2.1.1 General setup

Let G be a split semisimple group over a finite field $\mathbb{F}_q$ . Let T be a Cartan subgroup split over $\mathbb{F}_q$ , B a Borel subgroup containing T, and U its unipotent radical. Let $X = G/U$ be the basic affine space associated to G considered as a variety over $\mathbb{F}_q$ . Let W be the Weyl group W. We let S denote the set of simple reflections in W. Writing $q = p^m$ for some prime number p, we choose $\ell$ to be a prime with $\ell \neq p$ .

2.1.2 $\ell$ -adic sheaves, Tate twists, and Grothendieck groups

We work with the category of perverse sheaves $\mathrm{Perv}(G/U)$ of mixed $\ell$ -adic perverse sheaves on the basic affine space $G/U$ , and more generally with the constructible derived category $D^b(G/U)$ of mixed $\ell$ -adic sheaves on $G/U$ . We choose an isomorphism $\overline{\mathbb{Q}}_{\ell} \cong \mathbb{C}$ and work with $\mathbb{C}$ going forward. Pick a square root $q^{\frac{1}{2}}$ of q in $\mathbb{C}$ once and for all, and define the half-integer Tate twist $(\tfrac{1}{2})$ on $D^b(G/U)$ . We then view $K_0(G/U) = K_0(D^b(G/U))$ as a $\mathbb{Z}[v, v^{-1}]$ -module where $v^{-1}$ acts by $(\tfrac{1}{2})$ . When $\mathcal{C}$ is any category for which $K_0(\mathcal{C})$ is a $\mathbb{Z}[v, v^{-1}]$ -module, we denote by $K_0(\mathcal{C})\otimes \mathbb{C}$ the specialization $K_0(\mathcal{C}) \otimes_{\mathbb{Z}[v, v^{-1}]} \mathbb{C}$ at $v = q^{\frac{1}{2}}$ . We use $\mathbb{D}$ to denote the Verdier duality functor. We let ${}^pH^i$ be the perverse cohomology functors for any $i \in \mathbb{Z}$ .

We also choose once and for all a nontrivial additive character $\psi : \mathbb{F}_q \to \overline{\mathbb{Q}_{\ell}}$ , and let $\mathcal{L}_{\psi}$ be the corresponding Artin–Schreier sheaf on $\mathbb{G}_a$ .

The variety $G/U$ comes with a natural Frobenius morphism $\mathrm{Fr} : G/U \to G/U$ ; we can then consider the category of Weil sheaves $\mathrm{Perv}_{\mathbb{F}_q}(G/U)$ on $G/U$ , i.e. sheaves $K \in \mathrm{Perv}(G/U)$ equipped with a natural isomorphism $\mathrm{Fr}^*K \cong K$ .

2.1.3 Elements of G indexed by W

For every simple root $\alpha_s$ of G corresponding to the simple reflection s, we fix an isomorphism of the corresponding one-parameter subgroup $U_s \subset U$ with the additive group $\mathbb{G}_{a}$ . This uniquely defines a homomorphism $\rho_s : SL_{2} \to G$ which induces the given isomorphism of $\mathbb{G}_{a}$ (embedded in $SL_{2}$ as upper-triangular matrices) with $U_s$ ; then let

\begin{align*} n_s & = \rho_s \begin{pmatrix} 0 & 1\\-1 & 0 \end{pmatrix}.\end{align*}

Writing a reduced word $w = s_{i_1} \dots s_{i_k}$ for any $w \in W$ , we set $n_w = n_{s_{i_1}} \dots n_{s_{i_k}}$ , and one can check that this does not depend on the reduced word. We also define for any $s \in S$ the subtorus $T_s \subset T$ obtained from the image of the coroot $\alpha_s^\vee$ and define $T_w$ for any $w \in W$ to be the product of all $T_s$ ( $s \in S$ ) for which $s \leq w$ in the Bruhat order.

2.2.1 Fourier transforms on $\mathrm{Perv}(G/U)$

In [Reference Kazhdan and LaumonKL88] and [Reference PolishchukPol01], to each $w \in W$ the authors assign an element of $D^b(G/U\times G/U)$ which, up to shift, is perverse and irreducible. Following [Reference PolishchukPol01], let $X(w) \subset G/U \times G/U$ be the subvariety of pairs $(gU, g'U) \subset (G/U)^2$ such that $g^{-1}g' \in Un_wT_wU$ . There is a canonical projection $\mathrm{pr}_w : X(w) \to T_w$ sending (gU, g’U) to the unique $t_w \in T_w$ such that $g^{-1}g' \in Un_wt_wU$ . In the case where $w = s \in S$ , the morphism $\mathrm{pr}_s : X(s) \to T_s \cong \mathbb{G}_{m, k}$ extends to $\overline{\mathrm{pr}}_s : \overline{X(s)} \to \mathbb{G}_{a,k}$ and we have

\begin{align*} \overline{K(s)} = (-\overline{\mathrm{pr}}_s)^* \mathcal{L}_\psi,\end{align*}

and in the case of general $w \in W$ we have

\begin{align*} \overline{K(w)} & = \overline{K(s_{i_1})} * \dots * \overline{K(s_{i_k})}\end{align*}

whenever $w = s_{i_1} \dots s_{i_k}$ is a reduced expression, where $*$ denotes the convolution of sheaves on $G/U \times G/U$ as defined in [Reference Kazhdan and LaumonKL88]. One can take this as the definition of Kazhdan–Laumon sheaves, these being well defined by the proposition below, or refer to the explicit definition of $\overline{K(w)}$ which works for all $w \in W$ at once given in [Reference Kazhdan and LaumonKL88] or [Reference PolishchukPol01].

For any $s \in S$ , they note that the endofunctor $K \to K *\overline{K(s)}$ of $D^b(G/U)$ can be identified with a certain ‘symplectic Fourier–Deligne transform’ defined as follows. Let $P_s$ be the parabolic subgroup of G associated to s, and let $Q_s =[P_s, P_s]$ . The map $G/U \to G/Q_s$ has all fibers isomorphic to $\mathbb{A}^2 \setminus \{(0, 0)\}$ , and it is shown in [Reference Kazhdan and LaumonKL88, §2] that there exists a natural fiber bundle $\pi : V_s \to G/Q_s$ of rank 2 equipped with a G-invariant symplectic pairing which contains $G/U$ as an open set, with inclusion $j : G/U \to V_s$ with $\pi\circ j$ being the original projection $G/U \to G/Q_s$ . There is then a symplectic Fourier–Deligne transform $\tilde{\Phi}_s$ on $D^b(V_s)$ defined by

(1) \begin{align} \tilde{\Phi}_s(K) & = p_{2!}(\mathcal{L}\otimes p_1^*(K))[2](1),\end{align}

where the $p_i$ are the projections of the product $V_s \times_{G/Q_s} V_s$ on its factors, and $\mathcal{L} = \mathcal{L}_\psi(\langle, \rangle)$ is a smooth rank-one $\overline{\mathbb{Q}_\ell}$ -sheaf which is the pullback of the Artin–Schreier sheaf $\mathcal{L}_\psi$ under the morphism $\langle, \rangle$ ; cf. [Reference PolishchukPol01, §4]. We then define the endofunctor $\Phi_s$ of $D^b(G/U)$ by

\begin{align*} \Phi_s(K) & = j^*\tilde{\Phi}_sj_{!} K.\end{align*}

For any $w \in W$ , we let $\Phi_w = \Phi_{s_{i_1}} \dots\Phi_{s_{i_k}}$ where $s_{i_1} \dots s_{i_k}$ is a reduced expression for w as a product of simple reflections. The functors $\Phi_w$ are the gluing functors which Kazhdan and Laumon use to define the so-called glued categories $\mathcal{A}$ .

The functors $\Phi_w$ (respectively, $\Psi_w$ ) are each right (respectively, left) t-exact on $D^b(G/U)$ with respect to the perverse t-structure. For any $w \in W$ , let $\Phi_w^\circ = {}^pH^0\Phi_w$ and $\Psi_w^\circ = {}^pH^0\Psi_w$ , noting that $\Phi_w = L\Phi_w^\circ$ , $\Psi_w = R\Psi_w^\circ$ .

Proposition 2.4 [Reference PolishchukPol01, § 4.1]. For any $s \in S$ , there are natural morphisms $c_s : \Phi_s^2 \to \mathrm{Id}$ and $c_s' : \mathrm{Id} \to \Psi_s^2$ satisfying the associativity conditions

\begin{align*} \Phi_s \circ c_s & = c_s \circ \Phi_s : \Phi_s^3 \to \Phi_s,\quad \Psi \circ c_s' = c_s' \circ \Psi : \Psi_s \to \Psi_s^3. \end{align*}

the former coming from $c_s$ and the latter coming from our induction hypothesis.

2.2.2 Definition of the Kazhdan–Laumon category

Definition 2.6 [Reference Kazhdan and LaumonKL88, Reference PolishchukPol01]. The Kazhdan–Laumon category $\mathcal{A}$ has objects which are tuples $(A_w)_{w \in W}$ with $A_w \in \mathrm{Perv}(G/U)$ and equipped with morphisms

\begin{align*} \theta_{y,w} : \Phi_y^\circ A_w \to A_{yw} \end{align*}

for every $y, w \in W$ such that the diagram

commutes for any $y, y', w \in W$ .

A morphism f between objects $(A_w)_{w \in W}$ and $(B_w)_{w \in W}$ is a collection of morphisms $f_w : A_w \to B_w$ such that we have the following.

It is shown in [Reference PolishchukPol01] that this category is abelian, and that the functors $j_{w}^* : \mathcal{A} \to \mathrm{Perv}(G/U)$ defined by $j_{w}^*((A_w)_{w \in W}) = A_w$ are exact.

2.2.3 Definition of the w-twisted categories $\mathcal{A}_{w, \mathbb{F}_q}$

We note that the category $\mathcal{A}$ carries an action of the Weyl group W as follows. For any $w \in W$ , we let $\mathcal{F}_w : \mathcal{A} \to \mathcal{A}$ be the exact functor defined by right translation of the indices in the tuple, i.e. $\mathcal{F}_w((A_y)_{y\in W}) = (A_{yw})_{y \in W}$ .

For any two such objects $(A, \psi_A)$ and $(B, \psi_B)$ , we let $\mathrm{Hom}_{\mathcal{A}_{w, \mathbb{F}_q}}(A, B)$ be the set of morphisms $f \in \mathrm{Hom}_{\mathcal{A}}(A, B)$ such that $f \circ \psi_A = \psi_B \circ \mathcal{F}_w\mathrm{Fr}^*f$ .

2.2.4 Adjoint functors on $D^b(\mathcal{A})$

We then define a functor $j_{w!}^\circ : \mathrm{Perv}(G/U) \to \mathcal{A}$ by

\begin{align*} j_{w!}^\circ(K) & = (\Phi_{yw^{-1}}^\circ K)_{y \in W}. \end{align*}

One can check that the morphisms $\nu_{y', y}$ for $y, y' \in W$ introduced in Corollary 2.5 endow the tuple $(\Phi_{yw^{-1}}^\circ K)_{y \in W}$ with the structure morphisms required to define an object of $\mathcal{A}$ . We let $j_{w!}$ be the left-derived functor to $j_{w!}^\circ$ .

Proposition 2.11 [Reference PolishchukPol01, Proposition 7.1.2]. For any $w \in W$ , there is an adjunction $(j_{w!}^\circ, j_w^*)$ . Further, the functor $j_{w!} : D^b(G/U) \to D^b(\mathcal{A})$ has the property that

(2) \begin{align} {}^pH^i(j_{w!}(K)) & = ({}^pH^i\Phi_{yw^{-1}}(K))_{y \in W}, \end{align}

and there is also an adjunction $(j_{w!}, j_w^*)$ on derived categories.

Analogously, acting instead by the functors $\Psi_{yw^{-1}}$ in (2) defines a right adjoint $j_{w*}^\circ$ to $j_w^*$ and its right-derived functor $j_{w*}$ in the very same way, as is also explained in [Reference PolishchukPol01].

2.2.5 The functor $\iota$

Definition 2.12. We define an endofunctor $\iota$ of $\mathcal{A}$ by

\begin{align*} \iota((A_w)_{w \in W}) & = (\Phi_{w_0}^\circ A_{w_0w})_{w \in W} \end{align*}

with the structure morphisms described in [Reference PolishchukPol01, §7.2].

It is shown in loc. cit. that we can also abuse notation and view $\iota$ as a functor on $D^b(\mathcal{A})$ (by replacing $\Phi_{w_0}^\circ$ with $\Phi_{w_0}$ ); for our purposes we will only need the functor $\iota^2$ , which we will later describe as an endofunctor of $D^b(\mathcal{A})$ more conceptually in Lemma 6.2.

2.2.6 Objects of the form $j_{w!}(A)$

Proof. The adjunction in Proposition 2.11 gives that, for any $A \in \mathcal{A}$ ,

\begin{align*} \mathrm{Ext}_{\mathcal{A}}^\bullet(j_{w!}(B), A) & = \mathrm{Ext}_{\mathrm{Perv}(G/U)}^\bullet(B, j_w^*A), \end{align*}

and the latter is a finite-dimensional vector space because the category $\mathrm{Perv}(G/U)$ has finite cohomological dimension.

2.2.7 Polynomials and localization

Fix the Weyl group W and choose $w_0$ its longest element.

Definition 2.14. Define $P(x, v), \tilde{P}(x, v) \in \mathbb{Z}[x, v, v^{-1}]$ by

\begin{align*} P(x, v) & = \prod_{i=0}^{\ell(w_0)} (x - v^{2i}),\\ \tilde{P}(x, v) & = \prod_{i=1}^{\ell(w_0)} (x - v^{2i}), \end{align*}

and let $p(v) = \tilde{P}(1, v)$ .

3.1.1 The monodromic Hecke category

In [Reference GouttardGou21], the author defines for any $\mathcal{L}$ a category $\mathcal{P}_{\mathcal{L}}$ (called $D^b_{(B)}(G/U)_{t}$ in loc. cit., where t is a parameter determined by $\mathcal{L}$ ) such that for $\mathcal{L}$ trivial, this reduces to the familiar derived category $D^b_{(B)}(G/U)$ of B-constructible sheaves on $G/U$ . Elements of $\mathcal{P}_{\mathcal{L}}$ are $\mathcal{L}$ -monodromic with respect to the right action of T, while their left monodromies may correspond to any character sheaf in the W-orbit of $\mathcal{L}$ .

For any $w \in W$ , this category contains monodromic versions ${}_{w\mathcal{L}}\Delta(w)_{\mathcal{L}}$ and ${}_{w\mathcal{L}}\nabla(w)_{\mathcal{L}}$ of standard and costandard sheaves; we emphasize that extensions of such objects in this category may not be $\mathcal{L}$ -equivariant with respect to the right T-action even when they remain $\mathcal{L}$ -monodromic, so this category also contains monodromic versions of tilting objects $\mathcal{T}(w)_{\mathcal{L}}$ for any $w \in W$ .

We let ${}_{\mathcal{L}}\mathcal{P}_{\mathcal{L}}$ be the triangulated subcategory of objects generated by the standard and costandard objects corresponding to $w \in W_{\mathcal{L}}^\circ$ , each of whose objects must also be left-monodromic with respect to $\mathcal{L}$ .

3.1.2 Free-monodromic Hecke categories

In [Reference Bezrukavnikov and YunBY13], in the unipotent monodromy case where $\mathcal{L}$ is trivial, the authors define a category formed from a certain completion of $D^b_{\mathcal{L}}(G/U)$ called the category of unipotently free-monodromic sheaves.

In [Reference GouttardGou21], the case of nonunipotent monodromy was treated carefully. In loc. cit., the author defines a category $\hat{\mathcal{P}}_{\mathcal{L}}$ (which is called $\hat{D}^b_{(B)}(G/U)_{t}$ in loc. cit. where t is a parameter determined by $\mathcal{L}$ ), which is a certain completion of the category $\mathcal{P}_{\mathcal{L}}$ defined in Section 3.1.1, equipped with a monoidal structure which we also denote by $*$ in this context.

This category contains objects $\varepsilon_{n,\mathcal{L}}$ and $\hat{\delta}_{\mathcal{L}}$ introduced in [Reference Bezrukavnikov and TolmachovBT22, Corollary 5.3.3]. The object $\hat{\delta}_{\mathcal{L}}$ is the monoidal unit for the convolution product on $\hat{\mathcal{P}}_{\mathcal{L}}$ , whereas convolution with the objects $\varepsilon_{n, \mathcal{L}}$ can be thought of as a sort of projection to the subcategory of objects whose corresponding ‘logarithmic monodromy operator’ is nilpotent of order at most n; cf. [Reference Bezrukavnikov and YunBY13, Appendix A] for an explanation of this perspective.

Finally, we recall that $\hat{\mathcal{P}}_{\mathcal{L}}$ contains for all $w \in W$ free-monodromic versions ${}_{w\mathcal{L}}\hat{\Delta}(w)_{\mathcal{L}}$ , ${}_{w\mathcal{L}}\hat{\nabla}(w)_{\mathcal{\mathcal{L}}}$ of standard and costandard sheaves; cf. [Reference Bezrukavnikov and YunBY13] for the unipotent case, [Reference Lusztig and YunLY20] for a description of these objects in $\mathcal{P}_{\mathcal{L}}$ for arbitrary $\mathcal{L}$ , and [Reference GouttardGou21] for their free-monodromic versions in $\hat{\mathcal{P}}_{\mathcal{L}}$ . We define the subcategory ${}_{\mathcal{L}}\hat{\mathcal{P}}_{\mathcal{L}}$ analogously to the subcategory ${}_{\mathcal{L}}\mathcal{P}_{\mathcal{L}}$ of $\mathcal{P}_{\mathcal{L}}$ .

3.1.3 The center of the monodromic Hecke category

To define the notion of categorical center which will be useful in this setting, we will closely follow the conventions of [Reference Bezrukavnikov, Ionov, Tolmachov and VarshavskyBITV23] in this section. We begin by recalling some definitions from loc. cit.

For $\mathfrak{o}$ a W-orbit in $\mathrm{Ch}(T)$ , we let $\mathcal{H}^{(1)}_{\mathfrak{o}}$ be the full subcategory of $\mathcal{H}^{(1)}$ consisting of monodromic sheaves with monodromies in $\mathfrak{o}$ , i.e.

\begin{align*} \mathcal{H}^{(1)}_{\mathfrak{o}} \cong \bigoplus_{\mathcal{L} \in \mathfrak{o}} \mathcal{H}^{(1)}_{\mathcal{L}}. \end{align*}

Following [Reference Bezrukavnikov, Ionov, Tolmachov and VarshavskyBITV23, 3.2], consider the diagram

where $\pi$ is the projection and q is the quotient of the map $q' : G \times G/U \to G/U \times G/U$ given by $q'(g, xU) = (xU, gxU)$ by the free right T-action, with respect to which q’ is equivariant.

Definition 3.4. The Harish-Chandra transform is the functor

\begin{align*} \mathfrak{hc} = q_! \circ \pi^*: D^b(G/_{\mathrm{Ad}} G) \to D^b(G\backslash \mathcal{Y}), \end{align*}

which is monoidal with respect to the natural convolution product on each side by [Reference GinsburgGin89]; cf. [Reference Bezrukavnikov, Ionov, Tolmachov and VarshavskyBITV23, 2.2] for a detailed exposition of these convolution products.

In [Reference Bezrukavnikov, Ionov, Tolmachov and VarshavskyBITV23, Theorem 5.3.4], the authors show that the functor $\mathfrak{hc}$ realizes the category $D^b_{\mathfrak{C}, \mathcal{L}}(G)$ as the center $\mathcal{Z}\mathcal{H}^{(1)}_{\mathfrak{o}}$ , with the notion of center in this case being defined in [Reference Bezrukavnikov, Ionov, Tolmachov and VarshavskyBITV23, §2.2]. They then show that the projection map $\mathcal{Z}\mathcal{H}^{(1)}_{\mathfrak{o}} \to \mathcal{Z}\mathcal{H}^{(1)}_{\mathcal{L}}$ is a monoidal equivalence. Therefore in Theorem 5.3.2 of loc. cit., the authors identify $D^b_{\mathfrak{C}, \mathcal{L}}(G)$ with the center of $\mathcal{H}^{(1)}_{\mathcal{L}}$ . The following proposition is a consequence of this theorem, and it will be an important tool in the present work.

Proposition 3.6. There is a well-defined convolution functor

\begin{align*} - * - : D_{\mathcal{L}}^b(G/U)\times D^b_{\mathfrak{C}, \mathcal{L}}(G) \to D_{\mathcal{L}}^b(G/U), \end{align*}

and for any $Z \in D^b_{\mathfrak{C}, \mathcal{L}}(G)$ , the functor $- * Z$ is central: it commutes with convolution by elements of ${}_{\mathcal{L}}\hat{\mathcal{P}}_{\mathcal{L}}$ or ${}_{\mathcal{L}}{\mathcal{P}}_{\mathcal{L}}$ , and the isomorphism realizing this commutativity satisfies the corresponding associativity constraints.

In Definition 3.5, clearly $D^b_{\mathfrak{C},\mathcal{L}}(G) = D^b_{\mathfrak{C},\mathcal{L}'}(G)$ as categories whenever $\mathcal{L}$ and $\mathcal{L}'$ are in the same W-orbit $\mathfrak{o}$ . However, the convolution action described in Proposition 3.6 passes through the identification of $D^b_{\mathfrak{C}, \mathcal{L}}(G)$ with the center of $\mathcal{H}^{(1)}_{\mathcal{L}}$ , and so by this convention, this action which we will use throughout the present paper depends on $\mathcal{L}$ itself and not just its orbit. To make this identification explicit, we use the following definition.

3.1.4 Two-sided cells and character sheaves

In this and subsequent sections, we use the notion of two-sided Kazhdan–Lusztig cells in the Weyl group; see, for example, [Reference WilliamsonWil03] for a clear exposition.

Two-sided Kazhdan–Lusztig cells give a filtration on the category $D^b_{\mathfrak{C}, \mathcal{L}}(G)$ (cf. [Reference LusztigLus86]), and so for each $\underline{c} \in \underline{C}_{\mathcal{L}}$ , there are triangulated subcategories $D^b_{\mathfrak{C}, \mathcal{L}}(G)_{\leq\underline{c}}$ and $D^b_{\mathfrak{C}, \mathcal{L}}(G)_{<\underline{c}}$ of $D^b_{\mathfrak{C}, \mathcal{L}}(G)$ . We then define $D^b_{\mathfrak{C}, \mathcal{L}}(G)_{\underline{c}}$ as the quotient category $D^b_{\mathfrak{C}, \mathcal{L}}(G)_{\leq \underline{c}}/D^b_{\mathfrak{C}, \mathcal{L}}(G)_{< \underline{c}}$ , referring to the unipotent case treated in [Reference Bezrukavnikov, Finkelberg and OstrikBFO12, §5] for details. Let $G_{\mathrm{ad}}$ be the adjoint quotient of G; the following is a consequence of the classification of irreducible character sheaves in terms of cells given in [Reference LusztigLus86]; cf. [Reference Bezrukavnikov, Finkelberg and OstrikBFO12, Corollary 5.4].

Proposition 3.9. For any $\mathcal{L} \in \mathrm{Ch}(T)$ ,

\begin{align*} K_0(D^b_{\mathfrak{C}, \mathcal{L}}(G_{\mathrm{ad}})) \cong \bigoplus_{\underline{c} \in \underline{C}_{\mathcal{L}}} K_0(D^b_{\mathfrak{C}, \mathcal{L}}(G_{\mathrm{ad}})_{\underline{c}}) \end{align*}

as vector spaces. Further, for any $\underline{c} \in \underline{C}_{\mathcal{L}}$ , the preimage of the subspace

\begin{align*} \bigoplus_{\underline{c}' \leq \underline{c}} K_0(D^b_{\mathfrak{C}, \mathcal{L}}{(G_{\mathrm{ad}})}_{\underline{c}'}) \end{align*}

in $K_0(D^b_{\mathfrak{C}, \mathcal{L}}(G_{\mathrm{ad}}))$ under this isomorphism is a monoidal ideal.

3.1.5 The big free-monodromic tilting object and $\mathbb{K}_{\mathcal{L}}$

In [Reference GouttardGou21, §9.4], the author defines free-monodromic tilting sheaves with general monodromy, analogous to those appearing in [Reference Bezrukavnikov and YunBY13] for the case of unipotent monodromy.

In [Reference Bezrukavnikov and TolmachovBT22], working in the case of unipotent monodromy, the authors construct from $\hat{\mathcal{T}}(w_0)$ an object they call $\mathbb{K}$ , defined as $\mathbb{K} = p^*p_!\hat{\mathcal{T}}(w_0)$ where $p : U\backslash G/U \to (U\backslash G/U)/T$ , where T acts on $U\backslash G / U$ by conjugation.

They also define a character sheaf $\Xi$ whose details are explained in [Reference Bezrukavnikov and TolmachovBT22, §1.4] obtained by averaging the derived pushforward of the constant sheaf on the regular locus of the unipotent variety of G to obtain an element of $D^b(G/_{\mathrm{Ad}} G)$ . They then show how to define the projection of $\Xi$ to the subcategory of unipotent character sheaves, and that $\mathbb{K}$ is obtained by applying the functor $\mathfrak{hc}$ to the resulting object.

We now define an analog for arbitrary monodromy generalizing the unipotent case.

By Proposition 3.6, this means that the functor $- * \mathbb{K}_{\mathcal{L}} : D_{\mathcal{L}}^b(G/U) \to D_{\mathcal{L}}^b(G/U)$ commutes with convolution by elements of ${}_{\mathcal{L}}\hat{\mathcal{P}}_{\mathcal{L}}$ .

3.1.6 Fourier transform and convolution with costandard sheaves

Proposition 3.13. Let $\mathcal{F} \in D^b_{\mathcal{L}}(G/U)$ where $\mathcal{L} \in \mathrm{Ch}(T)$ . Then, for any $s \in S$ ,

(3) \begin{align} \Phi_s(\mathcal{F}) & = \begin{cases} \mathcal{F} * {}_\mathcal{L}\hat{\nabla}(s)_{\mathcal{L}}(\tfrac{1}{2}), & s \in W_{\mathcal{L}}^\circ,\\ \mathcal{F} * {}_{\mathcal{L}}\hat{\nabla}(s)_{s\mathcal{L}}, & s \not\in W_{\mathcal{L}}^\circ. \end{cases} \end{align}

We note that the difference between the two cases in (3) boils down to the calculation of $R\Gamma_{c}(T, \mathcal{L}\otimes\mathcal{L}_{\psi})$ in rank two, in the case where $\mathcal{L}$ is trivial versus the case where it is nontrivial. As explained in [Reference DeligneDel77, Applications de la formule des traces aux sommes trigonométriques], this is always concentrated in cohomological degree one, but has weight 0 or 1 depending on whether $\mathcal{L}$ is trivial, hence the presence of the Tate twist in (3). The full computation is explained in more detail in [Reference Morton-FergusonMF24, Proposition 4.1].

4.1.1 The Kazhdan–Laumon category as (co)algebras over a (co)monad

We recall a result from [Reference Bezrukavnikov, Braverman and PositselskiiBBP02] exhibiting the Kazhdan–Laumon category $\mathcal{A}$ as the category of (co)algebras over a certain (co)monad on the underlying category $\mathcal{B} = \mathrm{Perv}(G/U)^{\oplus W}$ .

Definition 4.1. Let $\Psi^\circ : \mathcal{B} \to \mathcal{B}$ be the endofunctor defined for any $A = (A_w)_{w \in W} \in \mathcal{B}$ by

(4) \begin{align} (\Psi^\circ A)_w & = \oplus_{y \in W} \Psi_{wy^{-1}}^\circ A_y. \end{align}

It has right-derived functor $\Psi = R\Psi^\circ$ given by

\begin{align*} (\Psi A)_w & = \oplus_{y \in W} \Psi_{wy^{-1}} A_y. \end{align*}

Similarly, we define $\Phi^\circ : \mathcal{B} \to \mathcal{B}$ by

\begin{align*} (\Phi^\circ A)_w & = \oplus_{y \in W} \Phi_{wy^{-1}}^\circ A_y, \end{align*}

with left-derived functor $\Phi = L\Phi^\circ$ given by

\begin{align*} (\Phi A)_w & = \oplus_{y \in W} \Phi_{wy^{-1}} A_y. \end{align*}

4.1.2 Differential graded enhancements of derived categories

Given an abelian category $\mathcal{C}$ , let $C_{\mathrm{dg}}(\mathcal{C})$ be the dg category with objects complexes of sheaves and morphisms the usual complexes of maps between complexes. We define the dg derived category $D_{\mathrm{dg}}(\mathcal{C})$ to be the dg quotient of $C_{\mathrm{dg}}(\mathcal{C})$ by the full subcategory of acyclic objects; its homotopy category is the usual derived category $D(\mathcal{C})$ .

The bounded dg derived category $D^b_{\mathrm{dg}}(\mathcal{C})$ is defined to be the full dg subcategory of $D_{\mathrm{dg}}(\mathcal{C})$ consisting of objects which project to $D^b(\mathcal{C})$ when passing to the homotopy category.

4.1.3 Barr–Beck–Lurie for dg categories applied to the Kazhdan–Laumon category

We first state the following general result, which follows from the Barr–Beck–Lurie monadicity theorem.

Now suppose also that $F : \mathcal{C}^T \to \mathcal{C}$ is exact and also admits a right adjoint $F^R$ . Suppose also that the left- and right-derived functors $LF^L$ and $RF^R$ give adjunctions $(LF^L, F)$ and $(F, RF^R)$ of functors between the dg derived categories $D_{\mathrm{dg}}(\mathcal{C}^T)$ and $D_{\mathrm{dg}}(\mathcal{C})$ . Let $\tilde{T} = F \circ LF^L$ be the resulting dg monad, and $D_{\mathrm{dg}}(\mathcal{C})^{\tilde{T}}$ the dg category of algebras over the monad $\tilde{T}$ .

If $F : D_{\mathrm{dg}}(\mathcal{C}^T) \to D_{\mathrm{dg}}(\mathcal{C})$ is conservative, then there is a canonical equivalence of dg categories

\begin{align*} \tilde{F} : D_{\mathrm{dg}}(\mathcal{C}^T) \to D_{\mathrm{dg}}(\mathcal{C})^{\tilde{T}} \end{align*}

Note that the monad $\Phi^\circ : \mathcal{B} \to \mathcal{B}$ can be enhanced to a monad $\Phi = L\Phi^\circ : D_{\mathrm{dg}}^b(\mathcal{B}) \to D_{\mathrm{dg}}^b(\mathcal{B})$ ; i.e. the functor $\Phi$ has a dg enhancement (since it arises from the functors $\Phi_w$ which are themselves defined using the six-functor formalism). We can then consider the dg category $D_{\mathrm{dg}}^b(\mathcal{B})^{\Phi}$ of algebras over the monad $F \circ \Phi$ or the dg category $D_{\mathrm{dg}}^b(\mathcal{B})_{\Psi}$ of coalgebras over the comonad $F \circ \Psi$ . The following is an application of Proposition 4.3.

Indeed, we check this adjunction explicitly: this free algebra functor sends $(B_w)_{w \in W} \in D^b_{\mathrm{dg}}(\mathcal{B})$ to the direct sum $\oplus_{w \in W} Lj_{w!}^\circ(A_w)$ where $Lj_{w!}^\circ = j_{w!}$ is the left adjoint to $j_w^*$ by Proposition 2.11. It is then clear from the adjunction $(j_{w!}, j_{w}^*)$ that the forgetful functor is its left adjoint. Now notice that in a similar way, we observe that the forgetful functor F has a right adjoint: it follows from the opposite adjunction $(j_w^*, j_{w*})$ that the functor sending any $(B_w)_{w \in W} \in D^b_{\mathrm{dg}}(\mathcal{B})$ to $\oplus_{w \in W} j_{w*}(B_w)$ (where again $j_{w*} = Rj_{w*}^\circ$ ) is right-adjoint to F.

We are now in the setup of Proposition 4.3, and both the abelian and dg versions of the functor F are clearly conservative. So applying this proposition, we get a natural equivalence $D_{\mathrm{dg}}(\mathcal{B}^{\Phi^\circ}) \to D_{\mathrm{dg}}(\mathcal{B})^{\Phi}$ . To conclude, we then note that this then restricts to an equivalence of bounded derived categories $D_{\mathrm{dg}}^b(\mathcal{B}^{\Phi^\circ}) \to D_{\mathrm{dg}}^b(\mathcal{B})^{\Phi}$ since cohomology in both categories is computed in the underlying category $D(\mathcal{B})$ after forgetting the algebra structure.

Dually, in terms of the comonad $\Psi$ , we have that the dg categories $D_{\mathrm{dg}}^b(\mathcal{B}_{\Psi^\circ})$ and $D_{\mathrm{dg}}^b(\mathcal{B})_{\Psi}$ are equivalent.

4.2.1 Action of the center on the Kazhdan–Laumon dg category

For any $\mathcal{L} \in \mathrm{Ch}(T)$ , let $\mathcal{B}_{\mathcal{L}} = \oplus_{w \in W}\mathrm{Perv}_{w\mathcal{L}}(G/U)$ .

By Proposition 3.6, for any $B \in {}_{\mathcal{L}}\hat{\mathcal{P}}_{\mathcal{L}}$ , the functors $- * Z * B$ and $- * B * Z$ on $\mathrm{Perv}_{\mathcal{L}}(G/U)$ are naturally isomorphic. The following lemma is a consequence of this fact.

Proof. By Proposition 3.13, for any $w \in W$ , there exists some Tate twist (d) such that $((Z \circ \Psi)(A))_w(d)$ is isomorphic to

\begin{align*} & \bigoplus_{y \in W} (\Psi_{wy^{-1}} A_y) * {}_{w\mathcal{L}}\hat{\Delta} (w)_{\mathcal{L}} * Z * {}_{\mathcal{L}}\hat{\nabla}(w^{-1})_{w\mathcal{L}}\\ & \quad \cong A_y * {}_{y\mathcal{L}}\hat{\nabla}(yw^{-1})_{w\mathcal{L}} * {}_{w\mathcal{L}}\hat{\Delta}(w)_{\mathcal{L}} * Z * {}_{\mathcal{L}}\hat{\nabla}(w^{-1})_{w\mathcal{L}}\\ & \quad \cong A_y * {}_{y\mathcal{L}}\hat{\Delta}(y)_{\mathcal{L}} * {}_{\mathcal{L}}\hat{\nabla}(y^{-1}){}_{y\mathcal{L}} * {}_{y\mathcal{L}}\hat{\nabla}(yw^{-1})_{w\mathcal{L}} * {}_{w\mathcal{L}}\hat{\Delta}(w)_{\mathcal{L}} * Z * {}_{\mathcal{L}}\hat{\nabla}(w^{-1})_{w\mathcal{L}} \\ & \quad \cong A_y * {}_{y\mathcal{L}}\hat{\Delta}(y){}_{\mathcal{L}} * Z * {}_{\mathcal{L}}\hat{\nabla}(y^{-1}){}_{y\mathcal{L}} * {}_{y\mathcal{L}}\hat{\nabla}(yw^{-1}){}_{w\mathcal{L}} * {}_{w\mathcal{L}}\hat{\Delta}(w){}_{\mathcal{L}} * {}_{\mathcal{L}}\hat{\nabla}(w^{-1}){}_{w\mathcal{L}}\\ & \quad \cong A_y * {}_{y\mathcal{L}}\hat{\Delta}(y){}_{\mathcal{L}} * Z * {}_{\mathcal{L}}\hat{\nabla}(y^{-1}){}_{y\mathcal{L}} * {}_{y\mathcal{L}}\hat{\nabla}(yw^{-1}){}_{w\mathcal{L}} \\ & \quad \cong \oplus_{y \in W} \Psi_{wy^{-1}}(Z(A))(d)\\ & \quad \cong ((\Psi \circ Z)(A))_w(d), \end{align*}

and these isomorphisms across all $w \in W$ assemble together to give a natural isomorphism $Z \circ \Psi \cong \Psi \circ Z$ . A conceptual explanation for the existence of this isomorphism appears in [Reference Lusztig and YunLY20, Lemma 11.12].

In the following, let $\pi_{\mathrm{ad}} : G \to G_{\mathrm{ad}}$ be the natural map from G to its adjoint quotient.

For any $B \in D^b_{\mathrm{dg}}(\mathcal{B}_\mathcal{L})_{\Psi}$ and $Z \in D^b_{\mathfrak{C},\mathcal{L}}(G_{\mathrm{ad}})$ , we define the new object $B * Z$ by $F(B) * Z$ as a tuple of elements of $\mathrm{Perv}_{\mathcal{L}}(G/U)$ where F is the forgetful functor. We then define the coalgebra structure map $B * Z \to \Psi (B * Z)$ by the composition

\begin{align*} B * Z & \to \Psi(B) * Z \cong \Psi(B * Z) \end{align*}

where the first map comes from the coalgebra structure on B and the subsequent isomorphism is the natural isomorphism described in Lemma 4.6. One can then check that this defines a coalgebra structure on $B * Z$ .

We note that we get a similar action of $D^b_{\mathfrak{C},\mathcal{L}}(G_{\mathrm{ad}})$ on $D^b(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}})$ since any object Z here is a sheaf on the variety $G_{\mathrm{ad}}$ and therefore has a natural isomorphism $\mathrm{Fr}^* Z \to Z$ ; this means $- * Z$ commutes with $\mathrm{Fr}^*$ , and therefore the action in Proposition 4.7 also gives an action in the setting of w-twisted Weil sheaves.

5.1.1 Definition of the categories $\mathcal{A}_{w, \mathbb{F}_q}^I$

We will now define certain parabolic analogs of the Kazhdan–Laumon category which correspond to subsets $I \subset S$ . We first recall the definition of the categories $\mathcal{A}_{W_I}$ from [Reference PolishchukPol01], build on this definition to define the categories $\mathcal{A}^I$ which we will need in the present work, and finally define $\mathcal{A}^I_{w, \mathbb{F}_q}$ for any $w \in W$ .

For any $J \subset I \subset S$ , and any right coset $W_Jx$ of $W_J$ in $W_I$ , there is a restriction functor $j_{W_Jx}^{W_I*} : \mathcal{A}_{W_I} \to \mathcal{A}_{W_J}$ remembering only the tuple elements and morphisms for $w \in W_Jx$ , $s \in J$ . When $I = S$ , we write only $j_{W_Jx}^*$ , and in this case we omit superscripts similarly for all functors introduced in this section.

Alternatively, $\mathcal{A}^I = \oplus_{W_I \backslash W} \mathcal{A}_{W_I}$ with a reindexing by W on the tuples in this category.

Just like $\mathcal{A}$ , the category $\mathcal{A}^I$ admits an action of W by functors $\{\mathcal{F}_w\}_{w \in W}$ which are defined by $\mathcal{F}_w((A_y)_{y \in W}) = (A_{yw})_{w \in W}$ . We define the category $\mathcal{A}_{w, \mathbb{F}_q}^I$ as in Definition 2.8 but with $\mathcal{A}$ replaced by $\mathcal{A}^I$ .

5.1.2 Adjunctions between these categories

For any $J \subset I$ , there is an obvious restriction functor $j_{I,J}^* : \mathcal{A}^I \to \mathcal{A}^J$ which is the identity on objects but which remembers only the morphisms $\Phi_y^\circ A_w \to A_{yw}$ for $y \in W_J$ . As in Proposition 2.11, there is an analogous derived version of this morphism and the following adjunction.

Proposition 5.4. For any $J \subset I$ , the functor $j_{I,J}^*$ admits a left adjoint $j_{I,J!}$ . For any $A \in \mathcal{A}^J$ , $j_{I,J!}^\circ(A)$ is the direct sum

(5) \begin{align} \bigoplus_{x \in W_J\backslash W_I} j^{W_I\circ}_{W_Jx!}((A_w)_{w \in W_Jx}),\end{align}

where $(A_w)_{w \in W_Jx}$ is considered as an element of $\mathcal{A}_{W_J}$ . Further, the adjoint pair $(j_{I,J!}^\circ, j_{I,J}^*)$ gives also an adjunction between $\mathcal{A}^I_{w, \mathbb{F}_q}$ and $\mathcal{A}^J_{w, \mathbb{F}_q}$ .

Proof. The direct sum in (5) has the natural structure of an object of $\mathcal{A}^I$ , as each summand object $j_{W_Jx!}^{W_I \circ}(A_w)_{w \in W_J x}$ has such a structure by Proposition 5.2. We then note that for any such $A \in \mathcal{A}^J$ and any $B \in \mathcal{A}^I$ ,

\begin{align*} & \mathrm{Hom}_{\mathcal{A}^I} \biggl(\bigoplus_{x \in W_J\backslash W_I} j_{W_Jx!}^{I\circ}((A_w)_{w \in W_Jx}),B\biggr)\\[10pt]& \quad \cong \bigoplus_{x \in W_J \backslash W_I} \mathrm{Hom}_{\mathcal{A}^I}(j_{W_J x!}^\circ((A_w)_{w \in W_J x}), B)\\[10pt]& \quad \cong \bigoplus_{x \in W_J \backslash W_I} \mathrm{Hom}_{\mathcal{A}_{W_J x}}((A_w)_{w \in W_J x}, j_{W_J x}^* B)\\[10pt]& \quad \cong \mathrm{Hom}_{\mathcal{A}^J}((A_w)_{w \in W}, \oplus_{x \in W_J \backslash W_I}j_{W_J x}^*B)\\[10pt]&\quad = \mathrm{Hom}_{\mathcal{A}^J}((A_w)_{w \in W}, j_{I,J}^*B).\end{align*}

To see that the adjoint pair $(j_{I,J!}^\circ, j_{I,J}^*)$ gives also an adjunction between $\mathcal{A}^I_{w, \mathbb{F}_q}$ and $\mathcal{A}^J_{w, \mathbb{F}_q}$ , we note that the morphisms on both sides of the equation above which are compatible with the morphism $\psi_A : \mathcal{F}_w\mathrm{Fr}^* A \to A$ are preserved by these isomorphisms.

5.2.1 Polishchuk’s canonical complex

For a fixed $A \in \mathcal{A}$ and a choice of $J \subset S$ , Polishchuk writes $A(J) = j_{S - J!}j_{S - J}^*A$ . In [Reference PolishchukPol01, §7.1], Polishchuk explains that for any $A \in \mathcal{A}$ , adjunction of parabolic pushforward and pullback functors $(j_{W_Ix!}, j_{W_Ix}^*)$ gives a canonical morphism $A(J) \to A(J')$ whenever $J \subset J'$ . He then defines the complex $C_\bullet(A)$ as

where $n = |S|$ .

Recalling the morphism $\iota : \mathcal{A} \to \mathcal{A}$ sending $(A_w)_{w \in W}$ to $(\Phi_{w_0}^\circ A_{w_0w})_{w \in W}$ , he describes natural morphisms $C_0 \to A$ and $\iota(A) \to C_{n-1}$ , and shows that

\begin{align*} H_i(C_\bullet(A)) & = \begin{cases} A, & i = 0,\\ 0, & i \neq 0, n - 1,\\ \iota(A), & i = n-1. \end{cases}\end{align*}

He then describes a 2n-term complex $\tilde{C}_\bullet$ formed from attaching $C_\bullet(\iota A)$ to $C_\bullet(A)$ via the maps $C_0(\iota A) \to \iota A \to C_{n-1}(A)$ with the property that

\begin{align*} H_i(\tilde{C}_{\bullet}(A)) & = \begin{cases} A, & i = 0,\\ 0, & i \neq 0, 2n - 1,\\ \iota^2(A), & i = 2n - 1. \end{cases}\end{align*}

5.2.2 Compatibility of the complex with w-twisted structure

Proof. For any k, components of the map $C_k(A) \to C_{k-1}(A)$ are each maps of the form

(6) \begin{align} j_{S-J!}j_{S-J}^*A \to j_{S-J'!}j_{S-J'}^*A, \end{align}

where $J' \subset J \subset S$ are such that $|J| = k$ , $|J'| = k - 1$ , which we now describe. By adjunction the data of such a map is equivalent to a map

(7) \begin{align} j_{S-J}^*A \to j_{S-J}^*j_{S-J'!}j_{S-J'}^*A, \end{align}

and compatibility with the w-twisted structure is preserved under this adjunction. By the definition of $j_{S-J'!}$ , we have that

\begin{align*} j_{S-J}^*j_{S-J'!}j_{S-J'}^*A & \cong \bigoplus_{x \in W_{S-J}\backslash W} j_{S-J}^*j_{W_{S-J}x!}j_{S-J'}^*A, \end{align*}

and one can check that the map in (6) appearing in the definition of Polishchuk’s complex in [Reference PolishchukPol01] comes in (7) from a natural injection in $\mathcal{A}^{S-J}$ from $j_{S-J}^*A$ into this direct sum defined by sending the yth tuple entry to the yth tuple entry in the direct summand corresponding to the unique x for which $y \in W_{S-J}x$ . It is straightforward to check that this injection preserves w-twisted Weil structures on both sides coming from the w-twisted Weil structure on A, and therefore the map in (6) is a map in $\mathcal{A}_{w, \mathbb{F}_q}$ .

5.2.3 Parabolic canonical complexes

We remark that the content appearing in §5.2.1 can be generalized to provide complexes in $\mathcal{A}^I_{w, \mathbb{F}_q}$ for any $I \subset S$ with $|I| = k$ and any $w \in W$ . Namely, if we fix $I \subset S$ , $w \in W$ , and $A \in \mathcal{A}_{w, \mathbb{F}_q}^I$ , we let $A^I(J) = j_{S-J!}^Ij_{S-J}^{I*}A$ whenever $J \supset S - I$ , and we define the complex $C^I_\bullet(A)$ by

indexed such that $C_{k-1}$ is the first term and $C_0$ is the last term in the above. The results in §5.2.1 then still hold, giving a version of the canonical complex associated to an object $A \in \mathcal{A}_{w, \mathbb{F}_q}^I$ .

5.2.4 Equations in the Grothendieck group

The following is a consequence of the fact that the full twist is central in the braid group, along with the fact that the symplectic Fourier transforms $\Phi_w$ form a braid action.

Lemma 5.6. For any $J \subset I \subset S$ there is a natural isomorphism

\begin{align*} j_{J!}^I\circ \iota^2 \cong \iota^2\circ j_{J!}^I \end{align*}

of functors from $D^b(\mathcal{A}^J)$ to $D^b(\mathcal{A}^I)$ .

Theorem 5.7. For any $w \in W$ and $A \in \mathcal{A}_{w, \mathbb{F}_q}$ the element

\begin{align*} (\iota^2 - 1)^n[A] \in K_0(\mathcal{A}_{w, \mathbb{F}_q}) \end{align*}

lies in $V^{\mathrm{fp}}$ , for $n = |S|$ .

Proof. By the derived category version of the canonical complex construction in combination with the observation about its homology provided in §5.2.1, for any $A \in \mathcal{A}_{w, \mathbb{F}_q}$ we have the following equation in $K_0(\mathcal{A}_{w, \mathbb{F}_q})$ :

\begin{align*} [\iota A] + (-1)^{|I| - 1}[A] & = \sum_{\substack{I'\\ I \subset I' \subsetneq S}} (-1)^{|I'|} [j_{J!}^Ij_{J}^{I*} A] \end{align*}

(cf. the proof of [Reference PolishchukPol01, Theorem 11.5.1] where the analogous equation is used in the case where $w = e$ , $I = S$ ). By the ‘doubled’ canonical complex $\tilde{C}_{\bullet}(A)$ and the description of its homology in §5.2.1, this means that $[\iota^2 A] - [A]$ is a linear combination of elements lying in the image of the functors $j_{J!}^Ij_{J}^{I*}$ for $J \subsetneq I$ .

Now by induction on the $|I|$ appearing in the equation above, it follows from Lemma 5.6 that $(\iota^2 - 1)^n[A]$ is a linear combination of elements lying in the image of the functors $j_{\varnothing!}j_{\varnothing}^*$ . We have that, for any $B \in \mathcal{A}^\varnothing_{w, \mathbb{F}_q}$ ,

\begin{align*} j_{\varnothing!}j_{\varnothing}^* = \oplus_{y \in W} j_{y!}B_y,\end{align*}

and each of these direct summands has finite cohomological dimension by Proposition 2.13, completing the proof of the theorem.

6.1.1 The full twist

Definition 6.1. Let $\mathcal{L} \in \mathrm{Ch}(T)$ . Then we define the element

\begin{align*} \mathrm{FT}_{\mathcal{L}} & = {}_{\mathcal{L}}\hat{\nabla}(w_0)_{w_0\mathcal{L}} * {}_{w_0\mathcal{L}}\hat{\nabla}(w_0)_{\mathcal{L}} \end{align*}

of ${}_{\mathcal{L}}\hat{\mathcal{P}}_{\mathcal{L}}$ . The object $\mathrm{FT}_{\mathcal{L}}$ admits a central structure and comes from a character sheaf in $D^b_{\mathfrak{C}, \mathcal{L}}(G_{\mathrm{ad}})$ via pullback composed with $\mathfrak{hc}_{\mathcal{L}}$ ; see [Reference Bezrukavnikov and TolmachovBT22] for an explicit description of this character sheaf in the unipotent case, whose proof can be adapted similarly for arbitrary monodromy. As with $\mathbb{K}_{\mathcal{L}}$ , we will identify $\mathrm{FT}_{\mathcal{L}}$ with its underlying character sheaf in $D^b_{\mathfrak{C}, \mathcal{L}}(G_{\mathrm{ad}})$ , and by $- * \mathrm{FT}_{\mathcal{L}}$ we will denote the action as described in Proposition 4.7.

We note that to properly extend the result in [Reference Bezrukavnikov and TolmachovBT22] to the case of nonunipotent monodromy, one needs an analog of [Reference Bezrukavnikov and TolmachovBT22, Theorem 5.6.4] (the statement that the Harish-Chandra transform composed with the long intertwining functor, i.e. the $*$ -version of the Radon transform, commutes with Verdier duality) adapted to the $\ell$ -adic setting. Before loc. cit., this result appeared for unipotent monodromy in [Reference Chen and Yom DinCYD17, Corollary 7.9]. As the authors of [Reference Bezrukavnikov, Ionov, Tolmachov and VarshavskyBITV23] note, the methods of [Reference Chen and Yom DinCYD17] can be adapted to treat the case of general monodromy. They also suggest that a new uniform proof will appear in upcoming work. Finally, note also that [Reference Bezrukavnikov, Finkelberg and OstrikBFO12, Corollary 3.4] treats the case of general monodromy in the characteristic-zero setting.

By the definition of $\mathrm{FT}_{\mathcal{L}}$ combined with Proposition 3.13, we obtain the following lemma.

6.1.2 The action of the full twist on cells

Proposition 6.3. For any $\underline{c} \in \underline{C}_{\mathcal{L}}$ , there exists an integer $d_{\mathcal{L}}(\underline{c})$ between 0 and $2\ell(w_0)$ such that for any $a \in K_0(D^b_{\mathfrak{C}, \mathcal{L}}(G)_{\underline{c}})$ ,

(8) \begin{align} [\mathrm{FT}_{\mathcal{L}}] * a = v^{d_{\mathcal{L}}(\underline{c})}a.\end{align}

To compute the value of d, we can pass back along the Harish-Chandra transform and work in the category $\mathcal{H}_{\mathcal{L}}^{(1)}$ . The Grothendieck ring $K_0(\mathcal{H}_{\mathcal{L}}^{(1)})$ is the monodromic Hecke algebra $\mathcal{H}_{\mathcal{L}}$ . By [Reference Lusztig and YunLY20], this is isomorphic to the usual Hecke algebra associated to the group $W_{\mathcal{L}}^\circ \subset W$ , with $\mathrm{FT}_{\mathcal{L}}$ being identified with the usual full-twist $\tilde{T}_{w_{0,\mathcal{L}}}^2$ . By [Reference LusztigLus84, 5.12.2], the full twist in the usual Hecke algebra acts on the cell subquotient module of the Hecke algebra corresponding to a cell $\underline{c}$ by the scalar $v^{d(\underline{c})}$ , where $d(\underline{c})$ is described in loc. cit. Passing this fact back along the monodromic-equivariant isomorphism from [Reference Lusztig and YunLY20], the result follows.

6.1.3 $\mathbb{K}_{\mathcal{L}}$ in the top cell subquotient

Definition 6.6. For any $\mathcal{L} \in \mathrm{Ch}(T)$ , let $q_{\mathcal{L}}(v)$ be the Poincaré polynomial

\[q_{\mathcal{L}}(v) = \sum_{w \in W} (-v^2)^{\ell(w)}\]

of the group $W_{\mathcal{L}}^\circ$ . Note by the Chevalley–Solomon formula that $q_{\mathcal{L}}(v)$ can be expressed as the product of some linear factors each of which is a factor of $(v^{2i} - 1)$ for some $1 \leq i \leq \ell(w_0)$ .¹

Combining of [Reference YunYun09, Theorem 5.3.1] with the expression of standard objects in terms of irreducible objects via inverse Kazhdan–Lusztig polynomials, we compute that the multiplicity of $\hat{\mathrm{IC}(e)}_{\mathcal{L}}$ is exactly

\begin{align*} \sum_{w \in W_{\mathcal{L}}^\circ} (-v^2)^{\ell(w_0) - \ell(w)} = \sum_{w \in W_{\mathcal{L}}^\circ} (-v^2)^{\ell(w)} = q_{\mathcal{L}}(v). \end{align*}

For the next proposition, we recall the definition of the character sheaves $\varepsilon_{n,\mathcal{L}}$ from §3.1.2.

Proposition 6.8. For any n, the element

(9) \begin{align} [\varepsilon_{n,\mathcal{L}} * \mathbb{K}_{\mathcal{L}}] - (v^2 - 1)^{\mathrm{rank}(T)}q_{\mathcal{L}}(v)[\varepsilon_{n,\mathcal{L}}]\end{align}

of $K_0(D^b_{\mathfrak{C}}(G_{\mathrm{ad}}))$ lies in the subspace $K_0(D^b_{\mathfrak{C},< \underline{c}_{e}}(G_{\mathrm{ad}}))$ .

By [Reference Bezrukavnikov and TolmachovBT22], $[\mathbb{K}_{\mathcal{L}}] = (v^2 - 1)^{\mathrm{rank}(T)}[\hat{\mathcal{T}}_{\mathcal{L}}]$ in the full Grothendieck group $K_0(\hat{\mathcal{P}}_{\mathcal{L}})$ . Note that in the corresponding top cell subquotient module for the monodromic Hecke algebra $K_0(\mathcal{H}_{\mathcal{L}}^{(1)})$ , the equation $[\hat{\mathcal{T}}_{\mathcal{L}}] - q_{\mathcal{L}}(v)[\hat{\delta}_{\mathcal{L}}]$ holds by Lemma 6.7. This means that $q'(v) = (v^2 - 1)^{\mathrm{rank}(T)}q_{\mathcal{L}}(v)$ is the only value for which $[\varepsilon_{n,\mathcal{L}} * {\mathbb{K}_{\mathcal{L}}}] - q'(v)[\varepsilon_{n,\mathcal{L}}]$ lies in a lower cell submodule of $K_0(D^b_{\mathfrak{C},\mathcal{L}}(G_{\mathrm{ad}}))$ , and therefore $[\varepsilon_{n,\mathcal{L}} * \mathbb{K}_{\mathcal{L}}] = (v^2 - 1)^{\mathrm{rank}(T)}q_{\mathcal{L}}(v)[\varepsilon_{n,\mathcal{L}}]$ .

6.1.4 Convolution with $\mathbb{K}_{\mathcal{L}}$ for Kazhdan–Laumon objects

Lemma 6.9. For any $s \in S$ , the map

\begin{align*} c_s * \mathbb{K}_{\mathcal{L}}: {}_{\mathcal{L}}\hat{\nabla}(s)_{s\mathcal{L}} * {}_{s\mathcal{L}}\hat{\nabla}(s)_{\mathcal{L}} * \mathbb{K}_{\mathcal{L}} \to \mathbb{K}_{\mathcal{L}} \end{align*}

is an isomorphism.

It is then enough to show that $p * \mathbb{K}_{\mathcal{L}}$ and $i' * \mathbb{K}_{\mathcal{L}}$ are each isomorphisms. This follows from the fact that $\hat{\mathrm{IC}}(s)_{\mathcal{L}} * \mathbb{K}_{\mathcal{L}} = 0$ . Indeed, $\hat{\mathrm{IC}}(s)_{\mathcal{L}} * \mathbb{K} = 0$ if and only if $\hat{\mathrm{IC}}(s)_{\mathcal{L}} * \hat{\mathcal{T}}_{\mathcal{L}} = 0$ , which follows from the fact that its class in the Grothendieck is zero combined with the fact that $\hat{\mathcal{T}}_{\mathcal{L}}$ is convolution-exact, since it is tilting.

Corollary 6.10. For any $A \in \mathcal{A}_{w,\mathbb{F}_q}$ , the object $A * \mathbb{K}_{\mathcal{L}}$ of $\mathcal{A}_{w,\mathbb{F}_q}$ has the property that for any $y, z \in W$ , the composition

\begin{align*} \theta_{z^{-1},zy} \circ (\Phi_{z^{-1}}^\circ\theta_{z,y}) : A_y * \mathbb{K}_{\mathcal{L}} \to A_y * \mathbb{K}_{\mathcal{L}} \end{align*}

is an isomorphism.

where the unlabeled arrows are the isomorphisms given by the central structure on $\mathbb{K}_{\mathcal{L}}$ ; these extend to similar central morphisms for conjugates of $\mathbb{K}_{\mathcal{L}}$ by standard/costandard sheaves by the same argument as in [Reference Lusztig and YunLY20, Lemma 11.12].

We note that since the second and third morphisms above clearly commute with these central morphisms, the above morphism agrees with the composition

where the first morphism is again the isomorphism coming from centrality of $\mathbb{K}_{\mathcal{L}}$ The composition of the last two morphisms in the sequence above must, by the definition of Kazhdan–Laumon categories, be equal to the morphism

\[A_y * c_z * {}_{y\mathcal{L}}\hat{\Delta}(y)_{\mathcal{L}} * \mathbb{K}_{\mathcal{L}} * {}_{\mathcal{L}}\hat{\nabla}(y^{-1})_{y\mathcal{L}},\]

where $c_z$ is the morphism ${}_{y\mathcal{L}}\hat{\nabla}(z^{-1})_{zy\mathcal{L}} * {}_{zy\mathcal{L}}\hat{\nabla}(z)_{y\mathcal{L}} \to \mathrm{Id}$ which is obtained by applying the morphisms $c_s$ successively for every simple reflection s in a reduced expression for z. By the functoriality of the central morphisms discussed above, they also commute with this $c_z$ , and so the entire composition above actually agrees with the morphism

\[A_y * {}_{y\mathcal{L}}\hat{\Delta}(y)_{\mathcal{L}} * \mathbb{K}_{\mathcal{L}} * {}_{\mathcal{L}}\hat{\nabla}(y^{-1})_{y\mathcal{L}} * c_z.\]

By our inductive definition of $c_z$ along with the same argument as in Lemma 6.9, $\mathbb{K}_{\mathcal{L}} * {}_{\mathcal{L}}\hat{\nabla}(y^{-1})_{y\mathcal{L}} * c_z$ is an isomorphism, and therefore $\theta_{z^{-1},zy} \circ (\Phi_{z^{-1}}^\circ \theta_{z,y})$ must be, too.

Proof. We can forget the w-twisted Weil structure on $A * \mathbb{K}_{\mathcal{L}}$ and consider it as an object in $\mathcal{A}$ . In $\mathcal{A}$ , the adjunction $(j_{e!}, j_{e}^*)$ gives a morphism

\begin{align*} a : j_{e!}j_e^*(A * \mathbb{K}_{\mathcal{L}}) \to A * \mathbb{K}_{\mathcal{L}}. \end{align*}

We claim that this is an isomorphism.

It is enough to show that the component morphisms $a_y : j_y^*j_{e!}j_e^*(A * \mathbb{K}_{\mathcal{L}}) \to j_y^*(A * \mathbb{K}_{\mathcal{L}})$ are each isomorphisms. By definition, these are the structure morphisms $\theta_{y,e} : \Phi_y^\circ(A * \mathbb{K}_{\mathcal{L}})_e \to (A * \mathbb{K}_{\mathcal{L}})_y$ .

By Corollary 6.10, the morphisms

\begin{align*} \theta_{y^{-1},y}\circ (\Phi_{y^{-1}}^\circ \theta_{y,e}) & : \Phi_{y{-1}}^\circ \Phi_y^\circ A_e \to A_e,\\ \theta_{y,e} \circ (\Phi_{y^{-1}}^\circ\theta_{y^{-1},y}) & : \Phi_{y}^\circ\Phi_{y^{-1}}^\circ A_{y} \to A_y \end{align*}

are both isomorphisms. This tells us that $\theta_{y,e}$ is both a monomorphism and an epimorphism.

This means $a : j_{e!}j_e^*(A * \mathbb{K}_{\mathcal{L}}) \to A * \mathbb{K}_{\mathcal{L}}$ is an isomorphism as we claimed. The proposition then follows since all objects in the image of $j_{e!}$ have finite cohomological dimension by Proposition 2.13.

6.2.1 Proof of Theorem 1.5 by the action of the full twist on cells

Lemma 6.12. For any $a \in K_0(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}})$ , there exists some $r \geq 1$ for which

\begin{align*} P^r(\mathrm{FT}_{\mathcal{L}}, v) \cdot a = 0 \end{align*}

in $K_0(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}})$ , where $P^r$ is the polynomial for which $P^r(x, v) = P(x, v)^r$ for any x.

Further, if $a \in K_0(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}})_{< \underline{c}_{e}}$ , then

\begin{align*} \tilde{P}^r(\mathrm{FT}_{\mathcal{L}}, v) \cdot a = 0. \end{align*}

Indeed, since by Propositions 6.3 and 3.9, the eigenvalues of the map $[\mathrm{FT}_{\mathcal{L}}] * -$ on $K_0(D^b_{\mathfrak{C},\mathcal{L}}(G_{\mathrm{ad}}))$ are each of the form $v^{2i}$ for $0 \leq i \leq \ell(w_0)$ and of the form $v^{2i}$ for $1 \leq i \leq \ell(w_0)$ on $K_0(D^b_{\mathfrak{C},\mathcal{L}}(G_{\mathrm{ad}})_{< \underline{c}_{e}})$ , we must only choose r to be the maximum multiplicity occurring in the characteristic polynomial of $[\mathrm{FT}_{\mathcal{L}}] * -$ on $K_0(D^b_{\mathfrak{C},\mathcal{L}}(G_{\mathrm{ad}}))$ , since each degree-one term of this characteristic polynomial is a factor of P(x, v) (respectively, $\tilde{P}(x, v)$ ) by definition of the latter. Choosing r in this way, we get that the polynomials in the lemma vanish, as desired.

The following two lemmas will be used in the proofs of Corollary 6.15 and Theorem 1.4.

Lemma 6.13. Suppose A is a $\mathbb{Z}[v]$ -module equipped with a $\mathbb{Z}[v]$ -linear endomorphism $T : A \to A$ . Let $Q(x, v) \in \mathbb{Z}[x, v]$ be any polynomial. Then if $a \in A$ satisfies

\begin{align*} Q(T, v) \cdot a & = 0, \end{align*}

then Q(1, v) a lies in the $\mathbb{Z}[T, v]$ -span of $(T - 1)\cdot a$ .

Proof. The polynomial $Q(x, v) - Q(1, v) \in \mathbb{Z}[x, v]$ lies in the ideal $(x - 1)$ , so we can write

\begin{align*} Q(1, v) & = Q(x, v) - \tilde{Q}(x, v)(x - 1) \end{align*}

for some polynomial $\tilde{Q}(x, v) \in \mathbb{Z}[x, v]$ . Now setting $x = T$ and applying both sides to a, we get

\begin{align*} Q(1, v)a & = Q(T, v)\cdot a - \tilde{Q}(T, v)(T - 1)\cdot a\\ & = -\tilde{Q}(T, v)(T - 1)\cdot a, \end{align*}

which is in the $\mathbb{Z}[T, v]$ -span of $(T - 1)\cdot a$ .

Proof. Suppose a satisfies $\tilde{P}(\iota^2, v)^k \cdot a = 0$ . By Theorem 5.7, we also have $(\iota^2 - 1)^{n} \cdot a \in V^{\mathrm{fp}}$ . We claim that this implies that

\[(\iota^2 - 1)^{n-j} \cdot a \in V^{\mathrm{fp}}_{p(v)}\]

for all $0 \leq j \leq n$ . Indeed, suppose for induction that

\[(\iota^2 - 1)^{n-j+1}\cdot a \in V^{\mathrm{fp}}_{p(v)},\]

and let $a' = (\iota^2 - 1)^{n-j} \cdot a$ . Then

\[(\iota^2 - 1)\cdot a' \in V^{\mathrm{fp}}_{p(v)}\]

and $\tilde{P}(\iota^2, v)^k \cdot a' = 0$ , so by Lemma 6.13 (applied to $T = \iota^2$ , $Q(x, v) = \tilde{P}(x, v)^k$ ), we have that $p(v)^k\cdot a' = \tilde{P}(1, v)^k\cdot a'$ is in the $\mathbb{Z}[\iota^2, v, v^{-1}]$ -span of $(\iota^2 - 1) \cdot a'$ , and therefore lies in $V^{\mathrm{fp}}_{p(v)}$ . Dividing by $p(v)^k$ gives $a' \in V^{\mathrm{fp}}_{p(v)}$ , and so we can proceed by induction until $j = n$ where we conclude that $a \in V^{\mathrm{fp}}_{p(v)}$ .

We now conclude the proof of Theorem 1.5, which states that for any character sheaf $\mathcal{L}$ of T and element $w \in W$ , the localization of the $\mathbb{Z}[v, v^{-1}]$ -module $K_0(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}})$ at p(v) is spanned by classes of objects of finite projective dimension in $\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}}$ .

Proof of Theorem 1.5. Let $A \in \mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}}$ . By Proposition 6.8,

(10) \begin{align} [A * \mathbb{K}_{\mathcal{L}}] - (v^2 - 1)^{\mathrm{rank}(T)}q_{\mathcal{L}}(v)[A] \end{align}

is an element of $K_0(D^b(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}}))_{< \underline{c}_e}$

By Corollary 6.15, this means $(v^2 - 1)^{\mathrm{rank}(T)}([A * \mathbb{K}_{\mathcal{L}}] - q_{\mathcal{L}}(v)[A]) \in V^{\mathrm{fp}}_{p(v)}$ .

By Proposition 6.11, $A * \mathbb{K}_{\mathcal{L}}$ itself has finite projective dimension, and so in combination with equation (10), this means $(v^2 - 1)^{\mathrm{rank}(T)}q_{\mathcal{L}}(v)p(v)[A] \in V^{\mathrm{fp}}_{p(v)}$ . Note that by Definition 6.6 each degree-one factor of $q_{\mathcal{L}}(v)$ is also a factor of p(v), so we can divide by $q_{\mathcal{L}}(v)$ in the localization $V^{\mathrm{fp}}_{p(v)}$ to get $[A] \in V^{\mathrm{fp}}_{p(v)}$ , completing the proof.

7.1.1 Polishchuk’s description of $K_0(\mathcal{A}_{\mathbb{F}_q})$

A crucial tool which we will use in the proof of Theorem 1.4 is the following description of $K_0(\mathcal{A}_{\mathbb{F}_q})$ provided by Polishchuk.

Theorem 7.1 [Reference PolishchukPol01, Proposition 3.4.1]. The map

\begin{align*} K_0(\mathcal{A}_{\mathbb{F}_q}) & \to \bigoplus_{w \in W} K_0(\mathrm{Perv}_{\mathbb{F}_q}(G/U)) \end{align*}

induced by the functor $\oplus_{w \in W} j_{w}^*$ is injective. Its image is the subset

\[\{(a_w)_{w \in W} \in K_0(\mathrm{Perv}_{\mathbb{F}_q}(G/U)) ~|~a_{sw} - \Phi_s a_w \in \mathrm{im}(\Phi_s^2 - 1), s \in S, w\in W\}.\]

7.1.2 Recalling [Reference Morton-FergusonMF24]

In [Reference Morton-FergusonMF24], we study the subalgebra $\mathrm{KL}(v)$ of endomorphisms of $K_0(G/U)$ generated by the symplectic Fourier transforms $\{\Phi_s\}_{s \in S}$ . By § 2.2.1, this is the same as the subalgebra of $K_0(G/U \times G/U)$ generated under convolution by classes of Kazhdan–Laumon sheaves; we denote the generator of $\mathrm{KL}(v)$ corresponding to $w \in W$ by $\mathsf{a}_w$ . In this section, we use the monodromic Hecke algebras $\mathcal{H}_{\mathcal{L}}$ and $\mathcal{H}_{\mathfrak{o}}$ with the standard generators $\tilde{T}_w$ as defined in [Reference Lusztig and YunLY20]; see [Reference Morton-FergusonMF24] for a more precise outline of our conventions.

In [Reference Morton-FergusonMF24, §4], we show that for any character sheaf $\mathcal{L}$ with W-orbit $\mathfrak{o}$ , there is a surjection $\pi_{\mathcal{L}} : \mathrm{KL}(v) \to \mathcal{H}_{\mathfrak{o}}$ . The following result follows from the main result of loc. cit. which explicitly identifies the algebra $\mathrm{KL}(v)$ as a subalgebra of a generic-parameter version of the Yokonuma–Hecke algebra.

1. (i) If $w_1, w_2 \in W$ , then $\pi_{\mathcal{L}}(\mathsf{a}_{w_1}\mathsf{a}_{w_2}) = \pi_{w_2\mathcal{L}}(\mathsf{a}_{w_1})\pi_{\mathcal{L}}(\mathsf{a}_{w_2})$ in $\mathcal{H}_{\mathfrak{o}}$ .

2. (ii) If $w \in W_{\mathcal{L}}^\circ$ , then $\pi_{\mathcal{L}}(\mathsf{a}_w) = \tilde{T}_{w} \in \mathcal{H}_{\mathfrak{o}}$ .

3. (iii) If $s\in S$ is not in $W_{\mathcal{L}}^\circ$ , then $\pi_{\mathcal{L}}(\mathsf{a}_s^2) = 1$ .

Finally, the morphism $\prod_{\mathcal{L}} \pi_{\mathcal{L}}$ is injective, so if $a \in \mathrm{KL}(v)$ is such that $\pi_{\mathcal{L}}(a) = 0$ for all character sheaves $\mathcal{L}$ , then $a = 0$ .

Lemma 7.3. For any $\mathcal{L}$ with W-orbit $\mathfrak{o}$ , the algebra morphism $\pi_{\mathcal{L}} : \mathrm{KL}(v) \to \mathcal{H}_{\mathfrak{o}}$ is such that

\begin{align*} \pi_{\mathcal{L}}(\mathsf{a}_{w_0}^2) & = \tilde{T}_{w_{0,\mathcal{L}}}^2, \end{align*}

where $w_{0,\mathcal{L}}$ is the longest element of $W_{\mathcal{L}}^\circ$ .

Indeed, if we had $y^{-1}sy \in W_{\mathcal{L}}^\circ$ , then we since $w_{0,\mathcal{L}}$ dominates all elements of $W_{\mathcal{L}}^\circ$ in the Bruhat order, we could pick $z \in W$ such that $y^{-1}syz = w_{0,\mathcal{L}}$ with $\ell(y^{-1}sy) + \ell(z) = \ell(w_0,\mathcal{L})$ . But then we would have $syz = yw_{0,\mathcal{L}}$ with $\ell(sy) + \ell(z) = \ell(syz) = \ell(yw_{\mathcal{L},0})$ . But since $\ell(sy) < \ell(y)$ and $\ell(z) \leq \ell(w_{0,\mathcal{L}})$ , this is impossible.

Now choose some y for which $yw_{0,\mathcal{L}} = w_0$ with $\ell(y) + \ell(w_{0,\mathcal{L}})$ . We will show that $\pi_{\mathcal{L}}(\mathsf{a}_{w_0}^2) = \tilde{T}_{w_{0,\mathcal{L}}}^2$ by induction on the length of y. Choosing $s \in S$ such that $\ell(sy) < \ell(y)$ , this induction hypothesis along with the fact proved in the previous paragraph gives that

\begin{align*} \pi_{\mathcal{L}}(\mathsf{a}_{w_0}^2) & = \pi_{\mathcal{L}}(\mathsf{a}_{w_{0,\mathcal{L}}}\mathsf{a}_{y^{-1}}\mathsf{a}_y\mathsf{a}_{w_{0,\mathcal{L}}})\\ & = \pi_{\mathcal{L}}(\mathsf{a}_{w_{0,\mathcal{L}}}\mathsf{a}_{(sy)^{-1}}\mathsf{a}_s^2\mathsf{a}_{sy}\mathsf{a}_{w_{0,\mathcal{L}}})\\ & = \pi_{sy\mathcal{L}}(\mathsf{a}_{w_{0,\mathcal{L}}}\mathsf{a}_{(sy)^{-1}})\pi_{sy\mathcal{L}}(\mathsf{a}_s^2)\pi_{\mathcal{L}}(\mathsf{a}_{sy}\mathsf{a}_{w_{0,\mathcal{L}}})\\ & = \pi_{\mathcal{L}}(\mathsf{a}_{w_{0,\mathcal{L}}}\mathsf{a}_{(sy)^{-1}}\mathsf{a}_{sy}\mathsf{a}_{w_{0,\mathcal{L}}})\\ & = \pi_{\mathcal{L}}(\mathsf{a}_{syw_{0,\mathcal{L}}}^2)\\ & = \tilde{T}_{w_{0,\mathcal{L}}}^2.\end{align*}

7.1.3 The action of the full twist on $\mathrm{Perv}_{\mathbb{F}_q}(G/U)$

Now recall that the endomorphism $\Phi_{w_0} : K_0(G/U) \to K_0(G/U)$ agrees with right convolution with the Kazhdan–Laumon sheaf $\overline{K(w_0)}$ . Since the convolution $D^b(G/U) \times D^b(G/U \times G/U) \to D^b(G/U)$ is a triangulated functor, it is enough to show that $P([\overline{K(w_0)} * \overline{K(w_0)}], v) = 0$ in $K_0(G/U \times G/U)$ .

Letting $1_{\mathcal{L}}$ be the idempotent in $\mathcal{H}_{\mathfrak{o}}$ corresponding to the $\mathcal{L}$ -monodromic subalgebra, in loc. cit. we show that

\begin{align*} \pi_{\mathcal{L}}(\mathsf{a}_s) & = \begin{cases} -v\tilde{T}_{s}^{-1}1_{\mathcal{L}}, & s \in W_{\mathcal{L}}^\circ,\\ -\tilde{T}_s1_{\mathcal{L}}, & s \not\in W_{\mathcal{L}}^\circ. \end{cases} \end{align*}

It is a straightforward calculation from the above to show that $\pi_{\mathcal{L}}([K(w_0) * K(w_0)]) = v^{2\ell(y)}\tilde{T}_{y}^{-2}1_{\mathcal{L}} \in \mathcal{H}_{\mathcal{L}}$ , where y is the longest element of $W_{\mathcal{L}}^\circ$ . By the main result of [Reference Lusztig and YunLY20], the monodromic Hecke algebra $\mathcal{H}_{\mathcal{L}}$ is isomorphic to $\mathcal{H}_{W_{\mathcal{L}}^\circ}$ , with full twists on each side being identified as in Lemma 7.3. By [Reference LusztigLus84, §5.12.2] which identifies the eigenvalues of the full twist in the regular representation, we have that $P(-v^{2\ell(y)}\tilde{T}_{y}^2, v) = 0$ in $\mathcal{H}_{\mathcal{L}}$ .

Finally, we note that by Proposition 7.2, if a polynomial is satisfied by $\pi_{\mathcal{L}}([\overline{K(w_0)} * \overline{K(w_0)}])$ in each $\mathcal{H}_{\mathfrak{o}}$ , then it is also satisfied in $K_0(G/U \times G/U)$ ; this completes the proof of the proposition.

7.2.1 Proof of Theorem 1.4

Let $a \in K_0(\mathcal{A}_{\mathbb{F}_q})$ . We can write $p(v) = \tilde{P}(1, v) = \tilde{P}(x, v) + r(x, v)(x - 1)$ for some $r(x, v) \in \mathbb{Z}[x, v]$ . So if

\begin{align*} a_0 & = \tilde{P}(\iota^2, v)a,\\ a_1 & = r(\iota^2, v)(\iota^2 - 1)a,\end{align*}

then $a_0 + a_1 = p(v)a$ , so it suffices to show that $a_0, a_1 \in V^{\mathrm{fp}}_{p(v)}(\mathcal{A}_{\mathbb{F}_q})$ .

First we show this for $a_0$ . We claim that for any $w \in W$ and $s \in S$ , $\Phi_{s}^2((a_0)_w) = (a_0)_w$ . Since $a_0 = \tilde{P}(\Phi_{w_0}^2, v)a$ , this will follow from the fact that for any $s \in I$ , $(\Phi_s^2 - 1)\tilde{P}(\Phi_{w_0}^2, v) = 0$ as an endomorphism of $K_0(G/U)$ . By Proposition 7.2, this relation holds if and only if it holds after applying $\pi_{\mathcal{L}}$ for any character sheaf $\mathcal{L}$ . By Lemma 7.3, this reduces to showing that if $\tilde{P}(\tilde{T}_{w_0}^2, v) \neq 0$ , then $(\tilde{T}_s^2 - 1)\tilde{P}(\tilde{T}_{w_0}^2, v) = 0$ for any $s \in S$ . We can rephrase this as saying that if the full twist acts by the eigenvalue 1, then so does $\tilde{T}_{s}^2$ for every $s \in S$ . Indeed, this follows from the classification of irreducible representations of Hecke algebras, and it is shown directly in [Reference PolishchukPol01, §11.5.3].

Now note that by Theorem 7.1, we have

(11) \begin{align} (a_0)_{sw} - \Phi_s(a_0)_w = (\Phi_s^2 - 1)b\end{align}

for some b. The relation $(\Phi_s + v^2)(\Phi_s^2 - 1) = 0$ from [Reference PolishchukPol01, Proposition 6.2.1] can be rewritten as $\Phi_s(\Phi_s^2 - 1) = -v^2(\Phi_s^2 - 1)$ , so when we apply $(\Phi_s^2 - 1)$ to the right-hand side of (11), we get

\begin{align*} (\Phi_s^2 - 1)^2b & = \Phi_s^2(\Phi_s^2 - 1)b - (\Phi_s^2 - 1)b\\ & = v^4(\Phi_s^2 - 1)b - (\Phi_s^2 - 1)b\\ & = (v^4 - 1)(\Phi_s^2 - 1)b.\end{align*}

This means that by applying $(\Phi_s^2 - 1)$ to both sides of (11), and using the fact that $\Phi_s^2((a_0)_{w'}) = (a_0)_{w'}$ for $w' = w$ and $w' = sw$ , we then get

\begin{align*} 0 & = (\Phi_s^2 - 1)((a_0)_{sw} - \Phi_s(a_0)_w)\\ & = (\Phi_s^2 - 1)^2b\\ & =(v^4 - 1)(\Phi_{s}^2 - 1)b,\\ & = (v^4 - 1)((a_0)_{sw} - \Phi_s(a_0)_w), \end{align*}

so $(v^4 - 1)((a_0)_{sw} - \Phi_s(a_0)_w) = 0$ , meaning $(a_0)_{sw} = \Phi_s(a_0)_w$ . This means $a_0 = j_{w!}j_w^*(a_0)$ in $K_0(\mathcal{A}_{\mathbb{F}_q})$ for any $w \in W$ , and so $a_0 \in V^{\mathrm{fp}}_{p(v)}$ .

Now it only remains to observe that $a_1 \in V^{\mathrm{fp}}_{p(v)}$ . By Proposition 7.4 and the definition of $a_1$ , $\tilde{P}(\iota^2, v)a_1 = 0$ . By Lemma 6.14, $a_1 \in V^{\mathrm{fp}}_{p(v)}$ . Then since $a_0$ and $a_1$ both lie in $V^{\mathrm{fp}}_{p(v)}$ and $p(v)a = a_0 + a_1$ , we have that $a \in V^{\mathrm{fp}}_{p(v)}$ .

8.1.1 The original proposal in [Reference Kazhdan and LaumonKL88]

In [Reference Kazhdan and LaumonKL88, §3], the proposed construction of representations is as follows. The authors begin by making Conjecture 1.1, which we now know to be false by Bezrukavnikov and Polishchuk’s appendix to [Reference PolishchukPol01]. However, for objects of finite projective dimension, one can still define the Grothendieck–Lefschetz-type pairing in the manner they describe

First, they define a Verdier duality functor $\mathbb{D} : \mathcal{A}_\psi \to \mathcal{A}_{\psi^{-1}}$ , where $\mathcal{A}_{\psi} = \mathcal{A}$ as we have been using it throughout this paper, while $\mathcal{A}_{\psi^{-1}}$ is the same category but using the additive character $\psi^{-1}$ instead of $\psi$ (where $\psi$ is the additive character we chose in §2.1.2).

They note that for any $A \in\mathcal{A}_{w, \mathbb{F}_q}$ and $B \in (\mathcal{A}_{\psi^{-1}})_{w, \mathbb{F}_q}$ and any $i \in \mathbb{Z}$ , the isomorphisms $\psi_A : \mathcal{F}_w\mathrm{Fr}^*A \to A$ and $\psi_B : \mathcal{F}_w\mathrm{Fr}^*B \to B$ give an endomorphism $\psi_{A,B}^i$ of the vector space $\mathrm{Ext}_{\mathcal{A}}^i(A, \mathbb{D}B)$ given by the composition

\begin{align*} & \mathrm{Ext}_{\mathcal{A}}^i(A, \mathbb{D}B) \rightarrow \mathrm{Ext}_{\mathcal{A}}^i(\mathcal{F}_w\mathrm{Fr}^*A, \mathcal{F}_{w}\mathrm{Fr}^*\mathbb{D}B)\\ & \quad\rightarrow \mathrm{Ext}_{\mathcal{A}}^i(\mathcal{F}_w A, \mathcal{F}_w \mathbb{D}B) \rightarrow \mathrm{Ext}_{\mathcal{A}}^i(A, \mathbb{D}B)\end{align*}

where the first map arises from the morphisms $\psi_A$ and $\psi_B$ , the next from the canonical isomorphisms $\mathrm{Fr}^*A \to A$ and $\mathrm{Fr}^*B \to B$ , and the last from the fact that $\mathcal{F}_w$ is invertible. This map is also described explicitly in [Reference Braverman and PolishchukBP98, § 4.3.1].

We can then define, for A having finite projective dimension and arbitrary B, the value

\begin{align*} \langle [A], [B]\rangle = \sum_{i\in \mathbb{Z}} (-1)^i \mathrm{tr}(\psi^i_{A,B}, \mathrm{Ext}^i_{\mathcal{A}}(A, \mathbb{D}B)).\end{align*}

This is clearly well defined at the level of Grothendieck groups. We will now explain how to use the result in Theorem 1.5 to extend Kazhdan and Laumon’s pairing, which as of this point is only defined for objects of finite projective dimension, to the full Grothendieck group in the monodromic case.

It is a straightforward computation that for any such A and B, we have

\begin{align*} \langle [A(-\tfrac{1}{2})], [B]\rangle = q^{\frac{1}{2}}\langle [A], [B]\rangle,\end{align*}

so the pairing is $\mathbb{Z}[v, v^{-1}]$ -linear where $\mathbb{Z}[v, v^{-1}]$ acts on the target field such that v is multiplication by $q^{\frac{1}{2}}$ .

8.1.2 A pairing on $K_0(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}}) \otimes \mathbb{C}$

We can do the same construction on the monodromic Kazhdan–Laumon category $\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}}$ and its Grothendieck group. Then using $\mathbb{Z}[v, v^{-1}]$ -linearity, the above definition gives us a pairing which is well defined on elements of $V^{\mathrm{fp}}$ .

Now we note that the polynomial p(v) evaluated at $v = q^{\frac{1}{2}}$ is nonzero, so we can extend this pairing linearly to the localization $V^{\mathrm{fp}}_{p(v)}$ . This then gives that the pairing is well defined on all of $V^{\mathrm{fp}}_{p(v)} \otimes \mathbb{C}$ , where in the tensor product we send $v \mapsto q^{\frac{1}{2}}$ . But, by Theorem 1.5, $V^{\mathrm{fp}}_{p(v)} \otimes \mathbb{C}$ is all of $K_0(\mathcal{A}_{w, \mathbb{F}_q}^{\mathcal{L}}) \otimes \mathbb{C}$ , so we can indeed define the pairing on this entire vector space; we will now explain how to use this to construct the Kazhdan–Laumon representations.