro stacks.

Note that $wB/B$ is stabilized by $U\cap wUw^{-1}\supset U(m(w))$ where

$$m(w)=\operatorname{max}\{\operatorname{ht}(\alpha)\,|\,\alpha\in \Delta_+\cap w^{-1}(\Delta_-)\}+1.$$

Note that m(w) is finite, since $|\Delta_+\cap w^{-1}(\Delta_-)|=\ell(w).$ For $n\geqslant 0$ , let

$$m(n)=\operatorname{max}\{m(w)\,|\,w\in W, \ell(w)\leqslant n\},$$

which is again finite. Then U(m(n)) stabilizes all points $wB/B$ for $\ell(w)\leqslant n.$ Since $U(m(n))\subset B$ is a normal subgroup, it acts trivially on $(G/B)_{\leqslant n}.$ Hence, the action of B on $(G/B)_{\leqslant n}$ factors through the finite-dimensional group $B/U(m(n))$ . Moreover, $\lim_{m\geqslant m(n)} B/U(m)= B$ and we see that

$$(B\backslash G/B)_{\leqslant n}=\lim_{m\geqslant m(n)} (B/U(m)\backslash (G/B)_{\leqslant n})$$

is a linearly reductive pro-stack whose transition maps are affine space bundles with fiber $U(m)/U(m+1)$ . The morphisms $(B\backslash G/B)_{\leqslant n}\to (B\backslash G/B)_{\leqslant n+1}$ are pro-closed immersions and we see that

$$B\backslash G/B=\operatorname{colim}_n (B\backslash G/B)_{\leqslant n}$$

has the structure of a linearly reductive ind-pro stack.

2.4 Affine Kac–Moody groups

Let $\mathring{\mathcal {D}}=(\mathring{X}, \{\alpha_i\}_{i\in \mathring{I}}, \{\alpha_i^{\vee}\}_{i\in \mathring{I}})$ be a classical root datum associated to an indecomposable Cartan matrix $\mathring{A}$ for $\mathring{I}=\{1,\dots,\ell\}$ . Associated to this, we obtain a reductive group $\mathring{G}\supset \mathring{B}\supset \mathring{T}$ with Borel subgroup and maximal torus.

The root datum $\mathring{\mathcal {D}}$ is free and cofree. Moreover, it is of simply connected type if the derived subgroup $[\mathring G,\mathring G]$ is simply connected and of adjoint type if the center $Z(\mathring G)$ is connected.

Let $\alpha_0=-\theta\in \mathring X$ and $\alpha_0^\vee=-\theta^\vee\in \mathring X^\vee$ , where $\theta\in \mathring \Delta$ is the highest root in the root system of $\mathring{\mathcal D}.$ We obtain the Kac–Moody root datum $\mathcal D=(\mathring X, \{\alpha_i\}_{i\in I}, \{\alpha_i^{\vee}\}_{i\in I})$ associated to the extended Cartan matrix A, for $I=\{0,\dots,\ell\}$ . If $\mathring{G}$ is simply connected, the associated Kac–Moody group $G_{\mathcal D}$ is the loop group of $\mathring{G}$ with k-points

$$G_{\mathcal D}(k)=\mathring{G}(k((t,t^{-1}))).$$

To obtain a free root datum, let $X=\mathring X\oplus \mathbb{Z} \delta$ and $X^\vee=\mathring X^\vee \oplus \mathbb{Z} d$ , where $\langle \delta, d\rangle=1$ and $\delta$ and d are zero on $\mathring X^\vee$ and $\mathring X$ , respectively. Moreover, modify the definition of the root $\alpha_0$ in $\mathcal D$ to $\alpha_0=-\theta+\delta\in X$ . The Kac–Moody root datum $\mathcal D_{\operatorname{free}}=(X, \{\alpha_i\}_{i\in I}, \{\alpha_i^{\vee}\}_{i\in I})$ is free. If $\mathring{G}$ is simply connected, the associated Kac–Moody group $G_{\mathcal D_{\operatorname{free}}}$ is the extension of the loop group by loop rotations with k-points

$$G_{\mathcal D_{\operatorname{free}}}(k)=\mathring{G}(k((t,t^{-1})))\rtimes \mathbb{G}_m(k).$$

Here, $z\in \mathbb{G}_m(k)$ acts on $P(t)\in k((t,t^{-1}))$ via $z\cdot P(t)=P(zt)$ . If $\mathring{\mathcal {D}}$ is of simply connected type then so is $\mathcal D_{\operatorname{free}}.$

Dually, to obtain a cofree root datum, let $X=\mathring X\oplus \mathbb{Z} K^*$ and $X^\vee=\mathring X^\vee \oplus \mathbb{Z} K$ , where $\langle K^*, K\rangle=1$ and $K^*$ and K are zero on $\mathring X^\vee$ and $\mathring X$ , respectively. Moreover, modify the definition of the coroot $\alpha_0^\vee$ in $\mathcal D$ to $\alpha_0^\vee=-\theta^\vee+K\in X^\vee.$ The Kac–Moody root datum $\mathcal D_{\operatorname{cofree}}=(X, \{\alpha_i\}_{i\in I}, \{\alpha_i^{\vee}\}_{i\in I})$ is cofree. If $\mathring{G}$ is simply connected, the associated Kac–Moody group $G_{\mathcal D_{\operatorname{cofree}}}$ is a central extension of the loop group of $\mathring G$ . If $\mathring{\mathcal {D}}$ is of adjoint type then so is $\mathcal D_{\operatorname{cofree}}.$

2.5 K-theory Soergel bimodules

We now explain how to attach a category of so-called K-theory Soergel bimodules to a Kac–Moody root datum $\mathcal {D}$ . Denote by $R=\mathbb{Z}[X]$ the character ring. We use the exponential notation $e^\lambda\in R$ for $\lambda \in X.$ The ring R inherits a natural action of the Weyl group $W=W(\mathcal {D})$ . For a simple reflection $s\in W$ we consider the R-bimodule

$$\operatorname{BS}(s)=R\otimes_{R^{s}}R,$$

where $R^{s}$ are the s-invariants in R.

We also record the following useful fact. For $w\in W$ , we denote by $R_w$ the twisted R-bimodule with $r\cdot m\cdot r'=rw(r')m$ for $r,r'\in R$ and $m\in R_w.$

Lemma 2.2. For $w,w'\in W$ we have $R_w\otimes_R R_{w'}=R_{ww'}$ . If $\mathcal {D}$ is free, then in the abelian category of R-bimodules we have

$$\operatorname{Hom}_{R\otimes R}(R_{w'}, R_{w}) = \begin{cases} R & \text{if } w = w', \\ 0 & \text{if } w \neq w'.\end{cases}$$

We now collect some helpful properties on K-theory Soergel bimodules. For this, let $i\in I$ , $s=s_i$ , $\alpha=\alpha_i$ and $\alpha^\vee= \alpha^\vee_i.$ The Demazure operator, see [Reference DemazureDem75], associated to s is defined by

$$D_s\colon R\to R,\, e^\lambda\mapsto \frac{e^\lambda-e^{s( \lambda)-\alpha}}{1-e^{-\alpha}}.$$

The operator is $R^s$ -linear and restricts to the identity on $R^s$ . We also consider the operator

$$D^-_s=1-D_s\colon R\to R,\, e^\lambda\mapsto \frac{e^{\lambda}-e^{s(\lambda)}}{1-e^{\alpha}}.$$

We obtain the following special case of a theorem of Pittie and Steinberg, see [Reference SteinbergSte75].

Lemma 2.3. Assume that there is a $\varpi\in X$ such that $\langle \varpi, \alpha^\vee\rangle=1$ . Then as an $R^s$ -module, $R=R^s\oplus e^\varpi R^s$ with splitting given by the isomorphism

$$(D_s, D^-_s): R\to R^s\oplus e^{-\varpi} R^s.$$

This yields a splitting of the Bott–Samelson module

$$\operatorname{BS}(s)=R\otimes_{R^s}R=(R\otimes 1)\oplus (R\otimes e^{-\varpi})$$

as a left R-module.

We now consider the map of left R-modules

$$\operatorname{gr}=(\operatorname{gr}_e,\operatorname{gr}_s): \operatorname{BS}(s)=R\otimes_{R^s}R\to R_e\oplus R_s,\, r\otimes r'\mapsto (rr', rs(r')).$$

Lemma 2.4. Assume that there is a $\varpi\in X$ such that $\langle \varpi, \alpha^\vee\rangle=1$ . Then, $\operatorname{gr}$ is injective and its image, as a left R-module, is

$$\operatorname{im}(\operatorname{gr})=R(1,1)\oplus R(e^{-\varpi},e^{-s\varpi}).$$

This implies using $\varpi-s\varpi=\alpha$ that there is the following map of short exact sequences of left R-modules.

We also make use of the following fact.

3.1 Quotient stacks

Fix a base scheme $\operatorname{pt}=\operatorname{Spec}(k)$ for an algebraically closed field k of characteristic zero. A stack $\mathcal X/k$ is linearly reductive if it can be represented as a quotient stack $X/G$ where:

1. (i) $G/k$ is a linearly reductive group;

2. (ii) $X/k$ is a G-quasi-projective scheme, that is, it admits an embedding in a projectivized representation of G, and is of finite type.

This definition includes quotients $X/G$ for linear algebraic groups $G\subset \operatorname{GL}_n$ which are not linearly reductive, since $X/G\cong X\times^G\operatorname{GL}_n/\operatorname{GL}_n$ and $\operatorname{GL}_n$ is linearly reductive.

We denote by $\operatorname{B} G={\operatorname{pt}}/G$ the classifying stack of a group G.

3.2 K-motives on stacks

For a linearly reductive stack $\mathcal X$ we denote by $\operatorname{SH}(\mathcal X)$ the stable motivic homotopy category associated to $\mathcal X$ as defined in [Reference HoyoisHoy17].

By [Reference HoyoisHoy20], there is a ring spectrum $\operatorname{KGL}_{\mathcal X}$ in $\operatorname{SH}(\mathcal X)$ representing $\mathbb A^1$ -homotopy invariant algebraic K-theory. The category of K-motives on $\mathcal X$ is defined by

$$\operatorname{DK}(\mathcal X)=\operatorname{Mod}_{\operatorname{KGL}_{\mathcal X}}(\operatorname{SH}(\mathcal X)).$$

In [Reference HoyoisHoy17, Reference Khan and RaviKR24] it is shown that there is a functor between the category of correspondences of linearly reductive stacks with quasi-projective morphisms

\begin{align*} \operatorname{Corr}(\operatorname{Stk})&\to \operatorname{Pr}: \mathcal X \mapsto \operatorname{DK}(\mathcal X),\, \big(\mathcal X\xleftarrow{f} \mathcal Z \xrightarrow{g} \mathcal Y\big)\mapsto g_!f^*.\end{align*}

Thus, there are the following functors:

1. (i) $f^*,f_*,f_!,f^!$ for quasi-projective morphisms;

2. (ii) and bifunctors $\mathcal {H}{\operatorname{om}},\otimes.$

These fulfill the usual axioms of a six functor formalism, such as base change, localization and projection formulae. We note that for f smooth, there is an equivalence $f^*\simeq f^!$ . This is related to Bott periodicity for K-theory.

The category of K-motives computes algebraic K-theory and G-theory. For a stack $\mathcal X$ denote by $K(\mathcal X)$ and $G(\mathcal X)$ the K-theory spectra of the category of perfect complexes and coherent sheaves, respectively. If $\mathcal X/k$ is a smooth linearly reductive stack and $f:\mathcal Y\to\mathcal X$ is a quasi-projective map of linearly reductive stacks, there are equivalences of spectra (in the first case of ring spectra)

(1) \begin{align} \operatorname{Map}_{\operatorname{DK}(\mathcal X)}(\mathbb{1}, \mathbb{1}) &\simeq \operatorname{K}(\mathcal X) ,\end{align}

(2) \begin{align}\operatorname{Map}_{\operatorname{DK}(\mathcal Y)}(\mathbb{1},f^!\mathbb{1})& \simeq \operatorname{G}(\mathcal Y),\end{align}

see [Reference HoyoisHoy20, below Definition 5.1] and [Reference HoyoisHoy20, Remark 5.7].

3.3 Reduced K-motives

Since we are concerned with the Hecke category of a split reductive/Kac–Moody group, which is already defined over the integers, the constructions carried out here should be insensitive to the choice of base scheme $\operatorname{pt}=\operatorname{Spec}(k).$ In particular, the higher algebraic K-theory $K_{\gt 0}(\operatorname{pt})$ is irrelevant for us.

To remove the higher K-groups, we pass to the category of reduced K-motives

$$\operatorname{DK}_{{\operatorname{r}}}(\mathcal X)=\operatorname{DK}(\mathcal X)\otimes_{\operatorname{K}(\operatorname{pt})}\operatorname{K}_{0}(\operatorname{pt})$$

which is the Lurie tensor product of $\operatorname{DK}(\mathcal X)$ over the K-theory spectrum $\operatorname{K}(\operatorname{pt})$ with $\operatorname{K}_0(\operatorname{pt})=\mathbb{Z}$ . Here, the action of $\operatorname{K}(\operatorname{pt})$ on $\operatorname{DK}(\mathcal X)$ arises in the following way. We use that $\mathcal X$ is a linear reductive stack and hence of the form $X/G$ and denote by $f:X/G\to \operatorname{B} G$ the projection, which is quasi-projective by assumption. Then we obtain a pullback map $\operatorname{K}(\operatorname{pt})\to \operatorname{K}(\operatorname{B} G)=\operatorname{Map}_{\operatorname{DK}(\operatorname{B} G)}(\mathbb{1},\mathbb{1})\stackrel{f^*}{\to} \operatorname{Map}_{\operatorname{DK}(\mathcal X)}(\mathbb{1},\mathbb{1})$ . The latter ring spectrum naturally acts on $\operatorname{DK}(\mathcal X)$ .

The precise definition is explained in [Reference Eberhardt and ScholbachES23] in the context of motivic sheaves $\operatorname{DM}$ and in [Reference EberhardtEbe24a, 2.3] for K-motives $\operatorname{DK}$ . There it is shown that $\operatorname{DK}_{{\operatorname{r}}}$ inherits the six functor formalism from $\operatorname{DK}.$ Furthermore, $\operatorname{DK}_{{\operatorname{r}}}$ computes reduced K-theory and reduced G-theory

(3) \begin{align} \operatorname{Map}_{\operatorname{DK}_{{\operatorname{r}}}(\mathcal X)}(\mathbb{1},\mathbb{1}) &\simeq \operatorname{K}(\mathcal X)_{\operatorname{r}} = \operatorname{K}(\mathcal X)\otimes_{\operatorname{K}(\operatorname{pt})}\operatorname{K}_{0}(\operatorname{pt}) ,\end{align}

(4) \begin{align}\operatorname{Map}_{\operatorname{DK}_{{\operatorname{r}}}(\mathcal Y)}(\mathbb{1},f^!\mathbb{1})& \simeq \operatorname{G}(\mathcal Y)_{\operatorname{r}} = \operatorname{G}(\mathcal Y)\otimes_{\operatorname{K}(\operatorname{pt})}\operatorname{K}_{0}(\operatorname{pt}),\end{align}

for $\mathcal X/k$ smooth and $f:\mathcal Y\to\mathcal X$ a quasi-projective map of linearly reductive stacks. Reduced G-theory is well-behaved for sufficiently ‘cellular’ stacks.

The categories of (reduced) K-motives $\operatorname{DK}(\mathcal X)$ and $\operatorname{DK}_{{\operatorname{r}}}(\mathcal X)$ are too large for our purposes, since they, for example, contain the information about the (reduced) G-theory of $\mathcal Y$ for all quasi-projective maps $f:\mathcal Y\to \mathcal X$ . For this reason, we will often restrict our attention to the stable subcategories generated by the tensor unit $\mathbb{1}$ which we think of as categories of locally constant objects.

1. (i) For $\mathcal X=\operatorname{B} U$ , where U is a unipotent group, the stable subcategory generated by the constant object

   $$\operatorname{DK}_{{\operatorname{r}}}(\operatorname{B} U)\supset \langle \mathbb{1} \rangle_{\operatorname{stb}}\stackrel{\sim}{\to}\operatorname{D}^b(\mathbb{Z})$$
   is equivalent to the bounded derived category of $\mathbb{Z}$ -modules. This follows from the fact that $K_0(\operatorname{B} U) \cong \mathbb{Z}$ .

2. (ii) For $\mathcal X=\operatorname{B} T$ , where T is a torus, the stable subcategory generated by the constant object

   $$\operatorname{DK}_{{\operatorname{r}}}(\operatorname{B} T)\supset \langle \mathbb{1} \rangle_{\operatorname{stb}}\stackrel{\sim}{\to}\operatorname{D}^b(\mathbb{Z}[X(T)])$$
   is equivalent to the bounded derived category of $\mathbb{Z}[X(T)]$ -modules, where X(T) is the character lattice of T. Here, we use that $K_0(\operatorname{B} T)=\mathbb{Z}[X(T)].$

3.4 Extension to certain ind-pro-stacks

We now explain how to extend the formalism to certain ind-pro-stacks, keeping the Hecke stack $B\backslash G/B$ of a Kac–Moody group in mind.

We can extend the formalism of K-motives to these stacks in the following way.

Definition 3.5. The category of (reduced) K-motives on a linearly reductive pro-stack $\mathcal X=\lim_{m\geqslant 0} \mathcal X_m$ is defined as the limit

$$\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)=\lim_{m}{}_{\hspace{-0.1em}*}\operatorname{DK}_{({\operatorname{r}})}(\mathcal X_m)$$

along the $*$ -pushforwards of the transition maps.

Lemma 3.7. For a linearly reductive pro-stack $\mathcal X=\lim_{m\geqslant 0} \mathcal X_m$ , there is an equivalence

$$\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)=\lim_{m}{}_{\hspace{-0.1em}*}\operatorname{DK}_{({\operatorname{r}})}(\mathcal X_m)\simeq \underset{m}{\operatorname{colim}}^!\operatorname{DK}_{({\operatorname{r}})}(\mathcal X_m)$$

with the colimit in $\operatorname{Pr}$ along the $!$ -pullback maps. Moreover, $\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)$ is compactly generated and inherits the functors $\otimes$ , $\mathcal {H}{\operatorname{om}}$ and for pro-quasi-projective maps $f:\mathcal X\to \mathcal Y$ the functors $f_*$ , $f^!$ and $f^*$ .

Proof. The first statement follows since there is an adjunction

$$p_m^*=p_m^!\colon \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_{m-1})\leftrightarrows \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_{m}): p_{m,*}$$

and the comparison of limits and colimit, see [Reference LurieLur09, 5.5.3.3]. The existence of the functors follows from the compatibility with the (co)limits.

To also define a $!$ -pushforward functor, we need the following restriction.

Using base change, we obtain the following.

We also use the following statement that is relevant for quotients by pro-unipotent groups.

Lemma 3.10. Let $\mathcal X=\lim_{m\geqslant 0} \mathcal X_m$ be a linearly reductive pro-stack where we assume additionally that the transition maps $p_m$ are torsors over vector bundles. In this case $p_m^!=p_m^*$ is fully faithful and hence the natural insertion map

$$\operatorname{ins}_0: \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_0)\to \operatorname{DK}_{({\operatorname{r}})}(\mathcal X)$$

is fully faithful as well.

Now we extend our formalism to ind-pro-stacks.

Definition 3.12. The category of (reduced) K-motives on a linearly reductive ind-pro-stack $\mathcal X=\operatorname{colim} \mathcal X_n$ is defined as the colimit in $\operatorname{Pr}$

$$\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)=\underset{m}{\operatorname{colim}}{}_{\hspace{0.05em}*}\operatorname{DK}_{({\operatorname{r}})}(\mathcal X_m)$$

along the $*$ -pushforward maps.

Lemma 3.14. For a linearly reductive ind-pro-stack $\mathcal X=\operatorname{colim} \mathcal X_n$ , there is an equivalence of categories

$$\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)=\underset{m}{\operatorname{colim}}{}_{\hspace{0.05em}*}\operatorname{DK}_{({\operatorname{r}})}(\mathcal X_m)\simeq \lim_{m}{}^{\hspace{-0.1em}!}\operatorname{DK}_{({\operatorname{r}})}(\mathcal X_m)$$

with the limit along the $!$ -pullbacks. Moreover, $\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)$ inherits the functors $\otimes$ , $\mathcal {H}{\operatorname{om}}$ and for ind-pro-quasi-projective morphisms $f:\mathcal X\to \mathcal Y$ the functors $f_*$ , $f^!.$

Proof. This works as in Lemma 3.7 using that there is an adjunction

\begin{align*}\iota_{n,*}&=\iota_{n,!}\colon \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_{n})\leftrightarrows \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_{n+1}): \iota_{n}^!.\end{align*}

Similarly to Lemma 3.10 we obtain the following.

Lemma 3.17. If in a linearly reductive ind-pro-stack $\mathcal X=\operatorname{colim} \mathcal X_n$ the transition maps $\iota_n$ are pro-closed immersions, $\iota_{n,*}$ is fully faithful and hence the insertion maps

$$\operatorname{ins}_n: \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_n)\to \operatorname{DK}_{({\operatorname{r}})}(\mathcal X)$$

are fully faithful.

In summary, let $\mathcal X=\operatorname{colim}_n \lim_m \mathcal X_{n,m}$ be an ind-pro stack with a presentation of the form:

1. (i) $\mathcal X_{n,m}$ is a linearly reductive stack;

2. (ii) $p_{n,m}: \mathcal X_{n,m}\to \mathcal X_{n,m-1}$ is a torsor under a vector bundle; and

3. (iii) $\iota_{n,m}:\mathcal X_{n,m}\to \mathcal X_{n+1,\varphi(n,m)}$ is a closed immersion.

Then we can consider the category of reduced K-motives $\operatorname{DK}_{({\operatorname{r}})}(\mathcal X)$ on $\mathcal X$ and for each n there are fully faithful insertion functors

$$\operatorname{ins}_n\operatorname{ins}_0: \operatorname{DK}_{({\operatorname{r}})}(\mathcal X_{0,n})\to \operatorname{DK}_{({\operatorname{r}})}(\mathcal X).$$

For most purposes, we can hence restrict ourselves to considering K-motives on the linearly reductive stacks $\mathcal X_{0,n}.$

4.1 K-motives on the Hecke stack

Recall that k is an algebraically closed field of characteristic 0 and $\operatorname{pt}=\operatorname{Spec}(k)$ . Denote by $G\supset B\supset T$ the Kac–Moody group together with a Borel subgroup and maximal torus associated to the Kac–Moody datum $\mathcal {D}$ , see §§ 2.1 and 2.2. Denote by $W\supset S$ the Weyl group with the set of simple reflections.

The Hecke stack $B\backslash G/B$ is ind-pro-linearly reductive, see § 2.3. We can hence consider the category of reduced K-motives $\operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B)$ , see § 3.4. For $w\in W$ , we denote by $i_{w}: B\backslash BwB/B\to B\backslash G/B$ the inclusion. We denote by

$$\Delta_w=i_{w,!}\mathbb{1}, \nabla_w=i_{w,*}\mathbb{1}\in \operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B)$$

the corresponding standard and costandard objects.

Definition 4.1. The K-theoretic Hecke category is the stable subcategory generated by the standard objects

$$\mathcal {H}=\mathcal {H}^K_{\mathcal D}=\left\langle\Delta_w\,|\,w\in W\right\rangle_{\operatorname{stb}}\subset \operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B).$$

4.2 Monoidal structure

The Hecke stack is a monoid in the category of correspondences of ind-pro-linearly reductive stacks with multiplication given by the following convolution diagram.

(5)

The maps $p_1,p_2$ and m are ind-pro-proper with fiber $G/B.$ The inclusion map $i_e: B\backslash B /B\to B\backslash G/B$ is the unit. This turns $\operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B)$ into a monoidal category: There are two monoidal products defined by

$$A* B = m_*(p_1^*A\otimes p_2^*B)=m_!(p_1^*A\otimes p_2^*B)$$

and

$$ A*^! B = m_*(p_1^!A\otimes p_2^!B)=m_!(p_1^!A\otimes p_2^!B)$$

for $A,B\in \operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B)$ . For both products, the unit object is $i_{e,*}\mathbb{1}$ .

It will turn out that $*$ and $*^!$ are equivalent when restricted to the Hecke category $\mathcal {H}$ , see Theorem 4.17. This is immediate if G is a reductive group, since then $G/B$ is smooth and $p_i^*\cong p_i^!$ for $i=1,2.$

We now show that the monoidal structures restrict to $\mathcal {H}.$ For $w\in W$ , denote

$$\mathcal H_{\leqslant w}=\left\langle\Delta_x\,|\,x\in W, x\leqslant w\right\rangle_{\operatorname{stb}}\subset \mathcal {H}.$$

For $s\in S,$ let $P_s=B\cup BsB$ be the corresponding standard parabolic and $k_s:B\backslash P_s/B\to B\backslash G/B$ the inclusion. Denote $E_s=k_{s,!}\mathbb{1}_s$ and $E_e=\Delta_e.$

1. (i) $\Delta_x*\Delta_y=\Delta_{xy}$ and $\nabla_x*^!\nabla_y=\nabla_{xy}$ if $\ell(xy)=\ell(x)+\ell(y);$

2. (ii) $\left\langle E_e,E_s\right\rangle=\left\langle\Delta_e,\Delta_s\right\rangle=\left\langle\nabla_e,\nabla_s\right\rangle=\mathcal H_{\leqslant s};$

3. (iii) $E_s* E_s=E_s*^! E_s\cong E_s \oplus E_s$ ; and

4. (iv) $\mathcal H_{\leqslant s}* \mathcal H_{\leqslant s}=\mathcal H_{\leqslant s}*^! \mathcal H_{\leqslant s}\subset \mathcal H_{\leqslant s}.$

Part (ii) follows from the localization cofiber sequences

induced by the decomposition $P_s=B\cup BsB$ .

Part (iii) follows from the projective bundle formula applied to $m: B\backslash P_s\times^B P_s/B\to B\backslash P_s/B$ which has fiber $P_s/B\cong \mathbb{P}^1.$

Part (iv) follows from parts (ii) and (iii) and the fact that $\Delta_e=E_e$ is the unit for convolution.

We also need the following statement for standard parabolic subgroups.

4.3 Pure objects

We define the category of pure objects in $\mathcal {H}$ as the additive, idempotent-closed monoidal (with respect to $*$ ) subcategory

generated by the objects $E_s.$ For $\underline{x}=(s_1,\dots,s_n)\in S^n$ we abbreviate the associated Bott–Samelson K-motive by

$$E_{\underline{x}}=E_{s_1}*\cdots* E_{s_n}.$$

Lemma 4.6. For a sequence $\underline{x}=(s_1,\dots,s_n)\in S^n,$ denote

$$\pi_{\underline{x}}:\operatorname{BS}_{\underline{x}}=B\backslash P_{s_1}\times^B\cdots\times^BP_{s_n}/B\to B\backslash G/B.$$

Then there are isomorphisms

$$E_{\underline{x}}=E_{s_1}*\cdots* E_{s_n}\cong\pi_{\underline{x},!}\mathbb{1}=\pi_{\underline{x},*}\mathbb{1}\cong E_{s_1}*^!\cdots*^! E_{s_n}.$$

Corollary 4.8. The category of pure objects is generated by the Bott–Samelson objects $E_{\underline{x}}$ as an additive and idempotent closed subcategory and one may replace $*$ by $*^!$ in its definition

Corollary 4.9. Pure objects stably generate the Hecke category:

$$\left\langle\mathcal {H}_{\operatorname{pure}}\right\rangle_{\operatorname{stb}}=\mathcal {H}.$$

4.4 Pointwise purity and formality

The pure objects in the Hecke category fulfill an additional pointwise purity property which allows to show a formality result. The following definitions and statements are a variation of [Reference Soergel and WendtSW18, § 6].

There is the following equivalent description of pointwise pure objects.

for $w_k\in W$ and $k=1,\dots n.$ Dually, pointwise $*$ -pure objects are precisely the objects that admit a cofiltration whose associated graded are standard objects.

By assumption, $i_{w_k}^! E$ is a finite direct sum of constant objects $\mathbb{1}$ , so $i_{w_k,*}i_{w_k}^! E$ is a finite direct sum of costandard objects $\nabla_{w_k}=i_{w_k,*}\mathbb{1}$ . The desired filtration can hence be obtained by refinement.

Now assume that E admits a filtration by objects $E^k$ whose associated graded are costandard objects. We show that E is pointwise $!$ -pure by induction on the length of the cofiltration. Let $w\in W$ and consider the following cofiber sequence:

By induction, $E^{k-1}$ is pointwise $!$ -pure, so $i_w^!E^{k-1}$ is a finite direct sum of objects $\mathbb{1}$ . If $w\neq w_k$ , then $i_w^!\nabla_{w_k}=0$ and, hence, $i_w^!E^k=i_w^!E^{k-1}$ is also a finite direct sum of objects $\mathbb{1}$ . If $w=w_k$ , then $i_w^!\nabla_{w_k}=\mathbb{1}$ . Since

$$\operatorname{Hom}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash BwB/B)}(\mathbb{1},\mathbb{1}[1])=\pi_{-1}(K(B\backslash BwB/B)\otimes_{K(\operatorname{pt})}K_0(\operatorname{pt}))=0,$$

the cofiber sequence splits and $i_w^!E^k=i_w^!E^{k-1}\oplus \mathbb{1}$ is also a finite direct sum of objects $\mathbb{1}$ . Hence, $E^k$ is pointwise $!$ -pure.

The statement for pointwise $*$ -pure objects follows by dual arguments.

An important consequence is the following formality of mapping spaces of pointwise pure objects.

Proof. By base change and (3) we see that

\begin{align*} \operatorname{Map}(\Delta_x, \nabla_y)&=K(B\backslash (BxB\cap ByB)/B)\otimes_{K(\operatorname{pt})}K_0(\operatorname{pt})\\ &=K_0(B\backslash (BxB\cap ByB)/B) \end{align*}

is concentrated in degree zero for all $x,y\in W.$ The general statement follows by a double induction on the length of a cofiltration by standard modules of M and filtration by costandard modules of N (see Lemma 4.11) using the five lemma.

We now show that pure objects in the Hecke category are also pointwise pure.

Similar to the proof of Lemma 4.11 there is a filtration of $E_{\underline{x}}$ with associated graded of the form $i_{w,*}i_w^!\pi_{\underline{x},*}\mathbb{1}.$ The fiber $\pi_{\underline{x}}^{-1}(B\backslash BwB /B)$ admits a stratification whose fibers are affine bundles over $B\backslash BwB /B$ , see [Reference HainesHai, Reference HaerterichHae04]. Using this, we can refine the filtration such that the associated graded pieces are of the form $i_{w,*}p_*\mathbb{1}=\mathbb{1}$ where p is an affine bundle over $B\backslash BwB /B.$ Hence, the filtration has the desired form.

Since pure objects stably generate the Hecke category, see Corollary 4.9, with Proposition A.4 and Lemma 4.12 we obtain the following formality result for the K-theoretic Hecke category.

Corollary 4.14. There is an equivalence of monoidal categories between the K-theoretic Hecke category and the category of bounded chain complexes of pure objects in $\mathcal {H}$ :

$$\mathcal {H}\stackrel{\sim}{\to}\operatorname{Ch}^b(\mathcal {H}_{\operatorname{pure}}).$$

4.5 Duality

Our next goal is to show that the Hecke category $\mathcal {H}$ is rigid. For $M\in \operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B)$ we define

\begin{align*} \mathbb{D}(M)=\mathcal {H}{\operatorname{om}}(M,\omega)\quad \text{ and }\quad \mathbb{D}^{-}(M)=\mathbb{D}(\operatorname{inv}^*M),\end{align*}

where $\operatorname{inv}$ is the inversion map of G and $\omega=p^!(\mathbb{1})$ and $p :B\backslash G/B\to B\backslash {\operatorname{pt}}/B$ is the projection.

We collect some properties of the duality functors.

1. (i) There are isomorphism $\mathbb{D}(E_{\underline{x}})\cong E_{\underline{x}}$ and $\mathbb{D}^{-}(E_{\underline{x}})\cong E_{\underline{x}^{\operatorname{op}}}$ .

2. (ii) We have that $\mathbb{D}$ and $\mathbb{D}^{-}$ preserve $\mathcal H$ and $\mathbb{D}^2\cong \operatorname{id}\cong (\mathbb{D}^{-})^2$ on $\mathcal H.$

3. (iii) There is a natural map $\mathbb{D}(A)*^!\mathbb{D}(B)\to \mathbb{D}(A* B)$ that restricts to an isomorphism on $\mathcal H.$

4. (iv) There is a natural map $\mathbb{D}(A*^! B)\to \mathbb{D}(A)*\mathbb{D}(B)$ that restricts to an isomorphism on $\mathcal H.$

5. (v) There is a natural map $\mathbb{D}^-(A* B)\to \mathbb{D}^-(B)*^!\mathbb{D}^-(A)$ that restricts to an isomorphism on $\mathcal H.$

Part (ii) follows from part (i) using that $\mathcal {H}_{\operatorname{pure}}$ stably generates $\mathcal H.$

Parts (iii) and (iv) follow since $\mathbb{D}$ exchanges $!$ and $*$ and $\mathbb{D}^2=\operatorname{id}$ by part (ii). Similarly, part (v) follows by keeping track of the extra $\operatorname{inv}^*$ .

Lemma 4.16. Let $A,B,C\in\mathcal {H}$ , then there are natural isomorphisms

$$\operatorname{Hom}(A* B, C)=\operatorname{Hom}(A,C*^! \mathbb{D}^{-}(B))=\operatorname{Hom}(B,\mathbb{D}^{-}(A)*^! C).$$

For $A,B\in \mathcal {H}$ , using $\mathbb{D}^2=\operatorname{id}$ , there is a natural isomorphism

$$\operatorname{Hom}(A,B)\cong \operatorname{Hom}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash {\operatorname{pt}}/B)}(p_!(A\otimes \mathbb{D}(B)),\mathbb{1}).$$

Using this and Lemma 4.15 we can hence write

$$ \operatorname{Hom}(A* B, C)=\operatorname{Hom}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash {\operatorname{pt}}/B)}(p_!(A* B\otimes \mathbb{D}(C)),\mathbb{1}) $$

and

\begin{align*} \operatorname{Hom}(A,C*^! \mathbb{D}^{-}(B))&=\operatorname{Hom}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash {\operatorname{pt}}/B)}(p_!( A\otimes \mathbb{D}(C*^! \mathbb{D}^{-}(B))),\mathbb{1})\\&=\operatorname{Hom}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash {\operatorname{pt}}/B)}(p_!(A\otimes \mathbb{D}(C)* \operatorname{inv}^*(B)),\mathbb{1}).\end{align*}

Replacing $\mathbb{D}(C)$ by C, we hence need to show that there is a natural isomorphism

$$p_!((A* B)\otimes C)\cong p_!(A\otimes (C* \operatorname{inv}^*(B))).$$

Using that $m_!=m_*$ as well as the projection formula, we get

$$(A* B)\otimes C =m_*(p_1^*A\otimes p_2^*B\otimes m^*C)$$

and

$$A\otimes (C* \operatorname{inv}^*(B)) =m_*(m^*A\otimes p_1^*C\otimes p_2^*\operatorname{inv}^*B).$$

Consider the isomorphism $r:B\backslash G\times^B G/B, [x,y]\mapsto [xy,y^{-1}]$ . Then $p_1r=m$ , $p_2r=\operatorname{inv}p_2$ and $mr=p_1$ . Hence,

$$r^*(p_1^*A\otimes p_2^*B\otimes m^*C)=m^*A\otimes p_1^*C\otimes p_2^*\operatorname{inv}^*B.$$

Using $r_*r^*=\operatorname{id}$ , $pmr=pm$ , $m_!=m_*$ and $r_!=r_*$ , we obtain

\begin{align*} p_!((A* B)\otimes C)&=p_!m_*(p_1^*A\otimes p_2^*B\otimes m^*C)\\ &=p_!m_*r_*r^*(p_1^*A\otimes p_2^*B\otimes m^*C)\\ &=p_!m_*(m^*A\otimes p_1^*C\otimes p_2^*\operatorname{inv}^*B)\\ &=p_!(A\otimes (C* \operatorname{inv}^*(B))).\end{align*}

Proof. To show this we make use of [Reference Boyarchenko and DrinfeldBD13]. In the notation of [Reference Boyarchenko and DrinfeldBD13], the object $\Delta_e$ is dualizing in the category $\mathcal {H}$ with the monoidal structure given by the $*$ -convolution. The duality functor is $\mathbb{D}^-$ using Lemma 4.16 where we use that $\mathbb{D}^-$ is an anti-equivalence by Lemma 4.15. By [Reference Boyarchenko and DrinfeldBD13, 3.1] and using Lemma 4.15, there is a natural map

$$X* Y\to \mathbb{D}^-(\mathbb{D}^-Y* \mathbb{D}^-X)=X*^!Y.$$

The objects $E_s$ are left and right dualizable by Lemma 4.16 using that on $B\backslash P_s/B$ we have $*=*^!$ by Lemma 4.5. By Corollary 4.8, all objects in $\mathcal {H}_{\operatorname{pure}}$ are left and right dualizable as well. This implies by [Reference Boyarchenko and DrinfeldBD13, Lemma 3.4], that the natural map between $*$ and $*^!$ is an isomorphism on $\mathcal {H}$ . The statement now follows from [Reference Boyarchenko and DrinfeldBD13, Corollary 3.6].

4.6 Erweiterungssatz

We now give a Soergel description for the pure objects in the Hecke category. Let $R=\mathbb{Z}[X(T)]$ . Then by Lemma 3.2 we can identify

$$\operatorname{Map}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash{\operatorname{pt}}/B)}(\mathbb{1},\mathbb{1})=K_0(T\backslash{\operatorname{pt}}/T)=R\otimes R.$$

Consider the functor

$$\mathbb{K}: \operatorname{DK}_{{\operatorname{r}}}(B\backslash G/B)\to \operatorname{D}(R\otimes R),\, \mathbb{K}(M)=\operatorname{Map}_{\operatorname{DK}_{{\operatorname{r}}}(B\backslash{\operatorname{pt}} /B)}(\mathbb{1},p_*M).$$

Proof. The functor $p_*$ is lax monoidal when we equip $\operatorname{DK}_{{\operatorname{r}}}(B\backslash{\operatorname{pt}} /B)$ with the monoidal structure $\otimes$ . To see this, we use that there is a natural map

\begin{align*} p_*(-)\otimes p_*(-)&\to p_*p_{1,*}p_1^*(-)\otimes p_*p_{2,*}p_2^*(-)\\ &=\overline{p}_*p_1^*(-)\otimes \overline{p}_*p_2^*(-)\\ &\to \overline{p}_*(p_1^*(-)\otimes p_2^*(-))\\ &= p_*m_*(p_1^*(-)\otimes p_2^*(-))\\ &\stackrel{\sim}{\to} p_*m_!(p_1^*(-)\otimes p_2^*(-))=p_*(-* -) \end{align*}

where we use the notation of (5) and denote $\overline{p}: B\backslash G\times^B G/B\to B\backslash{\operatorname{pt}} /B.$ We then use that $\operatorname{Map}(\mathbb{1},-)$ is lax monoidal.

We now show that the functor $\mathbb{K}$ evaluated at Bott–Samelson K-motives ‘computes’ the K-theory of Bott–Samelson resolutions.

Lemma 4.20. Let $\underline{x}\in S^n$ , then there is an equivalence of R-bimodules

$$\mathbb{K}(E_{\underline{x}})\simeq K_0(\operatorname{BS}_{\underline{x}}).$$

Under an appropriate assumption on the root datum, the functor $\mathbb{K}$ is actually monoidal, when we equip $\operatorname{D}(R\otimes R)$ with the monoidal structure $-\otimes_R -$ .

1. (i) The lax monoidal structure yields an isomorphism

   $$\mathbb{K}(E_s* E_{\underline{x}})\stackrel{\sim}{\leftarrow}\mathbb{K}(E_s)\otimes_R\mathbb{K}(E_{\underline{x}})=R\otimes_{R^s}\mathbb{K}(E_{\underline{x}}).$$

2. (ii) When restricted to $\mathcal {H}$ , the functor $\mathbb{K}$ is monoidal, where $\mathcal {H}$ and $\operatorname{D}(R\otimes R)$ are equipped with the monoidal structures $*$ and $\otimes_R$ , respectively.

3. (iii) We have $\mathbb{K}(\nabla_w)=R_w$ and $\mathbb{K}(E_s)=R\otimes_{R^s}R$ .

Part (ii) is true when restricting $\mathbb{K}$ to the subcategory $\mathcal {H}_{\operatorname{pure}}$ by induction and part (i). Now use that $\mathcal {H}_{\operatorname{pure}}$ generates $\mathcal {H}$ as a stable category.

Part (iii) is shown in [Reference EberhardtEbe24b, § 4.4].

We are now ready to prove a K-theoretic version of Soergel’s Erweiterungssatz [Reference SoergelSoe90] for Kac–Moody groups. See [Reference EberhardtEbe24b] for the case of reductive groups.

Proof. Let $\underline{x}$ and $\underline{y}$ be sequences of simple reflections. It suffices to show that $\mathbb{K}$ induces an isomorphism

$$\operatorname{Hom}_{\mathcal {H}}(E_{\underline{x}}, E_{\underline{y}})\to\operatorname{Hom}_{\operatorname{D}(R\otimes R)}(\mathbb{K}(E_{\underline{y}}),\mathbb{K}(E_{\underline{x}})).$$

Since $\mathbb{K}$ is monoidal by Theorem 4.21 and the object $E_{\underline{y}}$ is left and right dualizable with dual $E_{\underline{y}^{op}}$ by Lemma 4.15 and Theorem 4.17, we obtain the commutative square

where $\underline{z}$ is the concatenation of $\operatorname{op}(\underline{x})$ and $\underline{y}$ . We hence need to show that the bottom vertical map in the commutative square is an isomorphism.

By Lemma 4.13, the object $E_{\underline{z}}$ is pointwise $!$ -pure. By Lemma 4.11, the object $E_{\underline{z}}$ admits a filtration whose associated graded are costandard objects, that is, there are objects $E^{k}\in \mathcal {H}$ for $k=0,\dots,n$ with $E^0=0$ and $E^n=E_{\underline{z}}$ and cofiber sequences of the form

for some $w_k\in W$ . We show by induction on k that the map

is an isomorphism for all k. Consider the maps of exact sequences associated to the above cofiber sequence

where we abbreviate $w=w_k$ and the morphisms in the top row are in $\mathcal {H}$ and those in the bottom row are in $\operatorname{D}(R\otimes R)$ . We now explain why the diagram has the claimed form and, hence, by the five lemma the middle vertical arrow is an isomorphism.

The left vertical arrow is an isomorphism by induction. For the right vertical arrow, we consider two cases. If $w=e$ , both source and target are isomorphic to R and the map is clearly an isomorphism. If $w\neq e$ , the source of the arrow is zero by base change since $\Delta_e=i_{e,!}\mathbb{1}$ and $\nabla_w=i_{w,*}\mathbb{1}$ and $B\cap BwB=\emptyset$ . Moreover, the target of the arrow is zero since $\mathbb{K}(\Delta_e)=R$ , $\mathbb{K}(\nabla_w)=R_w$ and $\operatorname{Hom}_{R\otimes R}(R,R_w)=0$ by Lemma 2.2. In this step we use that the Kac–Moody datum $\mathcal {D}$ is free.

The left-most term in the bottom row is $\operatorname{Hom}(R,R_{w}[-1])=0$ , since both R and $\operatorname{Hom}(R,R_{w})$ are in the heart of the standard t-structure on $\operatorname{D}(R\otimes R)$ . The right-most term in the top row is $\operatorname{Hom}(\Delta_e,E^{k-1})=0$ using Lemma 4.12 and that $E^{k-1}$ is pointwise $!$ -pure and $\Delta_e$ is pointwise $*$ -pure.

Corollary 4.23. Assume that the Kac–Moody datum $\mathcal D$ is free and of simply connected type. Then the functor $\mathbb{K}$ yields a monoidal equivalence between pure objects in the K-theoretic Hecke category and K-theory Soergel bimodules

$$\mathcal {H}_{\operatorname{pure}}\to \operatorname{SBim}^K_{\mathcal D}$$

mapping $E_s$ to $R\otimes_{R^s}R.$

4.7 Soergel-theoretic description of the K-theoretic Hecke category

By combining the Erweiterungssatz, see Theorem 4.22 and Corollary 4.23, and the formality result, see Corollary 4.14, we obtain the following ‘combinatorial’ description of the K-theoretic Hecke category.

Theorem 4.25. Assume that the Kac–Moody datum $\mathcal {D}$ is free and of simply connected type, then there is an equivalence of monoidal categories

$$\mathcal {H}^K_{\mathcal D}\to \operatorname{Ch}^b(\operatorname{SBim}^K_{\mathcal D})$$

between the K-theoretic Hecke category and the category of bounded chain complexes of K-theoretic Soergel bimodules associated to $\mathcal {D}.$

5.1 Sheaves on topological spaces

Let $\Lambda$ be a regular Noetherian ring of finite global dimension. For a locally compact Hausdorff topological space X we denote by $\operatorname{D}(X, \Lambda)$ the full derived category of sheaves of $\Lambda$ -modules on X. We denote by $\operatorname{Haus}$ the category of locally compact Hausdorff topological spaces.

Theorem 5.1 [P. Scholze, Six-functor formalisms, private communication, Lecture 7]. There exists a 6-functor formalism

\begin{align*}\operatorname{D}(-,\Lambda) : \operatorname{Corr}(\operatorname{Haus}) &\rightarrow \operatorname{Pr}_\Lambda \\X &\mapsto \operatorname{D}(X,\Lambda).\end{align*}

In particular, for all $X \in \operatorname{Haus}$ the category $\operatorname{D}(X,\Lambda)$ is a closed symmetric monoidal category. We denote the tensor product by $\otimes$ and the internal mapping spaces by $\mathcal {H}{\operatorname{om}}$ . For all $f : X \to Y$ , we have the usual functors $f^!, f^*, f_!$ and $f_*$ between $\operatorname{D}(X,\Lambda)$ and $\operatorname{D}(Y, \Lambda)$ .

Using [P. Scholze, Six-functor formalisms, private communication, Proposition 4.17] and [Reference MannMan22, Proposition A.5.16] we can extend the 6-functor formalism to stacks on the category $\operatorname{Haus}$ . We give some example of such stacks, which we use in the later sections.

1. (i) If $X = \varinjlim_i X_i$ is an increasing union of locally compact Hausdorff spaces where the transitions maps are closed immersions, then we have

   $$\operatorname{D}(X, \Lambda) = \varinjlim_{i} \operatorname{D}(X_i, \Lambda),$$
   where the transitions maps are with respect to the $!$ -pushforward along the inclusions.

2. (ii) If $X = Y/G$ is a quotient, then

   $$\operatorname{D}(X, \Lambda) = \varprojlim_{i} \operatorname{D}(Y^{\times_X i+1}, \Lambda),$$
   where the transition maps are relative to the $*$ -pullback and $Y^{\times_X i+1}=Y\times_X\dots\times_XY$ with $(i+1)$ copies of Y.

5.2 Three ways to present monodromic sheaves

Let H be a commutative Lie group with contractible universal cover which we denote by $\tilde{H}$ . In particular, $\tilde{H}$ sits in an extension

\begin{equation*}1 \rightarrow \pi_1(H, 1) \rightarrow \tilde{H} \rightarrow H \rightarrow 1.\end{equation*}

We denote ${{\Lambda_{H}}} = \Lambda[\pi_1(H, 1)]$ . There is a canonical map

\begin{equation*}\pi_1(H,1) \rightarrow ({\Lambda_{H}})^{\times},\end{equation*}

which defines a ${{\Lambda_{H}}}$ -local system $\mathcal L_H$ of rank one on H, called the free monodromic local system.

Proof. By definition of $\mathcal L_H$ , if $f : H \rightarrow H'$ is a morphism of groups then we have a morphism ${{\Lambda_{H}}} \rightarrow {{\Lambda_{H'}}}$ and an isomorphism $f^*\mathcal L_{H'} = \mathcal L_H \otimes_{{{\Lambda_{H}}}} {{\Lambda_{H'}}}$ . Applying this to the multiplication map which is a morphism since H is commutative, we have

\begin{align*}m^*\mathcal L_H &= \mathcal L_{H \times H} \otimes_{{\Lambda_{H \times H}}} {{\Lambda_{H}}} \\&= (\mathcal L_{H} \boxtimes_{\Lambda} \mathcal L_H) \otimes_{{\Lambda_{H}} \otimes {{\Lambda_{H}}}} {{\Lambda_{H}}} \\&= \mathcal L_H \boxtimes_{{{\Lambda_{H}}}} \mathcal L_H.\end{align*}

Given the data of a group H together with a multiplicative local system $\mathcal L_H$ , for all spaces X with an action of H, there is a well-defined category of $(H, \mathcal L_H)$ -equivariant sheaves on X. Let X be a topological space with an action of H, we denote by $\operatorname{D}(X/(H,\mathcal L_H), {{\Lambda_{H}}})$ the category of $(H,\mathcal L_H)$ -equivariant sheaves on X as defined in [Reference GaitsgoryGai20, § 1.5].

Similarly, we denote by $\operatorname{D}(X/\tilde{H}, \Lambda)$ the category of sheaves on the stack $X/\tilde{H}$ which is the same as the category of $\tilde{H}$ -equivariant sheaves on X. Since $\tilde{H}$ is contractible the forgetful functor $\operatorname{D}(X/\tilde{H}, \Lambda) \rightarrow \operatorname{D}(X, \Lambda)$ is fully faithful.

Proof. By definition of equivariant and twisted equivariant sheaves, their formation commutes with limits of categories and in particular satisfy descent. More precisely, if $X_* \rightarrow X$ is a simplicial H-resolution of X, then we have

\begin{equation*}\operatorname{D}(X/(H, \mathcal L_H), {{\Lambda_{H}}}) = \varprojlim_{[n] \in \Delta^{op}} \operatorname{D}(X_n/(H, \mathcal L_H), {{\Lambda_{H}}}).\end{equation*}

Taking the bar resolution of X provides us with a simplicial resolution of X such that for all $[n] \in \Delta$ the action of H on $X_n$ is free and the simplicial H-torsor $X_* \rightarrow X_*/H$ is trivial. Let $Y_*$ be a simplicial splitting, so that $X_* = Y_* \times H$ . Then we have

\begin{equation*}\operatorname{D}(X_n/(H, \mathcal L_H), {{\Lambda_{H}}}) = \operatorname{D}(Y_n, {{\Lambda_{H}}}) \otimes_{\operatorname{D}({{\Lambda_{H}}})} \operatorname{D}(H/(H, \mathcal L_H), {{\Lambda_{H}}}).\end{equation*}

Similarly, we have

\begin{equation*}\operatorname{D}(X/\tilde{H}, \Lambda) = \varprojlim_{\Delta^{op}} \operatorname{D}(X_n/\tilde{H}, \Lambda) = \varprojlim_{\Delta^{op}} \operatorname{D}(Y_n, \Lambda) \otimes_{\operatorname{D}(R)} \operatorname{D}(H/\tilde{H}, \Lambda).\end{equation*}

Taking limits, we can assume first that the action of H on X is free and then that $X = H$ . But now it is clear that there is an equivalence of categories $\operatorname{D}(H/(H, \mathcal L_H), {{\Lambda_{H}}}) = \operatorname{D}({{\Lambda_{H}}}) = \operatorname{D}(H/\tilde{H}, \Lambda)$ . Hence, we get that the forgetful functor induces an equivalence $\operatorname{D}(X/(H, \mathcal L_H), {{\Lambda_{H}}}) \rightarrow \operatorname{D}(X/\tilde{H}, \Lambda) \subset \operatorname{D}(X, \Lambda)$ , this proves parts (i) and (ii).

Let us now show part (iii). Consider the pair of adjoint functors $p_!: \operatorname{D}(X, \Lambda) \leftrightarrows \operatorname{D}(X/\tilde{H},\Lambda) : p^!$ where $p\colon X \to X/\tilde{H}$ is the quotient map. Since the pullback $p^!$ is conservative and continuous the category $\operatorname{D}(X/\tilde{H},\Lambda)$ is identified with the category of module over the monad $p^!p_!$ . Since $p^!$ is fully faithful this monad is idempotent, that is the natural map $p^!p_!p^!p_! \to p^!p_!$ is an isomorphism and the category of modules over it is the full subcategory of objects $A \in \operatorname{D}(X, \Lambda)$ such that the map $A \to p^!p_!A$ is an isomorphism. Thus, we have to show that A satisfies this condition if and only if the $*$ -pullback to all H-orbits are locally constant.

For all $x \in X$ , we denote by $o_x : H \to X, h \mapsto hx$ , the orbit map of x. Let $A \in \operatorname{D}(X, \Lambda)$ be such that for all x, the object $o_x^*A$ is locally constant on H. Let $x \in X$ and consider the following Cartesian diagram

where the right column is the quotient by $\tilde{H}$ of the first column. We then have

\begin{align*}o_x^*A &= p_x^!p_{x,!}o_x^*A \\&= p_x^!\tilde{o}_x^*p_{!}A \\&=o_x^*p^!p_!A.\end{align*}

Here, the first equality follows from the fact that $o_x^*A$ is locally constant on H. The second equality comes from base change and the last one from the smoothness of $\tilde{H}$ . Hence, for all $x \in X$ , the $*$ -fiber at x of the morphism $A \to p^!p_!A$ is an isomorphism, so the morphism itself is an isomorphism. Conversely, if the map $A \to p^!p_!A$ is an isomorphism, then pullback along $o_x$ shows via the same calculation that $o_x^*A$ is locally constant.

The case of $!$ -fibers in part (iv) is handled dually. Indeed, the functor $p^*$ is conservative and exhibits the category $\operatorname{D}(X/\tilde{H}, \Lambda)$ as the category of comodules over the comonad $p^*p_*$ (which is idempotent since $p^*$ is fully faithful). As before, if A is monodromic, then the map $p^*p_*A \to A$ is an isomorphism. Applying the functor $o_x^!$ to this map shows that $o_x^!A$ is locally constant on H. Conversely, let A be such that for all $x \in X$ , the object $o_x^!A$ is locally constant and A is constructible. Then arguing as before, we get that the $!$ -fiber at x of the map $p^*p_*A \to A$ is an isomorphism and since A is constructible, so is $p^*p_*A$ and the isomorphism can be detected on $!$ -fibers. Hence, the map $p^*Ap_*A \to A$ is an isomorphism and A is monodromic.

Lemma 5.5. There is a well-defined four-functor formalism (so $f^*,f_*,f_!,f^!$ ) on stacks with H-action

\begin{equation*}X \mapsto \operatorname{D}(X/\tilde{H}, \Lambda) = \operatorname{D}(X/(H, \mathcal L_H), {{\Lambda_{H}}}).\end{equation*}

Let $f : H_1 \to H_2$ be a surjective morphism of Lie groups which have contractible universal covers. Then we have a natural map ${\Lambda_{H_1}} \to {\Lambda_{H_2}}$ .

Lemma 5.6. Let X be a space with an action of $H_2$ , there is an equivalence of categories

\begin{equation*}\operatorname{D}(X/\tilde{H}_1, \Lambda) = \operatorname{D}(X/\tilde{H}_2, \Lambda).\end{equation*}

Moreover, we have

\begin{equation*}f_!(\mathcal L_{H_1} ) = \mathcal L_{H_2}[2 \dim(\ker(f))].\end{equation*}

For the second part of the lemma, we first note that we have a commutative diagram

where $\tilde{f}$ is the map induced by f on universal covers. Since both $\tilde{H}_1$ and $\tilde{H}_2$ are contractible it follows that $\ker(\tilde{f})$ is weakly contractible. Hence, we have $\tilde{f}_!\Lambda = \Lambda[-2\dim(\ker(\tilde{f}))] =\Lambda[-2\dim(\ker(f))]$ , this implies the lemma.

5.3 Duality on monodromic sheaves

Let X be a stack with an H-action. There are two natural Verdier type dualities that appear on free monodromic sheaves.

On $\operatorname{D}(X/(H, \mathcal L_H), {{\Lambda_{H}}})$ , we can take ${{\Lambda_{H}}}$ -linear Verdier duality. However, the Verdier dual of a $(H, \mathcal L_H)$ -equivariant sheaf is naturally $(H, \mathcal L_H^{\vee})$ -equivariant (where $\mathcal L_H^{\vee}$ denotes the ${{\Lambda_{H}}}$ -dual of $\mathcal L_H$ ). Let $\iota : {{\Lambda_{H}}} \rightarrow {{\Lambda_{H}}}$ be the morphism induced by the inversion map of H.

5.4 Constructible sheaves

From now on we restrict our topological spaces to topological spaces of the form $X(\mathbb C)$ where X is an algebraic variety of finite type over $\mathbb C$ . A morphism $X(\mathbb C) \to Y(\mathbb C)$ is called algebraic if it comes from a morphism of algebraic varieties $X \to Y$ .

1. (i) $\operatorname{D}_c(X, \Lambda)$ the category of constructible sheaves on X, this is the full subcategory of $\operatorname{D}(X,\Lambda)$ of sheaves A for which there exists a Whitney stratification such that A is locally constant with perfect $*$ -stalk along all strata;

2. (ii) $\operatorname{D}_{\mathcal S-wc}(X, \Lambda)$ the category of weakly constructible sheaves on X, this the full subcategory of $\operatorname{D}(X,\Lambda)$ of sheaves A for which there exists a stratification such that A is locally constant.

For a quotient stack $X = Y/G$ the categories of (weakly) constructible sheaves is the category of sheaves on X whose pullback to X is (weakly) constructible. More generally, for a category of twisted equivariant sheaves, the category of (weakly) constructible twisted equivariant sheaves on Y is the full subcategory of twisted equivariant sheaves such that the underlying sheaf on Y is (weakly) constructible.

Definition 5.12. Let H be a Lie group with contractible simply connected cover and X be an H-space. Then we define the category of constructible monodromic sheaves to be

\begin{equation*}\operatorname{D}_c(X, \Lambda)_{\operatorname{mon}} = \operatorname{D}_{\mathcal S-wc}(X/\tilde{H}, \Lambda) \cap \operatorname{D}_c(X/(H, \mathcal L), {{\Lambda_{H}}})\end{equation*}

of sheaves that are $\Lambda$ -weakly constructible but have perfect ${{\Lambda_{H}}}$ -stalks.

The next proposition is an immediate consequence of standard theorems for Verdier duality.

6.1 Monodromic sheaves on flag varieties

Denote by $G\supset B\supset T$ the Kac–Moody group together with a Borel subgroup and maximal torus associated to the Kac–Moody root datum $\mathcal D$ . Denote by $U\subset B$ the unipotent radical, by $W\supset S$ the Weyl group and simple reflections. In this section, we identify all the groups with their complex points equipped with the analytic topology.

On the stack $U \backslash G/U$ , there are three actions of tori that we can consider:

1. (i) the action of T acting by left translations;

2. (ii) the action of T acting by right translations; and

3. (iii) the action of $T \times T$ acting by left and right translations.

We denote by $\mathcal {H}^{\operatorname{left}}, \mathcal {H}^{\operatorname{right}}$ and $\mathcal {H}^{\operatorname{leftright}}$ the corresponding categories of constructible monodromic sheaves $\operatorname{D}_c(U\backslash G/U)_{\operatorname{mon}}$ , see Definition 5.12. There are obvious forgetful functors $\mathcal {H}^{\operatorname{left}} \xleftarrow{\operatorname{oblv}^{\operatorname{right}}} \mathcal {H}^{\operatorname{leftright}} \xrightarrow{\operatorname{oblv}^{\operatorname{left}}} \mathcal {H}^{\operatorname{right}}$ .

We now introduce standard, costandard and tilting perverse sheaves in $\mathcal {H}$ . Let $n \in N(T)$ be an element in the normalizer of T and denote by w its image in the Weyl group of G. The choice of n yields a decomposition of the Bruhat stratum $BwB \simeq U \times T \times U_w$ given by $(u,t,v) \mapsto utnv$ where $U_w = U \cap \operatorname{Ad}(n)U$ . We denote by $\nu_n : BwB \rightarrow T$ the projection onto T. This map is equivariant for the action of $U \times U$ acting trivially on the target, hence we have a well-defined map of stacks $\nu_n : U \backslash BwB/U \to T$ which we will also call $\nu_n$ .

1. (i) $\Delta_n = i_{w,!} \nu_n^*\mathcal L_T[\ell(w) + \dim(T)]$ the standard sheaf;

2. (ii) $\nabla_n = i_{w,*} \nu_n^*\mathcal L_T[\ell(w) + \dim(T)]$ the costandard sheaf.

The standard and costandard sheaves are monodromic constructible sheaves (using either the left or right T-action) as the sheaf $\mathcal L_T$ is an $\hat{R}$ -constructible sheaf on T. Since the inclusions $i_w$ are affine, they are perverse.

1. (i) a $\Delta$ -flag for A is a filtration such that the graded pieces are isomorphic to standard sheaves;

2. (ii) a $\nabla$ -flag for A is a filtration such that the graded pieces are isomorphic to costandard sheaves; and

3. (iii) the sheaf A is tilting if it has both a $\Delta$ -flag and a $\nabla$ -flag.

We denote by $\mathcal {H}_{\operatorname{tilt}}\subset \mathcal {H}$ the full subcategory of tilting sheaves.

6.2 Monoidal structure

We equip the category $\mathcal {H}$ with a convolution structure. Consider the following diagram.

The convolution product is defined in two steps. Let $A, B \in \mathcal {H}$ which we see as the category of $(T \times T, \mathcal L_{T \times T})$ -equivariant sheaves on $U \backslash G/U$ , then $p_1^*A \boxtimes_{\mathbb{Z}} p_2^*B$ is naturally a sheaf of $\hat{R}^{\otimes 4}$ -modules, and it is constructible as an $\hat{R}^{\otimes 4}$ -sheaf on $U \backslash G \times^U G/U$ . We define the convolution of A and B as

\begin{equation*}A * B = {\operatorname{For}}_{\hat{R}^{\otimes 2}}^{\hat{R}^{\otimes 4}}m_!(p_1^*A \otimes_{\mathbb{Z}} p_2^*B)[\dim T],\end{equation*}

where the functor ${\operatorname{For}}_{\hat{R}^{\otimes 2}}^{\hat{R}^{\otimes 4}}$ is the forgetful functor induced by the inclusion $\hat{R} \otimes \hat{R} \rightarrow \hat{R}^{\otimes 4}$ induced by the outer inclusions. Keeping track of the actions of T, there is a canonical $\hat{R} \otimes \hat{R}$ -linear isomorphism

\begin{equation*}A * B = m_!(p_1^*{\operatorname{For}}^{\operatorname{right}}A \otimes_{\mathbb{Z}} p_2^*{\operatorname{For}}^{\operatorname{left}}B)[\dim T].\end{equation*}

In particular, since both ${\operatorname{For}}^{\operatorname{right}}A$ and ${\operatorname{For}}^{\operatorname{left}}B$ are $\hat{R}$ -constructible, the sheaf $A * B$ is $\hat{R} \otimes \hat{R}$ -constructible. There is also a dual convolution defined as

\begin{equation*} A *^! B = m_*(p_1^!{\operatorname{For}}^{\operatorname{right}}A \otimes_{\mathbb{Z}} p_2^!{\operatorname{For}}^{\operatorname{left}}B).\end{equation*}

6.3 Duality

Recall that we introduced the functor $\mathbb{D}'$ as a replacement of Verdier duality for monodromic sheaves, see § 5.3. For $A \in \mathcal {H}$ , we introduce the following duality functor

\begin{equation*} \mathbb{D}^{-}(A) = \operatorname{inv}^*\mathbb{D}'(A)(\varepsilon)[-\dim T],\end{equation*}

where $(\varepsilon)$ is again the twist of the $\hat{R}$ -structure by induced by the inversion map. Let us collect the following standard properties of the dualities.

1. (i) $(\mathbb{D}')^2\cong \operatorname{id}\cong (\mathbb{D}^{-})^2$ on $\mathcal {H}$ ;

2. (ii) there are isomorphisms $\mathbb{D}'(\Delta_n)\cong \nabla_n$ and $\mathbb{D}^{-}(\Delta_n)\cong \nabla_{n^{-1}}$ ;

3. (iii) there is a natural isomorphism $\mathbb{D}(A)*^!\mathbb{D}(B)\to \mathbb{D}(A* B)$ and $\mathbb{D}(A*^! B)\to \mathbb{D}(A)*\mathbb{D}(B)$ ;

4. (iv) there is a natural isomorphism $\mathbb{D}^-(A* B)\to \mathbb{D}^-(B)*^!\mathbb{D}^-(A)$ .

Lemma 6.7. For $A, B, C \in \mathcal {H}$ , there are canonical isomorphisms

$$\operatorname{Hom}(A * B, C) = \operatorname{Hom}(A, C *^! \mathbb{D}^{-}(B)) = \operatorname{Hom}(B, \mathbb{D}^{-}(A) *^! C).$$

1. (i) $\Delta_n * \Delta_{n'} = \Delta_{nn'}$ ;

2. (ii) $\nabla_n *^! \nabla_{n'} = \nabla_{nn'}$ .

6.4 Rank-one calculations

The goal of this section is to perform the rank-one calculations. Most proofs will reduce to arguments of [Reference TaylorTay25]. In [Reference TaylorTay25], the author assumes the group G to be adjoint. From now on, we assume that G is of semisimple rank one of adjoint type so it has a connected center.

Recall that $\hat{R} = \mathbb{Z}[X_*(T)]$ is the group ring on the space of cocharacters of T. For $\lambda \in X_*$ , we denote by $e^{\lambda} \in \hat{R}$ the corresponding element. We denote by $\alpha^{\vee}$ the simple coroot of G and by $\alpha$ the simple root of G. We also fix as in § 2.5 a cocharacter $\varpi^{\vee}$ such that $\langle \varpi^{\vee}, \alpha \rangle = 1$ .

We now consider the category $\mathcal {H}$ for G and with the objects $\Delta_1 = \nabla_1$ , $\Delta_s$ and $\nabla_s$ (note that the last two objects are defined up to isomorphism).

Lemma 6.12. There is an isomorphism

$$\Delta_s * \nabla_s = \nabla_s * \Delta_s = \Delta_1.$$

1. (i) There are two short exact sequences

   $$0 \to \Delta_s \to T_s \to \Delta_1 \to 0$$
   and
   $$0 \to \nabla_1 \to T_s \to \nabla_s \to 0.$$

2. (ii) Any object K satisfying (i) is isomorphic to $T_s$ .

3. (iii) There is an isomorphism $\operatorname{End}(T_s) = \hat{R} \otimes_{\hat{R}^s} \hat{R}$ .

Proof. (i) The first point is shown as in [Reference TaylorTay25, Lemma 5.1], namely we have (noncanonical) isomorphisms

$$\operatorname{Ext}^1(\Delta_1, \Delta_s) = \operatorname{Ext}^1(\nabla_s, \nabla_1) = \hat{R}/(1 - e^{\alpha^{\vee}}).$$

The element 1 then yields an object $T_s$ with the two desired short exact sequences.

(ii) We can argue as in [Reference TaylorTay25, Lemma 5.2]. The extension K corresponds to an element $a \in \hat{R}/(1 - e^{\alpha^{\vee}})$ , it is then enough to check that this element is invertible in this ring. As the ring $\hat{R}/(1 - e^{\alpha^{\vee}})$ is reduced by Lemma 2.5, it is enough to show that the image of a in all residue fields of $\hat{R}/(1 - e^{\alpha^{\vee}})$ are nonzero.

Let $\mathfrak{p}$ be a prime ideal of $\hat{R}/(1 - e^{\alpha^{\vee}})$ , which we consider as a prime ideal of $\hat{R}$ . Tensoring the first exact sequence with $\otimes_{\hat{R}} \hat{R}/\mathfrak{p}$ remains exacts as $\Delta_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p}$ lies in perverse degree 0. Hence, we have a short exact sequence

$$0 \to \Delta_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \to K \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \to \Delta_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \to 0,$$

which corresponds to the image of a along the induced map $\operatorname{Ext}^1(\Delta_s, \Delta_1) \to \operatorname{Ext}^1(\Delta_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p}, \Delta_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p}) = \hat{R}/(1-e^{\alpha^{\vee}}) \otimes_{\hat{R}} \hat{R}/\mathfrak{p} = \hat{R}/\mathfrak{p}$ .

If we suppose that this extension is split, that is a is 0 in $\hat{R}/\mathfrak{p}$ , then we can find a splitting $K\otimes_{\hat{R}} \hat{R}/\mathfrak{p} = \Delta_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \oplus \Delta_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p}$ . Using the second short exact sequence, we get

$$0 \to \nabla_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \to \Delta_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \oplus \Delta_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \to \nabla_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p} \to 0.$$

As $\operatorname{Hom}(\Delta_1 \otimes_{\hat{R}} \hat{R}/\mathfrak{p},\nabla_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p}) = 0$ , the second map in the short exact sequence induces an isomorphism $\Delta_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p} = \nabla_s \otimes_{\hat{R}} \hat{R}/\mathfrak{p}$ ; however, this contradicts the assumption that $\mathfrak{p}$ was an ideal of $\hat{R}/(1 - e^{\alpha^{\vee}})$ , hence a is nonzero.

(iii) We argue as in [Reference TaylorTay25, Proposition 5.4]. First, the monodromy map and the functors $\operatorname{gr}$ of [Reference Bezrukavnikov and RicheBR22, 6.3] yield maps

$$\hat{R} \otimes_{\hat{R}^s} \hat{R} \to \operatorname{End}(T_s) \to \hat{R}_e \oplus \hat{R}_s.$$

We want to show that the first map is an isomorphism. Their composition is injective by Lemma 2.4, hence the first map is injective. The injectivity of the second map follows from [Reference Bezrukavnikov and RicheBR22, Corollary 6.3]. It is then enough to show that both algebras have the same image in $\hat{R}_e \oplus \hat{R}_s$ . Let $a \in \operatorname{End}(T_s)$ and denote by $a_e$ and $a_s$ its components in $\hat{R}_e \oplus \hat{R}_s$ . Since $\hat{R} \otimes_{\hat{R}^s} \hat{R} \to \hat{R}_e \oplus \hat{R}_s \to \hat{R}_e$ is surjective by Lemma 2.4, we can find $b \in \hat{R} \otimes_{\hat{R}^s} \hat{R}$ such that $b_e = -a_e$ . We consider the endomorphism $a + b$ of $T_s$ . By the same Lemma, it is enough to show that $a_s + b_s$ lie in the ideal $(1 - e^{\alpha})$ . Equivalently, it is enough to show that this element is zero in $\hat{R}/(1 - e^{\alpha})$ and since this ring is reduced, it is enough to show that this element vanishes in all residue fields of this ring. But the condition on $\varpi$ implies that after reducing modulo some prime ideal containing $1 - e^{\alpha}$ , the extension

$$0 \to \Delta_s \to T_s \to \Delta_1 \to 0$$

is not split. As $a_e + b_e = 0$ , the map $a + b : T_s \to T_s$ factors through $\Delta_s$ and after reducing modulo some ideal it can only be zero only if the extension splits.

Remark 6.14. As observed in [Reference TaylorTay25, Lemma 5.3], it follows from Lemma 6.13 by convolving the second exact sequence with $\Delta_s$ that there is a triangle

$$\Delta_s \to \Delta_s * T_s \to \Delta_s * \nabla_s.$$

Since $\Delta_s * \nabla_s = \Delta_1$ , this triangle is then a short exact sequence of perverse sheaves and exhibits that we have an isomorphism

$$\Delta_s * T_s = T_s.$$

6.5 Tilting sheaves

We now collect some results about convolution, duality and the existence of perverse tilting sheaves.

For this, denote by $\mathcal {H}^s\subset \mathcal {H}$ the Hecke category for the minimal parabolic corresponding to a simple reflection $s\in S.$ Then, by the discussion in Lemma 6.13, there is an indecomposable tilting sheaf $T_s\in \mathcal {H}^s$ such that the multiplicity of $\Delta_s$ in a $\Delta$ -flag of $T_s$ is 1. The sheaf $T_s$ is unique up to isomorphism and in this section we fix an arbitrary representative of this sheaf in its isomorphism class. Once we introduce the $\mathbb{V}$ -functor, we will have a canonical way of choosing such a representative.

Proof. We mimic the argument of [Reference Bezrukavnikov and YunBY13, Proposition 4.3.3]. We first show that for $w,w' \in W$ we have

$$\Delta_w * \Delta_{w'} \in \langle \Delta_v[\leqslant 0], v \in W \rangle \subset \mathcal {H}$$

and

$$\nabla_w * \nabla_{w'} \in \langle \nabla_v[\geqslant 0], v \in W \rangle \subset \mathcal {H},$$

where $\langle - \rangle$ denotes the full subcategory generated by extensions. Let us prove the first point, the second is proven in a similar way. We can argue by induction on $\ell(w)$ and reduce to the case $w = s$ is a simple reflection. Then if $\ell(sw') = \ell(w') + 1$ , we have $\Delta_s * \Delta_{w'} = \Delta_{sw'}$ . If $\ell(sw') = \ell(w') - 1$ , then we use the short exact sequence

\begin{equation*}0 \to \Delta_s \to T_s \to \Delta_1 \to 0\end{equation*}

from Lemma 6.13. After convolving with $\Delta_s$ , we get a triangle

\begin{equation*}\Delta_s * \Delta_s \to \Delta_s * T_s \to \Delta_s,\end{equation*}

hence $\Delta_s * \Delta_s \in \langle \Delta_s * T_s, \Delta_s[-1] \rangle$ . By Remark 6.14, we have $\Delta_s * T_s \simeq T_s$ hence $\Delta_s * \Delta_{w'}$ is a successive extension of $\Delta_{w'}[-1], \Delta_{sw'}$ and $\Delta_{w'}$ .

The sheaf $T * T'$ now belongs to the category $\langle \Delta_v[\leqslant 0], v \in W \rangle \cap \langle \nabla_v[\geqslant 0], v \in W \rangle$ . The rest of the argument of [Reference Bezrukavnikov and YunBY13] follows verbatim.