On the Induction of p-Cells

Abstract

We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson (2017) and Jensen (2020)) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right p-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the p-canonical basis.

1 Introduction

Let (W, S) be a Coxeter system and let ${\mathcal {H}_{(W,S)}}$ denote its Hecke algebra. In [22], Kazhdan and Lusztig introduced a preorder on the elements of W whose equivalence classes are known as right Kazhdan-Lusztig cells. These Kazhdan-Lusztig cells can be used to construct representations of ${\mathcal {H}_{(W,S)}}$, called cell modules. Since their introduction, cells have been extensively studied and there is a very rich theory describing them.

If the Coxeter group is crystallographic, we can replace the Kazhdan-Lusztig basis of ${\mathcal {H}_{(W,S)}}$ with the p-canonical basis (see [20]) which can be obtained as characters of the indecomposable parity complexes on a suitable Kac-Moody flag variety with coefficients in a field of characteristic p. For $p=0$, the p-canonical basis specializes to the Kazhdan-Lusztig basis by results of Kashiwara and Tanisaki [23].

Working with the p-canonical basis, we obtain a positive characteristic analogue of the Kazhdan-Lusztig cells, called p-cells. The main motivation for studying p-cells is that, other than providing a first approximation of the multiplicative structure of the p-canonical basis, they allow one to construct representations of the Hecke algebra (called cell modules) that naturally come with a canonical basis.

Another motivation comes from the study of coherent sheaves on the nilpotent cone. According to a recent conjecture by Achar, Hardesty and Riche [2, Conjecture 5.5] antispherical p-cells in affine Weyl groups are essentially Lusztig-Vogan cells, hence they correspond to vector bundle cells on nilpotent orbits. Moreover, these vector bundles can be constructed in terms of the $G_1$-cohomology of tilting modules, where $G_1$ denotes the first Frobenius kernel (cf. [1]).

The theory of p-cells has been initially developed by the first author in [16], where several properties of Kazhdan-Lusztig cells are generalized to p-cells. The following theorem is inspired by [5, Proposition 3.11] and [29, Proposition 1] and follows immediately from a key result in [16, Theorem 3.9].

Parabolic induction is well-known for Kazhdan-Lusztig cells. It was first proven by Barbasch and Vogan in [5, Proposition 3.15] (see also [25, (5.26.1)]) for a finite Weyl group W in the setting of primitive ideals for complex semi-simple Lie algebras. For arbitrary Coxeter groups, it was first conjectured by Roichman in [29, $S_{5}$]. Finally, Geck proved it in the general setting of cells for Hecke algebras with unequal parameters in [10].

In the following we will use the notation of the theorem. One may introduce the I-hybrid basis, which combines the p-canonical basis of ${\mathcal {H}_{(W_I, I)}}$ and the standard basis elements corresponding to $^I W$ to get a ${\mathbb {Z}}[v, v^{-1}]$-basis of ${\mathcal {H}_{(W, S)}}$. Geck’s proof is completely elementary and algebraic, combinatorial in nature. It is based on the study of the I-hybrid basis of the Hecke algebra (for $p = 0$) and its relation with the Kazhdan-Lusztig basis.

In contrast, our proof is based on a geometric argument. We use the categorification of the p-canonical basis via indecomposable parity complexes on the flag variety of a Kac-Moody group. First, we prove that the base change coefficients between the p-canonical and the I-hybrid basis are Laurent polynomials with non-negative coefficients. To achieve this, we relate the base change between these two bases to Braden’s hyperbolic localization functors [4]. This works because parity sheaves are well behaved with respect to the hyperbolic localization functors (see [17]). We remark that our argument is an adaptation of the work of Grojnowski and Haiman [12], which deals with mixed Hodge modules in characteristic 0, to the setting of parity sheaves.

The rest of the proof follows along the lines of Geck’s proof. It uses a hybrid order which combines the right p-cell preorder in $W_I$ with the Bruhat order on $^I W$. The crucial ingredient here is Proposition 3.13 which shows that the base change coefficients between the p-canonical basis and the I-hybrid basis are unitriangular with respect to the hybrid order.

1.1 Structure of the Paper

• Section 2 We recall the definition of the Hecke algebra of a crystallographic Coxeter system. Then we discuss its geometric categorification via parity sheaves. We also recall the definition of and the main result about Braden’s hyperbolic localization functors. Finally, we introduce the p-canonical basis and p-cells and mention some of their elementary properties.

• Section 3 We introduce a new basis, called I-hybrid basis, for the Hecke algebra, combining the p-canonical basis for a parabolic subgroup and the standard basis. Then we study positivity properties of the p-canonical basis with respect to the I-hybrid basis. Finally, we prove that induction for p-cells still works.

2 Background

2.1 Crystallographic Coxeter Systems and Their Hecke Algebras

Let S be a finite set and $(m_{s, t})_{s, t \in S}$ be a matrix with entries in ${\mathbb {Z}}_{\geqslant 0} \cup \{\infty \}$ such that $m_{s,s} = 1$ and $m_{s, t} = m_{t, s} \geqslant 2$ for all $s \ne t \in S$. Denote by W the group generated by S subject to the relations $(st)^{m_{s, t}} = 1$ for $s, t \in S$ with $m_{s, t} < \infty$. We say that (W, S) is a Coxeter system and W is a Coxeter group. The Coxeter group W comes equipped with a length function $\ell : W \rightarrow {\mathbb {Z}}_{\geqslant }0$ and the Bruhat order $\leqslant$ (see [15] for more details). A Coxeter system (W, S) is called crystallographic if $m_{s, t} \in \{2, 3, 4, 6, \infty \}$ for all $s \ne t \in S$. We denote the identity of W by Id.

From now on, fix a generalized Cartan matrix $A = (a_{i,j})_{i, j \in J}$ (see [24, $S_{1}$.1.1]). Let $(J, {X}, \{{\alpha _{i}}: i \in J\}, \{{\alpha _{i}^{\vee }}: i \in J \})$ be an associated Kac-Moody root datum (see [34, $S_{1}$.2] for the definition). Then X is a finitely generated free abelian group, and for $i \in J$ we have elements ${\alpha _{i}}\in X$ and ${\alpha _{i}^{\vee }}\in {{X^{\vee }}} = \text {Hom}_{\mathbb {Z}}({X},{\mathbb {Z}})$ respectively satisfying $a_{i,j} = {{\alpha _{i}^{\vee }}}({\alpha _{j}})$ for all $i, j \in J$.

We require our Kac-Moody root datum to satisfy the following assumptions (see [26, $S_{7}$.3.1]):

• 1. free, i.e., the set $\{ {\alpha _{i}}: i \in J\}$ is linearly independent over ${\mathbb {Z}}$ in X,

• 2. cofree, i.e., the set $\{ {{\alpha _{i}^{\vee }}}: i \in J \}$ is linearly independent over ${\mathbb {Z}}$ in ${{X^{\vee }}}$,

• 3. cotorsion free, i.e., ${{X^{\vee }}} / \sum _{i \in J} {\mathbb {Z}} {{\alpha _{i}^{\vee }}}$ is torsion-free,

• 4. X is of rank $2|J |- \text {rk}(A)$.

Remark 2.1

These assumptions were originally imposed by Mathieu and Kumar (see [26, Example 7.10] for the relation between a realization of A and a Kac-Moody root datum associated to A satisfying the assumptions (i)–(iv)). Even though they are not necessary for the construction of Kac-Moody groups (see [30, Remarque 3.5 and $S_{3}$.19] and [26, $S_{8}$.7]), we will need the first assumption in our proof.

To A we associate a crystallographic Coxeter system (W, S) as follows: choose a set of simple reflections S of cardinality $|J |$ and fix a bijection $S \overset{\sim }{\rightarrow }\ J$, $s \mapsto i_s$. For $s \ne t \in S$ we define $m_{s, t}$ to be 2, 3, 4, 6, or $\infty$ if $a_{i_s, i_t} a_{i_t, i_s}$ is 0, 1, 2, 3, or $\geqslant 4$ respectively. The group W is called the Weyl group of A.

The Hecke algebra $\mathcal {H} = \mathcal {H}_{(W, S)}$ associated to (W, S) is the free ${\mathbb {Z}}[v, v^{-1}]$-algebra with $\{{{H_{w}}} \; \vert \; w \in W \}$ as basis, called the standard basis, and multiplication determined by:

$$\begin{aligned} {{H_{s}}}^2&= (v^{-1} - v) {{H_{s}}} + 1 \qquad{} & {} \text {for all } s \in S, \\ {{H_{x}}} {{H_{y}}}&= {{H_{xy}}}{} & {} \text {if } \ell (x) + \ell (y) = \ell (xy) . \end{aligned}$$

There is a unique ${\mathbb {Z}}$-linear ring involution $\overline{(-)}$ on $\mathcal {H}$ satisfying $\overline{v} = v^{-1}$ and $\overline{{H_{x}}} = {{H_{x^{-1}}}}^{-1}$. The Kazhdan-Lusztig basis element ${{\underline{H}_{x}}}$ is the unique element in ${{H_{x}}} + \sum _{y < x} v{\mathbb {Z}}[v] H_y$ that is invariant under $\overline{(-)}$. This is Soergel’s normalization from [32] of a basis introduced originally in [22].

Let $\iota$ be the ${\mathbb {Z}}[v, v^{-1}]$-linear anti-involution on ${\mathcal {H}}$ satisfying $\iota ({{H_{s}}}) = {{H_{s}}}$ for $s \in S$ and thus $\iota ({{H_{x}}}) = \{{H_{x^{-1}}}$ for $x \in W$.

2.2 Parity Complexes on Kac-Moody Flag Varieties

In this section we want to recall some results about Kac-Moody flag varieties and parity complexes on them. The standard references for Kac-Moody flag varieties are [26] and [24].

To our Kac-Moody root datum with Cartan matrix A we can associate a (maximal) Kac-Moody group G with a canonical torus $T \subset G$. Mathieu actually constructs group ind-schemes over ${\mathbb {Z}}$ in [27] and [28], but for our purposes it is enough to consider their complex points as in [24, $S_{6}$.1] (see [26, Exercise 8.123] and [30, $S_{3}$.20] for comparisons of the two constructions).

Let W be the Weyl group of G, i.e., the Weyl group of the Cartan matrix A. Let ${{\Phi }}$ denote the set of roots of G and ${{\Phi }}_+$ the set of positive roots. We denote by ${{\Phi }}^{\text {re}}$ the set of real roots, i.e., roots that can be written as $w\cdot \alpha _i$, where $w\in W$ and $\alpha _i$ is a simple root.

Let $U^+$ be the (pro-)unipotent group obtained as the image of the positive part of the associated Kac-Moody Lie algebra under the exponential map defined in [24, $S_{6}$.1.1] (also denoted $\mathfrak {U}^{ma+}$ in [30, 3.1]). For every positive real root ${\alpha } \in {{\Phi }}^{\text {re}}_+$ there exists a one parameter subgroup $U_{{\alpha }} \subset U^+$ isomorphic to the additive group ${\mathbb {Z}}{G}_a$ (see [24, Example 6.1.5]). For every ${\alpha }$ we fix an isomorphism $u_{\alpha }: {\mathbb {G}}_a \overset{\sim }{\longrightarrow }\ U_{\alpha }$. Such an isomorphism is unique up to multiplication by a unit in ${\mathbb {C}}$ (acting on ${\mathbb {G}}_a$). The subgroup $U^+$ is normalized by the torus T. Moreover, every $U_{\alpha }$ for ${\alpha }$ a positive real root is normalized by T and we have $t u_{\alpha }(x) t^{-1} = u_{\alpha }(\alpha (t)x)$ for $t \in T$ and $x \in {\mathbb {C}}$.

The Borel subgroup B is a subgroup of G isomorphic to the semidirect product $B=T\ltimes U^+$. The set G/B may be equipped with the structure of an ind-projective ind-variety and it is called Kac-Moody flag variety (see [24, $S_{7}$.1]).

For any finitary subset $I \subseteq S$ we have the corresponding standard parabolic subgroup $P_I$ containing B (see [24, Definitions 6.1.13 (4) and 6.1.18]). It admits a Levi decomposition $P_I = L_I \ltimes U_I$, where $L_I$ is a connected (finite dimensional) reductive group associated to $A_I$, the sub-Cartan matrix of A obtained by taking the rows and the columns indexed by I. We denote by ${{\Phi }}_I$ the roots of $L_I$. The group T is a connected algebraic torus, whereas G, B, $U^+$, $P_I$ and $U_I$ are all pro-algebraic groups. The following two examples cover the most important cases:

Example 2.2

If A is a Cartan matrix, then the Kac-Moody root datum is equivalent to a root datum in the ordinary sense, and G is the associated complex simply connected algebraic group, B a Borel subgroup, $T \subset B$ a maximal torus and $U^+$ the unipotent radical of B. In this case, $P_I$ is a standard parabolic subgroup and $L_I$ the corresponding Levi.

Example 2.3

Assume that $A= (a_{i,j})$ is a Cartan matrix of size $l-1$ and that the Kac-Moody root datum is simply connected (see [26, Example 7.11]). Let G be the corresponding semi-simple simply connected algebraic group. Moreover, one can add an l-th row and column to obtain a generalized Cartan matrix $\widetilde{A}$ as follows

$$\begin{aligned} a_{j,l}&= -\theta ({{\alpha _{j}^{\vee }}}),\\a_{l,j}&= -{{\alpha _{[j]}}}(\theta ^{\vee }) \quad \text {for } 1 \leqslant j < l,\\ a_{l,l}&= 2 \end{aligned}$$

where the $\alpha _i$ are the simple roots of G and $\theta$ the highest root. In this case, the Kac-Moody group $\widetilde{G}$ associated to $\widetilde{A}$ is of so-called untwisted affine type (see [24, $S_{1}$3.2]) and some Kac-Moody flag varieties associated to $\widetilde{G}$ admit an alternative description as partial affine flag varieties. Denote by $\mathcal {K} = {\mathbb {C}}((t))$ the field of complex Laurent series and by $\mathcal {O} = {\mathbb {C}}[t]$ the ring of complex power series. The Iwahori subgroup I is defined as the inverse image of a Borel subgroup $B \subset G({\mathbb {C}})$ under the evaluation map $G(\mathcal {O}) \rightarrow G({\mathbb {C}})$, $t \mapsto 0$. Then the affine flag variety $G(\mathcal {K}) / I$ and the affine Grassmannian $G(\mathcal {K}) / G(\mathcal {O})$ are isomorphic to the Kac-Moody flag variety $\widetilde{G} / \widetilde{P}_I$ for $I= \emptyset$ and $I = \{1, \dots , l-1\}$ respectively.

We will mostly be interested in the orbits of $P_I$ on ${{\mathcal {F}l_{G}}} = G / B$. Fix a finitary subset $I \subseteq S$ and the standard parabolic subgroup $W_I = \langle I \rangle \subseteq W$ generated by I. The group $P_I$ can also be realized as $P=BW_IB$. Each left $W_I$-coset contains a unique element of minimal length. We denote by $^{I}W$ the set of these minimal coset representatives. The action of $P_I$ on G/B induces the Bruhat decomposition

$$\begin{aligned} G/B= \bigcup _{w \in ^{}I W} {}^I Y_w \end{aligned}$$

where ${}^IY_w:= P_I w B / B$. In the following, we will usually denote ${}^{\emptyset } Y_x$ by $Y_x$. If $I = \emptyset$ each ${}^\emptyset Y_w$ is isomorphic to an affine space of dimension $\ell (w)$ (see [24, 7.4.16 Proposition]). For $w \in {}^I W$ the decomposition of ${}^I Y_w$ into B-orbits gives a cell decomposition

$$\begin{aligned} {}^I Y_w = \bigcup _{x \in W_I w} {}^{\emptyset } Y_x = \bigcup _{x \in W_I w} {\mathbb {C}}^{l(x)} \text {.} \end{aligned}$$

For $w\in W$ let ${{\Phi }}(w)=\{\alpha \in {{\Phi }}_+ \mid w^{-1}(\alpha )\in {{\Phi }}^-\}$. Let $U_{{{\Phi }}(w)}$ be the subgroup generated by $U_\alpha$, for $\alpha \in {{\Phi }}(w)$. Applying [24, Lemma 6.1.3] gives the following isomorphism (of varieties):

$$\begin{aligned} U_{{{\Phi }}(w)} \cong \prod _{\alpha \in {{\Phi }}(w)} U_\alpha . \end{aligned}$$

The multiplication induces an isomorphism (see [24, Exercise 6.2.E (1)] and [24, Proposition 7.1.15]):

$$\begin{aligned} U_{{{\Phi }}(w)}&\longrightarrow Y_w \text {,}\\ u&\longmapsto uwB . \nonumber \end{aligned}$$

(1)

In this paper, we will view ${\mathcal {F}l_{G}}:= G / B$ as ind-variety equipped with the algebraic stratification coming from the B-orbits. Fix a field k of characteristic p. We will consider $D^b_{B}({\mathcal {F}l_{G}})$, the B-equivariant bounded constructible derived category of sheaves of k-vector spaces on ${\mathcal {F}l_{G}}$.

Recall the notion of a parity complex introduced by Juteau, Mautner and Williamson in [18, $S_{2}$.2]. We will denote by ${\textbf{Parity}_{B}}{\mathcal {F}l_{G}})$ the full subcategory of $D^b_{B}({\mathcal {F}l_{G}})$ whose objects are parity complexes.

Theorem 2.4

For each $w \in W$ there exists up to isomorphism a unique indecomposable parity complex $\mathcal {E}(w) \in D^b_{B}(G/B)$ with support $\overline{Y_w}$ and restriction $\mathcal {E}(w)\vert _{Y_w} =\underline{k}_{Y_w} [\dim Y_w]$. Every indecomposable parity complex in $D^b_{B}(G/B)$ is up to shift isomorphic to $\mathcal {E}(w)$ for some $w \in W$.

The uniqueness up to isomorphism follows from [18, Proposition 4.3 and Theorem 2.12]. The existence of the indecomposable parity complexes $\mathcal {E}(w)$ is shown in [18, Theorem 4.6].¹ In the proof, the complex $\mathcal {E}(w)$ is constructed via the push-forward of the constant sheaf on a suitably chosen Bott-Samelson resolution.

2.3 The p-Canonical Basis and p-Cells

In this section, we recall the definition of the p-canonical basis in the geometric setting. For this we will need to define the character map:

Definition 2.5

Let $\mathcal {F}\in D^b_B({{\mathcal {F}l_{G}}})$. We can define an element $\text {ch}(\mathcal {F})\in \mathcal {H}$ via

$$\begin{aligned} \text {ch}(\mathcal {F}) = \sum _{\begin{array}{c} i\in {\mathbb {Z}}\\ x\in W \end{array}} \dim H^{i}(\mathcal {F}_x) v^{-i-\ell (x)} {{H_{x}}} \end{aligned}$$

where $\mathcal {F}_x$ denotes the stalk of $\mathcal {F}$ in xB/B. The map ch is called the character map and restricts to an isomorphism $\text {ch}: [{\textbf{Parity}_{B}}({{\mathcal {F}l_{G}}})] \overset{\sim }{\longrightarrow }\ \mathcal {H}$ of ${\mathbb {Z}}[v, v^{-1}]$-algebras where $[{\textbf{Parity}_{B}}({{\mathcal {F}l_{G}}})]$ denotes the split Grothendieck group of ${\textbf{Parity}_{B}}({{\mathcal {F}l_{G}}})$.

Definition 2.6

Define ${^{p}\underline{H}_{w}} = \text {ch}([\mathcal {E}(w)])$ for all $w \in W$. The set $\{{^{p}\underline{H}_{w}} \; \vert \; w \in W\}$ is a $\mathbb {Z}[v, v^{-1}]$-basis of the Hecke algebra $\mathcal {H}$, called the p-canonical or p-Kazhdan-Lusztig basis.

Remark 2.7

Using the transposed generalized Cartan matrix as input for the diagrammatic category of Soergel bimodules and the definition of the p-canonical basis given in [16, Definition 2.4], one obtains the same basis of $\mathcal {H}[(W, S)]$. This follows from the main result in [31, Part 3]. In other words, the various definitions of the p-canonical basis are consistent (up to taking Langlands’ dual input data).

We will need the following positivity property of the p-canonical basis which can be found in [20, Proposition 4.2 and its proof]:

Proposition 2.8

For all $x \in W$ we have:

• 1. ${^{p}\underline{H}_{x}} = {H_{x}} + \sum _{y < x} {^{p}h_{y, x}} {H_{y}}$ with ${^{p}h_{y, x}} \in {\mathbb {Z}}_{\geqslant 0}[v, v^{-1}]$,

• 2. $\iota ({^{p}\underline{H}_{x}}) = {^{p}\underline{H}_{x^{-1}}}$ and thus in particular ${^{p}h_{y, x}}= {^{p}h_{y^{-1}}}, _{x^{-1}}$.

The next result is obtained from [16, Corollary 3.10] by applying the anti-involution $\iota$ of $\mathcal {H}$ and using Proposition 2.8 (ii).

Proposition 2.9

Let $I \subseteq S$ be a finitary subset. Then for $x, y \in W_I$ and $z \in {{^{I} W}}$ we have:

$$\begin{aligned} {^{p}h_{yz, xz}}={^{p}h_{y,x}} \end{aligned}$$

Next, we recall the definition of p-cells from [16, $S_{3}$.1]. This notion is an obvious generalization of the notion of cells introduced by Kazhdan-Lusztig in [22].

Definition 2.10

For $h \in \mathcal {H}$ we say that ${{^{p}\underline{H}_{w}}}$ appears with non-zero coefficient in h if the coefficient of ${{^{p}\underline{H}_{w}}}$ is non-zero when expressing h in the p-canonical basis.

Define a preorder ${{\, \overset{p}{\underset{R}{\leqslant }} \,}}$ (resp. ${{\, \overset{p}{\underset{L}{\leqslant }} \,}}$) on W as follows: $x {{\, \overset{p}{\underset{R}{\leqslant }} \,}} y$ (resp. $x {{\, \overset{p}{\underset{L}{\leqslant }} \,}} y$) if and only if ${{^{p}\underline{H}_{x}}}$ appears with non-zero coefficient in ${{^{p}\underline{H}_{y}}} h$ (resp. $h{{^{p}\underline{H}_{y}}}$) for some $h \in \mathcal {H}$. Right (resp. left) p-cells are the equivalence classes in W with respect to ${{\, \overset{p}{\underset{R}{\leqslant }} \,}}$ (resp. ${{\, \overset{p}{\underset{L}{\leqslant }} \,}}$).

For more properties and results about p-cells, we refer the reader to [16].

2.4 Hyperbolic Localization

Let T be a complex torus and X a complex variety with an action of T. In this section, we let k be a field and D(X) denote the constructible derived category of sheaves of k-vector spaces on X. Moreover, we make the following assumption:

$$\begin{aligned} \begin{array}{cc}X~\text {has a covering by }T-\text {stable affine open subvarieties}.&\left( C\right) \end{array} \end{aligned}$$

Note that this assumption is automatically satisfied if X is normal by Sumihiro’s theorem (see [21, 33]).

Let $_{X}: {\mathbb {G}_m} \longrightarrow T$ be a cocharacter of T. We want to understand the hyperbolic localization with respect to the ${\mathbb {G}}_m$ action induced by $\chi$.

Denote by $Z:= X^{\chi } \subseteq X$ the variety of $\chi$-fixed points in X and let $Z_1, \dots , Z_m$ be its connected components. For $1 \leqslant i \leqslant m$ we will denote the attracting and repelling varieties of the component $Z_i$ by

$$\begin{aligned} Z_i^+ = \{ x \in X \; \vert \; \lim _{s \rightarrow 0} \chi (s)x \in Z_i \} \qquad \text { and}\qquad Z_i^- = \{ x \in X \; \vert \; \lim _{s \rightarrow \infty } \chi (s)x \in Z_i \} \end{aligned}$$

respectively. Let $Z^+$ (resp. $Z^-$) be the disjoint, disconnected union of the $Z_i^+$ (resp. $Z_i^-$) and define $f^{\pm }: Z \longrightarrow Z^{\pm }$ and $g^{\pm }: Z^{\pm } \rightarrow X$ via the component-wise inclusions. The attracting and repelling maps $p^{\pm }: Z^{\pm } \longrightarrow Z$ are defined via $p^+(x) = \lim _{s\rightarrow 0} \chi (s)x$ and $p^+(x) = \lim _{s\rightarrow \infty } \chi (s)x$. It follows from [14, Proposition 4.2] that these are algebraic maps.

Braden defines the hyperbolic localization functors $(-)^{!*}, (-)^{*!}: D(X) \longrightarrow D(Z)$ associated to the cocharacter $\chi$ for $\mathcal {F} \in D(X)$ as follows:

$$\begin{aligned} \mathcal {F}^{! *} := (f^+)^! (g^+)^{*} \mathcal {F}\text {,}\\ \mathcal {F}^{*!} := (f^-)^{*} (g^-)^! \mathcal {F}\text {.} \end{aligned}$$

We will use the following result (see [4, Theorem 1]):

Theorem 2.11

For any $\mathcal {F} \in D(X)$ there is a natural morphism $\iota _{\mathcal {F}}: \mathcal {F}^{*!} \longrightarrow \mathcal {F}^{! *}$. If $\mathcal {F}$ is weakly equivariant (e.g., comes from an object in the equivariant derived category), then

• 1. there are natural isomorphisms $\mathcal {F}^{! *} \cong (p^+)_! (g^+)^* \mathcal {F}$ and $\mathcal {F}^{*!} \cong (p^-)_{*} (g^-)^! \mathcal {F}$ and

• 2. $\iota _{\mathcal {F}}: \mathcal {F}^{*!} \longrightarrow \mathcal {F}^{! *}$ is an isomorphism.

Using this result, Braden proves for $k = {\mathbb {Q}}$ that the hyperbolic localization of the intersection cohomology complex $\textbf{IC}(X; {\mathbb {Q}})$ is a direct sum of shifted intersection cohomology complexes.

3 Induction of p-Cells

3.1 Positivity Properties

Recall that for our fixed finitary subset $I \subseteq S$ we denote the parabolic subgroup generated by I by $W_I$ and the minimal coset representatives by ${}^I W$. The multiplication induces a bijection

$$\begin{aligned} W_I \times {{{}^{I} W}}&\overset{\sim }{\longrightarrow }\ W\\ (x, y)&\longmapsto xy \end{aligned}$$

and we have $\ell (xy) = \ell (x) + \ell (y)$ if $(x, y) \in W_I \times {}^I W$ (see [3, Proposition 2.4.4]). The following I-hybrid basis interpolates between the standard basis for $I = \emptyset$ and the p-canonical basis for $I = S$.²

Lemma 3.1

The set $\{{{{}^{p}\underline{H}_{y}H_{x}}} \; \vert \; x \in W_I, y \in {{{}^{I} W}} \}$ is a ${\mathbb {Z}}[v,v^{-1}]$-basis of the Hecke algebra $\mathcal {H}$, called the I-hybrid basis.

Proof

Notice that ${{^{p}\underline{H}_{x}}}{H_{y}}\in {H_{xy}}+\sum _{z < xy} {\mathbb {Z}} \geqslant _{0} [v, v^{-1}] {H_{z}}$. Since the base change matrix with the standard basis is unitriangular, the set $\{{{{}^{p}\underline{H}_{x}H_{y}}} \; \vert \; x \in W_I, y \in {{{}^{I} W}} \}$ is also a basis.

$\square$

We introduce the following notation for the base change coefficients between the p-canonical and the I-hybrid bases

$$\begin{aligned} {{{}^{p}\underline{H}_{w}}} = \sum _{x \in W_I,\; y \in {{{}^{I} W}}} {{{}^{p} r}}^I_{xy,w} {{{}^{p}\underline{H}_{x}H_{y}}} \end{aligned}$$

with ${{{}^{p} r}}^I_{xy,w} \in {\mathbb {Z}}[v, v^{-1}]$ for $w \in W$, $x \in W_I$ and $y \in {{{}^{I} W}}$. The following result shows that these polynomials have in fact non-negative coefficients:

Proposition 3.2

For all $w \in W$, $x \in W_I$ and $y \in {{{}^{I} W}}$ the polynomial ${{{}^{p} r}}^I_{xy, w}$ lies in ${\mathbb {Z}}{\geqslant 0}[v, v^{-1}]$.

The main idea is to replace the purity arguments in [12, Theorem 3.2] by parity arguments. There is another possible approach using stratified mixed Tate motives (see Remark 3.10).

Before explaining the proof, we need to introduce some more notation. Recall the decomposition $P_I=L_I\ltimes U_I$ from Section 2.2. To simplify notation, we will write L for $L_I$ and P for $P_I$ from now on. Denote by $B_L$ the Borel subgroup of L induced by our choice of positive roots, so that we have $B_L = B \cap L$.

Lemma 3.3

For any $y \in {{{}^{I} W}}$ the stabilizer ${{{}^{y} B}}$ of yB satisfies $B_L = {{{}^{y} B}} \cap L$.

Proof

The group ${}^yB\cap L$ is the Borel subgroup of L corresponding to the system of positive roots $\{\alpha \in \Phi _I \mid y^{-1}(\alpha ) \in \Phi _I^+\}\subset \Phi _I$.

Since $y\in {}^IW$, we have $y^{-1}s > y^{-1}$ for all $s \in I$, so $y^{-1}({{\alpha _{s}}})\in {{\Phi }}^+$. Hence, for all ${\alpha } \in {{\Phi }}_I^+$ we have $y^{-1}({\alpha })\in {{\Phi }}^+$ and similarly for ${\alpha } \in {{\Phi }}_I^-$ we have $y^{-1}({\alpha })\in {{\Phi }}^-$. It follows that $\{\alpha \in {{\Phi }}_I \mid y^{-1}(\alpha ) \in \Phi _I^+\}={{\Phi }}_I^+$ and $B_L={}^yB\cap L$. $\square$

Hence, for any $y \in {{{}^{I} W}}$ we can realize ${{\mathcal {F}l_{L}}}$ inside ${{\mathcal {F}l_{G}}}$ via the isomorphism

$$\begin{aligned} i_y: {{\mathcal {F}l_{L}}}&\overset{\sim }{\longrightarrow }\ LyB/B \subseteq {{\mathcal {F}l_{G}}} \\ xB_L&\longmapsto xyB \end{aligned}$$

Denote by $X_w = \bigcup _{u \leqslant w} Y_u \subseteq {{\mathcal {F}l_{G}}}$ the Schubert variety associated to $w \in W$.

We fix a dominant cocharacter $\gamma : {\mathbb {C}}^{*} \longrightarrow T$ whose stabilizer in W is $W_I$ (this is possible because our realization is free). For each $w \in W$ the connected components of the fixed locus $X_w^{\gamma }$ are precisely the intersections $X_w \cap i_y({{\mathcal {F}l_{L}}})$ for $y \in {{{}^{I} W}}$ (note that $\gamma$ acts trivially by conjugation on a root subgroup $U_{\alpha }$ if and only if ${\alpha }$ lies in ${{\Phi }}_I$).

Lemma 3.4

Let $y\in {}^IW.$ Then $\gamma$ retracts ByB on yB, i.e.,

$$\begin{aligned} \lim _{t\rightarrow 0}\gamma (t) byB=yB \end{aligned}$$

for all $b\in B$.

Proof

Let ${{\Phi }}(y)=\{\beta _1, \beta _2, \ldots , \beta _k\}$. As in Lemma 3.3, we have $y^{-1}(\alpha )\in {{\Phi }}^+$ for any $\alpha \in {{\Phi }}_I^+$, hence ${{\Phi }}(y)\subset {{\Phi }}^+\setminus {{\Phi }}^+_I$. In particular, we have $\langle \beta ,\gamma \rangle >0$ for all $\beta \in {{\Phi }}(y)$.

By Eq. (1) we can write $byB=u_{\beta _1}(x_1)u_{\beta _2}(x_2)\ldots u_{\beta _k}(x_k)yB$. Then

$$\begin{aligned} \gamma (t)u_{\beta _1}(x_1)u_{\beta _2}(x_2)\ldots u_{\beta _k}(x_k)yB = u_{\beta _1}(t^{\langle \beta _1, \gamma \rangle }x_1)u_{\beta _2}( t^{\langle \beta _2, \gamma \rangle } x_2)\ldots u_{\beta _k}(t^{\langle \beta _k, \gamma \rangle } x_k)yB \end{aligned}$$

and $\lim _{t\rightarrow 0} \gamma (t)byB=yB$. $\square$

Lemma 3.5

The attracting map $\pi _y: PyB/B \longrightarrow LyB/B$ is induced by the group homomorphism $P\rightarrow L$. The attracting variety to $i_y({{\mathcal {F}l_{L}}})$ is $PyB/B = \bigcup _{x \in W_I} Y_{xy}$. Moreover, $\pi _y$ is an affine bundle, with fiber isomorphic to ${\mathbb {C}}^{\ell (y)}$.

Proof

By the previous lemma we have $ByB\subset \pi _y^{-1}(yB)$. Since $W_I$ stabilizes $\gamma$, the Levi subgroup L commutes with the image of $\gamma$. We can write any element $p\in P$ as $p=lu$, with $l\in L$ and $u\in U\subset B$. Now we have

$$\begin{aligned}\pi _y(pyB)=\lim _{t\rightarrow 0} \gamma (t)pyB=\lim _{t\rightarrow 0} l \gamma (t)uyB =lyB.\end{aligned}$$

The following diagram of L-equivariant maps is commutative

and the horizontal arrows are bijections since $\pi _y$ is an L-equivariant fiber bundle and

$$\begin{aligned} pr_1^{-1}(yB/B) \cong \pi _y^{-1}(yB/B)\cong ByB\cong {\mathbb {C}}^{\ell (y)}. \end{aligned}$$

$\square$

Denote by $j_y: PyB/B \longrightarrow {{\mathcal {F}l_{G}}}$ the inclusion. The following result gives a geometric interpretation of the coefficients occurring when expressing the class of an object in $D^b_B({{\mathcal {F}l_{G}}})$ in terms of the I-hybrid basis.

Lemma 3.6

For $x \in W_I$ and $y \in {{{}^{I} W}}$ we have:

• 1. For $\mathcal {F}\in D^b_{B_L}({{\mathcal {F}l_{L}}})$ we have

  $$\begin{aligned} \textrm{ch}\big ((j_{y})_! \pi _{y}^{*} (i_{y})_{*} \mathcal {F}\big )=v^{-\ell (y)}\textrm{ch}(\mathcal {F})H_y. \end{aligned}$$

  In this equation $\textrm{ch}(\mathcal {F}) \in \mathcal {H}_{({W_I, I})}$ is viewed as an element in $\mathcal {H}$ by identifying $\mathcal {H}_{({W_I, I})}$ as the subalgebra of $\mathcal {H}$ generated by ${H_{s}}$ for $s \in I$.

• 2. For $\mathcal {F} \in D^b_B({{\mathcal {F}l_{G}}})$ the coefficient of ${{^{p}\underline{H}_{x}}} {H_{y}}{y}$ in $\textrm{ch}(\mathcal {F})$ when expressed in the I-hybrid basis is $v^{\ell (y)}$ times the coefficient of ${{^{p}\underline{H}_{x}}}$ in $\textrm{ch}\big (i_y^{*} (\pi _y)_! j_y^{*} \mathcal {F}\big )$ when expressed in the p-canonical basis.

Proof

• 1. We have $\big ((j_{y})_! \pi _{y}^{*} (i_{y})_{*} \mathcal {F} \big )_z \ne 0$ only if $z \in W_Iy$. For $x \in W_I$ we have

  $$\begin{aligned} \big ( (j_{y})_! \pi _{y}^{*} (i_{y})_{*} \mathcal {F} \big )_{xy} = \left( (i_y)_{*} \mathcal {F} \right) _{xy} = \mathcal {F}_{x} \end{aligned}$$

  since $i_y$ is an isomorphism and $i_y^{-1}(xyB)=x B_L$. Hence, it follows that

  $$\begin{aligned} \textrm{ch} \left( (j_{y})_! \pi _{y}^{*} (i_{y})_{*} \mathcal {F} \right)&= \sum _{\begin{array}{c} i \in {\mathbb {Z}}\\ x \in W_I \end{array}} v^{-i -\ell (x) -\ell (y)} \text {dim} H^{i}(\mathcal {F}_{x}) {{H_{x}}}{{H_{y}}} \\&= v^{-\ell (y)} \textrm{ch}(\mathcal {F}) {{H_{y}}}{y}\text {.} \end{aligned}$$

• 2. The map $\pi _y$ is a topological fibration with fibers isomorphic to ${\mathbb {C}}^{\ell (y)}$. Since $\mathcal {F}$ is constant along the fibers of $\pi _y$ we have

  $$\begin{aligned} (j_{y})_! \pi _{y}^{*} (i_{y})_{*} i_y^{*} (\pi _y)_! j_y^{*} \mathcal {F} \cong (j_{y})_! \pi _{y}^{*} (\pi _y)_! j_y^{*} \mathcal {F} \cong (j_{y})_! j_y^{*} \mathcal {F} [-2\ell (y)] \text {.} \end{aligned}$$

  The flag variety ${{\mathcal {F}l_{G}}}$ is the disjoint union of PyB/B, for $y\in {}^I W$. It follows that

  $$\begin{aligned} \textrm{ch}(\mathcal {F})&=\sum _{y\in {}^IW}\textrm{ch} \left( (j_{y})_! j_y^{*} \mathcal {F} \right) \\&= \sum _{y\in {}^IW} v^{2\ell (y)} \textrm{ch} \left( {(j_{y})_! \pi _{y}^{*} (i_{y})_{*} i_y^{*} (\pi _y)_! j_y^{*} \mathcal {F}}\right) \\&= \sum _{y\in {}^IW} v^{\ell (y)} \textrm{ch}\left( { i_y^{*} (\pi _y)_! j_y^{*} \mathcal {F}}\right) H_y \end{aligned}$$

  where we used the first part for the last equality. We conclude by expressing $\textrm{ch}\left( {i_y^{*} (\pi _y)_! j_y^{*} \mathcal {F}} \right)$ in the p-canonical basis of $\mathcal {H}_{({W_I I})}.$ $\square$

Remark 3.7

In the proof of Lemma 3.6 an important role is played by the functor $(j_y)_!\pi _y^*(i_y)_*:D^b_{B_L}({{\mathcal {F}l_{L}}})\rightarrow D^b_B({{\mathcal {F}l_{G}}})$. If G is a reductive group, this functor is the Verdier dual of the geometric parabolic induction function defined in [7, Definition 3.3.1] which is the geometric counterpart of the parabolic induction functor from a Levi subalgebra in category $\mathcal {O}$.

The following result crucially relies on Braden’s hyperbolic localization:

Proposition 3.8

For any $y \in {{{}^{I} W}}$ the functor $i_y^{*}(\pi _y)_!j_y^{*}$ preserves parity complexes and thus restricts to a functor ${\textbf{Parity}_{B}}({{\mathcal {F}l_{G}}}) \longrightarrow {{\textbf{Parity}_{B_L}}}({{\mathcal {F}l_{L}}})$.

Proof

Theorem 2.11 implies that $\bigoplus _{y \in {{{}^{I} W}}} (\pi _y)_! j_y^{*}$ is isomorphic to Braden’s hyperbolic localization functor $(-)^{!*}: D^b_{B}({{\mathcal {F}l_{G}}}) \longrightarrow D^b_{B_L}({{\mathcal {F}l_{G}}}^{\gamma })$. A complex $\mathcal {F}$ on ${{\mathcal {F}l_{G}}}^{\gamma }$ is parity if and only if its restriction to any connected component is a parity complex, so it is enough to show that for every indecomposable parity complex $\mathcal {E}_w$ the complex $(\mathcal {E}_w)^{!*}$ is also parity.

As shown in the proof of [18, Theorem 4.6], every complex $\mathcal {E}_w$ occurs as a direct summand of $f_!\underline{k}_{BS}$, the push-forward of the constant sheaf on a suitably chosen Bott-Samelson resolution $f:BS\rightarrow {{\mathcal {F}l_{G}}}$.

In the proof of [19, Theorem 1.6], Juteau, Mautner and Williamson show that Bott-Samelson resolutions of Schubert varieties in Kac-Moody group satisfy the properties (1)–(3) of [19, $S_{2}$.2]. Hence, by [19, Propositions 2.2 and 2.4] we obtain that $(f_!\underline{k}_{BS})^{!*}$ is parity and, hence, so is its summand $(\mathcal {E}_w)^{!*}$.$\square$

Proof of Proposition 3.2

By Lemma 3.6(ii), the polynomial ${}^pr_{xy,w}^I$ is equal to $v^{\ell (y)}$ the coefficient of ${{{}^{p}\underline{H}_{x}}}$ in $\text {ch}\big (i_y^*(\pi _y)_!j_y^*\mathcal {E}_w\big )$. By Proposition 3.8, the complex $i_y^*(\pi _y)_!j_y^*\mathcal {E}_w$ is a parity complex on ${{\mathcal {F}l_{L}}}$, hence its character has positive coefficients when written in the p-canonical basis of $\mathcal {H}_{({W_I, I})}$. $\square$

Remark 3.9

For $z \in W$, $x \in W_I$ and $y \in {}^I W$ write

$$\begin{aligned} {{{}^{p}\underline{H}_{x}H_{y}}} \cdot {{{}^{p}\underline{H}_{z}}} = \sum _{u \in W_I,\; w \in {{{}^{I} W}}} {{{}^{p} d}}^{I, uw}_{xy, z} \;{{{}^{p}\underline{H}_{u}H_{w}}} \end{aligned}$$

(2)

with ${{{}^{p} d}}^{I, uw}_{xy,z} \in {\mathbb {Z}}[v, v^{-1}]$. Grojnowski and Haiman show in [12, Corollary 3.9] that the Laurent polynomials ${{{}^{p} d}}^{I, uw}_{xy,z}$ have non-negative coefficients for $p=0$. Recently Williamson [35] obtained, after specializing v to 1, a more general result which holds for a larger class of reflection subgroups of W.

Remark 3.10

Grojnowski and Haiman’s proof is based on purity arguments and is written in the language of mixed Hodge modules, which is not available for coefficients in a field of positive characteristic. More recently, Eberhardt, Kelly, and Scholbach [8, 9] introduced a formalism of stratified mixed Tate motives. This theory of mixed sheaves works for a more arbitrary ring of coefficients and has an appropriate six-functor formalism. In the setting of the flag variety and Bruhat stratification, they are equivalent to parity sheaves (see [9, Theorem 1.2(2)]). By using their results, one can directly apply the purity arguments used by Grojnowski and Haiman even in characteristic p. In particular, one can obtain with this approach a proof of the positivity in (2).

3.2 Main Result

The proof of the main result draws inspiration from [10]. Throughout, we are working with the right p-cell preorder instead of the left Kazhdan-Lusztig cell preorder. The analogous version for left p-cells can be obtained by applying the anti-involution $\iota$. The following result is the analogue of [10, Lemma 2.2]. Its proof works in our setting after replacing all Kazhdan-Lusztig related notions by the corresponding p-canonical analogues. We will rewrite the proof here for the sake of completeness.

Lemma 3.11

Let $\mathcal {J} \subseteq W_I$ be a subset such that

$$\begin{aligned} \{ u \in W_I \; \vert \; u {{\, \overset{p}{\underset{R}{\leqslant }} \,}} w \text { for some } w \in \mathcal {J} \} \subseteq \mathcal {J}. \end{aligned}$$

Let $\mathcal {M} = \langle {{^{p}\underline{H}_{x}}}{H_{y}} \; \vert \; x \in \mathcal {J}, y \in {{^{I} W}} \rangle _{{\mathbb {Z}}[v, v^{-1}]} \subseteq \mathcal {H}$. Then $\mathcal {M}$ is a right ideal in $\mathcal {H}$.

Proof

Since $\mathcal {H}$ is generated by ${{H_{s}}}{s}$ for $s \in S$ as an algebra, it is enough to check that ${^{p}\underline{H}_{x}}{H_{y}}{H_{s}} \in \mathcal {M}$ for all $x \in \mathcal {J}$ and $y \in {{{}^{I} W}}$. Deodhar’s lemma (see [13, Lemma 2.1.2]) shows that there are three cases to consider:

• 1. $ys \in {{^{I} W}}$ and $\ell (ys) > \ell (y)$. In this case, we have ${{^{p}\underline{H}_{x}}} {H_{y}} {H_{s}} = {{^{p}\underline{H}_{x}}} {H_{ys}} \in \mathcal {M}$ as desired.

• 2. $ys \in {{^{I} W}}$ and $\ell (ys) < \ell (y)$. Then ${{^{p}\underline{H}_{x}}} {H_{y}}{H_{s}}= {{^{p}\underline{H}_{x}}} ({H_{ys}} + (v^{-1} - v){H_{y}}) \in \mathcal {M}$.

• 3. $ys \notin {{^{I} W}}$. For $t = ysy^{-1} \in I$ it follows that $\ell (ys) = \ell (y) + 1 = \ell (ty)$ and thus

  $$\begin{aligned} {{^{p}\underline{H}_{x}}} {H_{y}} {H_{s}} = {{^{p}\underline{H}_{x}}} {H_{ys}} = {{^{p}\underline{H}_{x}}} {H_{ty}} = {{^{p}\underline{H}_{x}}} {H_{t}} {H_{y}}. \end{aligned}$$

  The definition of right p-cells implies that ${{^{p}\underline{H}_{x}}} {H_{t}}$ is a linear combination of terms ${{^{p}\underline{H}_{z}}}$ with $z {{\, \overset{p}{\underset{R}{\leqslant }} \,}} x$. Our assumption then ensures that ${{^{p}\underline{H}_{x}}} {H_{y}} {H_{s}}$ is a linear combination of terms ${{^{p}\underline{H}_{u}}} {{H_{y}}}$ with $u \in \mathcal {J}$. This concludes the proof. $\square$

As in [10, $S_{3}$] we introduce the following hybrid preorder on W:

Definition 3.12

Let $x, u \in W_I$ and $y, w \in {{{}^{I} W}}$. We write $xy {{\, \overset{p}{\underset{I, R}{\sqsubset }} \,}} uw$ if $x {{\, \overset{p}{\underset{R}{\leqslant }} \,}} u$ in the right p-cell preorder and $y < w$ in the Bruhat order. We write as well $xy {{\, \overset{p}{\underset{I, R}{\sqsubseteq }} \,}} uw$ if $xy {{\, \overset{p}{\underset{I, R}{\sqsubset }} \,}} uw$ or $xy = uw$.

A crucial ingredient in Geck’s proof is [10, Proposition 3.3]. For its proof Geck uses the characterization of the Kazhdan-Lusztig basis element ${\underline{H}_{w}}$ as the unique self-dual element in ${H_{x}} + \sum _{y < x} v{\mathbb {Z}}[v] {H_{y}}$. For this reason, his proof does not work for the p-canonical basis. Since Geck works in the unequal parameter case, he cannot rely on positivity properties which will allow us to conclude the following.

Proposition 3.13

We have for $x \in W_I$ and $y \in {{^{I} W}}$:

$$\begin{aligned} {{^{p}\underline{H}_{xy}}} = {{^{p}\underline{H}_{x}}} {H_{y}} + \sum _{\begin{array}{c} u \in W_I,\; w \in {{^{I} W}}\\ uw {{\, \overset{p}{\underset{I, R}{\sqsubset }} \,}} xy \end{array}} {{^{p} r}}^I_{uw, xy} {{^{p}\underline{H}_{u}}} {H_{w}} \end{aligned}$$

In particular, the polynomial ${{{}^{p} r}}^I_{uw, xy}$ vanishes unless $uw {{\, \overset{p}{\underset{I, R}{\sqsubseteq }} \,}} xy$.

Proof

Let $x \in W_I$ and $y \in {{{}^{I} W}}$. Consider for $\mathcal {J} = \{u \in W_I \; \vert \; u {{\, \overset{p}{\underset{R}{\leqslant }} \,}} x\}$ the right ideal $\mathcal {M} \subset {\mathcal {H}}$ constructed as in Lemma 3.11. Clearly, ${{{}^{p}\underline{H}_{x}}} {{{}^{p}\underline{H}_{y}}}$ lies in $\mathcal {M}$ (as ${{{}^{p}\underline{H}_{x}}} \in \mathcal {M}$). Since $\ell (xy) = \ell (x) + \ell (y)$, it follows that

$$\begin{aligned} {{^{p}\underline{H}_{x}}} {{^{p}\underline{H}_{y}}}&= {{^{p}\underline{H}_{xy}}} + \sum _{z < xy} m_z {{^{p}\underline{H}_{z}}}\end{aligned}$$

(3)

$$\begin{aligned}&=\sum _{u\in W_I,w\in {{^{I} W}}} {{^{p} r}}^I_{uw,xy}{{{}^{p}\underline{H}_{u}}}{{H_{w}}}+\sum _{z <xy}\sum _{u \in W_I,w\in {{^{I} }}W}m_z {{^{p} r}}^I_{uw,z}{{^{p}\underline{H}_{u}}}{H_{w}}. \end{aligned}$$

(4)

where the coefficients $m_z$ are in ${\mathbb {Z}}{\geqslant 0}[v,v^{-1}]$ by [20, Proposition 4.2(6)] and the coefficients ${{{}^{p} r^I}}_{uw,z}$ are in ${\mathbb {Z}}{\geqslant 0}[v,v^{-1}]$ by Proposition 3.2. From (4) we see that, for $u\in W_I$ and $w\in {{{}^{I} W}}W$, the coefficient of ${{{}^{p}\underline{H}_{u}}}{{H_{w}}}$ in ${{{}^{p}\underline{H}_{x}}}{{{}^{p}\underline{H}_{y}}}$ is ${{{}^{p} r}}^I_{uw,xy}+\sum _{m<xy} m_z{{{}^{p} r}}^I_{uw,z}$. If $u\not \le x$, this coefficient vanishes since ${{{}^{p}\underline{H}_{x}}}{{{}^{p}\underline{H}_{y}}}\in \mathcal {M}$, and by positivity we also get ${{{}^{p} r}}^I_{uw,xy}=0$. Therefore, we see that ${{{}^{p}\underline{H}_{xy}}}$ lies in $\mathcal {M}$ and we can express ${{{}^{p}\underline{H}_{xy}}}$ as follows

$$\begin{aligned} {{{}^{p}\underline{H}_{xy}}}&= \sum _{\begin{array}{c} u \in W_I,\; w \in {{{}^{I} W}}\\ u {{\, \overset{p}{\underset{R}{\leqslant }} \,}} x \end{array}} {{{}^{p} r}}^I_{uw, xy} {{{}^{p}\underline{H}_{u}}} {{H_{w}}} \end{aligned}$$

with ${{{}^{p} r}}^I_{uw, xy}\in {\mathbb {Z}}{\geqslant 0}[v,v^{-1}]$. It remains to show that ${{{}^{p} r}}^I_{uw, xy}=0$ unless $uw = xy$ or $w<y$. Expanding in the standard basis (see Proposition 2.8 (i)) we have

$$\begin{aligned} {{{}^{p}\underline{H}_{xy}}}= \sum _{\begin{array}{c} z \leqslant u \in W_I,\; w \in {{{}^{I} W}}\\ u {{\, \overset{p}{\underset{R}{\leqslant }} \,}} x \end{array}} {{{}^{p} r}}^I_{uw, xy} {{{}^{p} h}}_{z, u} {{H_{zw}}}. \end{aligned}$$

By comparing coefficients in the standard basis we get

$$\begin{aligned} {{{}^{p} h}}_{zw,xy}=\sum _{u\in W_I} {{{}^{p} r}}^I_{uw, xy} {{{}^{p} h}}_{z, u}. \end{aligned}$$

(5)

Since both the p-Kazhdan-Lusztig polynomials ${{{}^{p} h}}_{z, u}$ and the polynomials ${{{}^{p} r}}^I_{uw, xy}$ have non-negative coefficients, ${{{}^{p} r}}^I_{uw, xy}\ne 0$ for some $w \not \le y$ and $u \in W_I$ implies ${{{}^{p} h}}^I_{zw, xy}\ne 0$. But if $w \not \le y$, then also $zw\not \le xy$ because the quotient map $W\rightarrow {}^IW$ is a strict morphism of posets (see [6, Lemma 2.2]), contradicting Proposition 2.8(i).

Suppose $w=y$. Proposition 2.9 together with (5) implies that ${{{}^{p} r}}_{xy,xy}^I = 1$ and that ${{{}^{p} r}}_{uy,xy}^I$ vanishes for $u < x$. This concludes the proof. $\square$

Corollary 3.14

Let $\mathcal {J} \subseteq W_I$ and $\mathcal {M}$ be as in Lemma 3.11. Then we have

$$\begin{aligned} \mathcal {M} = \langle {{^{p}\underline{H}_{xy}}} \; \vert \; x \in \mathcal {J}, y \in {{^{I} W}} \rangle _{{\mathbb {Z}}[v, v^{-1}]}. \end{aligned}$$

Proof

Let $x \in \mathcal {J}$ and $y \in {{{}^{I} W}}$. Proposition 3.13 shows that ${{{}^{p}\underline{H}_{xy}}}$ lies in $\mathcal {M}$.

Moreover, Proposition 3.13 shows that in any total order refining the preorder ${{\, \overset{p}{\underset{I, R}{\sqsubseteq }} \,}}$ the base change matrix between the p-canonical basis and the I-hybrid basis is unitriangular. Since its inverse is of the same form, the claim follows. $\square$

The proof of our main result is analogous to [10, $S_{4}$]. We will give a complete proof for the reader’s convenience:

Theorem 3.15

Let $x, u \in W_I$ and $y, w \in {{{}^{I} W}}$. Then we have:

$$\begin{aligned} uw {{\, \overset{p}{\underset{R}{\leqslant }} \,}} xy \quad \Rightarrow \quad u {{\, \overset{p}{\underset{R}{\leqslant }} \,}} x \text {in} W_I \end{aligned}$$

In particular, the following holds:

$$\begin{aligned} uw {{\, \overset{p}{\underset{R}{\sim }} \,}} xy \quad \Rightarrow \quad u {{\, \overset{p}{\underset{R}{\sim }} \,}} x \text {in} W_I \end{aligned}$$

Proof

It is enough to consider the case where ${{}^{p}\underline{H}_{uw}}$ occurs with non-zero coefficient in ${{}^{p}\underline{H}_{xy}} {H_{s}}$ for some $s \in S$. Consider the set $\mathcal {J} = \{z \in W_I \; \vert \; z {\, \overset{p}{\underset{R}{\leqslant }} \,} x \}$. Since $\mathcal {J}$ satisfies the requirements in Lemma 3.11, it follows that $\mathcal {M} = \langle {{}^{p}\underline{H}_{a}} {H_{b}} \; \vert \; a \in \mathcal {J}, b \in {{{}^{I} W}}{I}{W} \rangle \subseteq \mathcal {H}$ is a right ideal.

Corollary 3.14 shows that ${{}^{p}\underline{H}_{xy}}$ lies in $\mathcal {M}$. Since $\mathcal {M}$ is a right ideal, the element ${{}^{p}\underline{H}_{xy}} {H_{s}}$ also lies in $\mathcal {M}$. From Corollary 3.14 it follows that we can write

$$\begin{aligned} {{}^{p}\underline{H}_{xy}} {H_{s}} = \sum _{a \in \mathcal {J},\; b \in {{{}^{I} W}}} \gamma _{xy, s}^{ab} {{}^{p}\underline{H}_{ab}} \end{aligned}$$

(6)

with $\gamma _{xy, s}^{ab} \in {\mathbb {Z}}[v, v^{-1}]$ for all $a \in \mathcal {J},\; b \in {{{}^{I} W}}$. Observe that the right-hand side of (6) is the expansion of ${{}^{p}\underline{H}_{xy}} {H_{s}}$ in the p-canonical basis. Therefore, our assumption that ${{}^{p}\underline{H}_{uw}}$ occurs with non-zero coefficient in ${{}^{p}\underline{H}_{xy}} {H_{s}}$ means that $u \in \mathcal {J}$. This in turn implies $u {\, \overset{p}{\underset{R}{\leqslant }} \,} x$ by the definition of $\mathcal {J}$ and finishes the proof of the theorem.$\square$

Before we can state our main result, we should recall the definition of a cell module.

Definition 3.16

For any right p-cell $C \subseteq W$, write $w {\, \overset{p}{\underset{R}{\leqslant }} \,} C$ (resp. $w {\, \overset{p}{\underset{R}{<}} \,} C$) if there exists an element $y \in C$ such that $w {\, \overset{p}{\underset{L}{\leqslant }} \,} y$ (and $w \not \in C$). By the definition of the right p-cell preorder, we can define right $\mathcal {H}$-modules

$$\begin{aligned} {\mathcal {H}_{{\, \overset{p}{\underset{R}{\leqslant }} \,} C}} := \bigoplus _{w {\, \overset{p}{\underset{R}{\leqslant }} \,} C} {\mathbb {Z}}[v, v^{-1}] {{}^{p}\underline{H}_{w}} \end{aligned}$$

and similarly ${\mathcal {H}_{{\, \overset{p}{\underset{R}{<}} \,} C}}$. Then the right p-cell module associated to C is defined as the right $\mathcal {H}$-module given by the quotient

$$\begin{aligned} {\mathcal {H}_{C}} := {\mathcal {H}_{{\, \overset{p}{\underset{R}{\leqslant }} \,} C}} / {\mathcal {H}_{{\, \overset{p}{\underset{R}{<}} \,} C}}. \end{aligned}$$

Theorem 3.17

(Parabolic induction for right p-cells). Let C be a right p-cell of $W_I$ and ${\mathcal {H}_{C}}$ the corresponding right cell module of ${\mathcal {H}_{(W_I, I)}}$. Then $C\cdot {{{}^{I} W}}$ is a union of right p-cells of W. The right module of ${\mathcal {H}_{(W, S)}}$ associated to $C\cdot {{{}^{I} W}}$ is isomorphic to ${\mathcal {H}_{C}} \otimes _{\mathcal {H}_{(W_I, I)}} {\mathcal {H}_{(W, S)}}$.

Proof

The first part is a corollary of Theorem 3.15 and the proof of the second part works exactly like the proof of [11, Lemma 5.2] which we report for completeness. Let

$$\begin{aligned} \mathcal {I}=\langle {{}^{p}\underline{H}_{x}}{H_{y}}\mid x{\, \overset{p}{\underset{R}{\leqslant }} \,} C, y \in {{{}^{I} }}W\rangle \quad \text { and }\quad {\hat{\mathcal {I}}}=\langle {{}^{p}\underline{H}_{x}}{H_{y}}\mid x{\, \overset{p}{\underset{R}{<}} \,} C, y \in {{{}^{I} }}W\rangle . \end{aligned}$$

Then ${\hat{\mathcal {I}}}\subset \mathcal {I}$ are right ideals of $\mathcal {H}_{(W,S)}$ by Lemma 3.11. The quotient $\mathcal {I}/{\hat{\mathcal {I}}}$ is a right $\mathcal {H}_{(W,S)}$-module and it is free as a ${\mathbb {Z}}[v,v^{-1}]$-module with a basis given by the residue classes of ${{}^{p}\underline{H}_{x}}{H_{y}}$, for $x\in C$ and $y\in {{{}^{I} W}}$. So we get an isomorphism of right $\mathcal {H}_{(W,S)}$-modules defined by

$$\begin{aligned} \mathcal {H}_{C}\otimes _{\mathcal {H}(W_I, I)} \mathcal {H}_{(W, S)}&\xrightarrow {\sim } \mathcal {I}/{\hat{\mathcal {I}}}\\ {{}^{p}\underline{H}_{x}}\otimes {H_{y}}&\mapsto {{}^{p}\underline{H}_{x}}{H_{y}}. \end{aligned}$$

On the other hand, by Corollary 3.14, we have

$$\begin{aligned} \mathcal {I}=\langle {{}^{p}\underline{H}_{xy}}\mid x{\, \overset{p}{\underset{R}{\leqslant }} \,} C, y \in {{}^{I} }\rangle \quad \text {and}\quad \hat{I}=\langle {{}^{p}\underline{H}_{xy}}\mid x{\, \overset{p}{\underset{R}{<}} \,} C, y \in {{}^{I} W}\rangle , \end{aligned}$$

and the quotient $\mathcal {I}/{\hat{\mathcal {I}}}$ also has a ${\mathbb {Z}}[v,v^{-1}]$-basis given by the residue classes of ${{}^{p}\underline{H}_{xy}}$, for $x\in C$ and $y\in {{}^{I} W}$. Hence, we get a second isomorphism of right $\mathcal {H}_{(W,S)}$-modules by

$$\begin{aligned} {\mathcal {H}_{C\cdot {{}^{I} }W}}&\xrightarrow {\sim } \mathcal {I}/\hat{\mathcal {I}}\\ {{}^{p}\underline{H}_{xy}}&\mapsto {{}^{p}\underline{H}_{xy}}. \end{aligned}$$

and the claim follows. $\square$

3.3 An Example: Finite Type $C_{3}$ for $p = 2$

In this section, we illustrate Theorem 3.15 in finite type $C_3$ where Kazhdan-Lusztig cells do not decompose into p-cells for $p = 2$ (see [16, $S_{3}$.4.3]). We will use the following Cartan matrix as input and label the simple reflections accordingly. One may obtain a Kac-Moody root datum satisfying our assumptions from any based root datum of the corresponding connected semi-simple simply connected algebraic group.

It should be noted that our input is Langlands’ dual to the one given in [16, $S_{3}$.4.3] as we work on the geometric side and not with diagrammatic Soergel bimodules.

Explicit computer calculation gives the following Kazhdan-Lusztig right cells (as in [16, $S_{3}$.4.3]):

$$\begin{aligned} C_0&= \{ \textrm{Id} \} \\ C_1&= \{ 1, 12, 121, 123 \} \\ C_2&= \{ 2, 21, 23, 212, 2123 \} \\ C_3&= \{ 3, 32, 321, 3212, 32123 \} \\ C_4&= \{ 13, 132, 1321 \}\\ C_5&= \{ 213, 2132, 21321 \} \\ C_6&= \{ 232, 2321, 23212 \} \\ C_{7}&= \{ 2121, 21213, 212132, 2121321, 21213213 \} \\ C_8&= \{ 1213, 12132, 121321 \} \\ C_9&= \{ 1232, 12321, 123212 \} \\ C_{10}&= \{ 13212, 132123, 1213212, 1232123, 12132123 \} \\ C_{11}&= \{ 21232, 212321, 2123212 \} \\ C_{12}&= \{ 232123, 232121, 2321213, 23212132 \} \\ C_{13}&= \{ w_0 \} \end{aligned}$$

For $p=2$ these right Kazhdan-Lusztig cells exhibit the following decomposition behavior into right p-cells:

$$\begin{aligned} C_2&= \underbrace{\{ 2, 23\}}_{{}^2 C_{2'}} \cup \underbrace{\{21, 212, 2123 \}}_{{}^2 C_{2''}} \\ C_3&= \underbrace{\{ 3, 32\}}_{{}^2 C_{3'}} \cup \underbrace{\{ 321, 3212, 32123 \}}_{{}^2 C_{3''}} \\ C_6 \cup C_{12}&= \underbrace{\{ 232 \}}_{{}^2 C_6} \cup \underbrace{\{ 2321, 23212, 232123 \}}_{{}^2 C_{6/12}} \cup \underbrace{\{ 232121, 2321213, 23212132 \}}_{{}^2 C_{12}} \\ C_i&= {}^2 C_i \text { for } i \in \{0, \dots , 13\} \setminus \{2,3,6,12\} \end{aligned}$$

The Hasse diagrams of the cell preorders look as follows. We display Kazhdan-Lusztig right cells on the left and right p-cells on the right. In these diagrams the cells that are depicted at one height form a two-sided cell.

We will choose $I = \{1, 2\}$ so that $W_I$ is a finite Coxeter group of type $B_2$ and we get

$$\begin{aligned} {}^I W = \{\textrm{Id}, 3, 32, 321, 3212, 32123 \}. \end{aligned}$$

For $p = 2$ the Kazhdan-Lusztig right cell $\{2, 21, 212\}$ in $W_I$ decomposes into two right p-cells $\{2\} \cup \{21, 212\}$ (see [16, $S_{3}$.4.1]).

For the right Kazhdan-Lusztig cells $C = \{2\}$ and $C' = \{21, 212\}$ in $W_I$ we get:

$$\begin{aligned} C \cdot {}^I W =&\underbrace{\{ 2, 23\}}_{{}^2 C_{2'}} \cup \underbrace{\{232 \}}_{{}^2 C_{6}} \cup \underbrace{\{2321, 23212, 232123 \}}_{{}^2 C_{6/12}} \\ C' \cdot {}^I W =&\underbrace{\{ 21, 212, 2123\}}_{{}^2 C_{2''}} \cup \underbrace{\{213, 2132, 21321 \}}_{{}^2 C_{5}} \cup \underbrace{\{232121, 2321213, 23212132 \}}_{{}^2 C_{12}} \\&\cup \underbrace{\{21232, 212321, 2123212 \}}_{{}^2 C_{11}} \end{aligned}$$

Observe that neither $C \cdot {}^I W$ nor $C' \cdot {}^I W$ is a union of right Kazhdan-Lusztig cells.