Parity-sheaves

  • Decomposition theorem fails in positive characteristics: IC sheaves are not nice to work with.
  • Multiplicities of IC in $\triangle$ and $\nabla$ are no longer always given by Kazhdan-Lusztig polynomials.
  • Parity sheaves are often IC sheaves, but in general are not necessarily perverse.

Theory of $p$-Kazhdan-Lusztig polynomials

  • We define $\mathrm{SBim}(W,V)$ in all characteristics.
  • We can already define $p$-Kazhdan-Lusztig polynomials by multiplicities of $B_w ^{\textrm{bim}}$ in $\underline{B}_y^{\textrm{bim}}$.

Example 2 (Type $B_2$). We have that $$ {} ^{p} \underline{H} _{sts} = \begin{cases} \underline{H} _{sts} + \underline{H} _s, &\text{if } p=2, \\ \underline{H} _{sts} , &\text{if } p\neq 2 . \end{cases} $$

Remark 3. When $p=0$ or for any fixed type, for $p>>0$, we have that ${} ^{p}\underline{H} _{sts} = \underline{H} _{sts}$.

Example 4 (Type A). We have that $ {} ^{p} \underline{H} _{w} = \underline{H} _{w}$ for $A _1, A _2,\dots,A _6$. In $A_7$, ${} ^{p}\underline{H} _{w} = \underline{H} _{w}$ unless $p = 2$. However, $p$-canonical basis in Type $A_m$ for $m \geq 8$ are not known completely. For example, 3-canonical basis is not KL basis in Type $ A _{11} $. It is open question that is 11 the first $ m $ for which they differ for $p=3$. Later we will discuss a way to produce pairs $(p,m)$, such that ${} ^{p}\underline{H} _{w} \neq \underline{H} _{w}$ for $A_m$.

Let $\mathbb{k}$ be a field, $D$ be a $\mathbb{k}$-linear triangulated category with a bounded t-structure, with heart $A$. We use $H$ for cohomology functors with respect to this t-structure.

Let $X$ be an object of $A$, $ \langle X \rangle _\triangle$ be the triangulated subcategory subcategory of $D$ generated by $X$.

Lemma 5 (Genreation). Assume that $\mathrm{End}(X)= \mathbb{k}$ and $\mathrm{Hom}(X,X[1])=0$. Then for $Y\in D$, the following are equivalent:

  1. $Y\in \langle X \rangle _\triangle$,
  2. for any $n\in \mathbb{Z}$, we have that $H^n(Y)$ is isomorphic to direct sum of copies of $X$.

Lemma 6. If $\mathrm{End}(X)=k$, and $\mathrm{Hom}(X,X[2n+1])=0$ for any $n\in\mathbb{Z} _{\geq 0}$. Then for any $Y\in D$, the following are equivalent:

  1. $Y\in \langle X \rangle _\triangle$ and $H^m = 0 $ for any $m$ odd ;
  2. There exists even integers $n_1, \dots, n_r$ and an isomorphism $$ Y\cong\oplus_{i=1}^r X[n_i].$$

Gerneral setting

Let $\mathbb{F}$ be an algebraically closed field, and an $\mathbb{F}$-algebraic variety $X$. We assume we are given a decomposition $$ X = \bigsqcup _{\lambda\in\Lambda} X _\lambda $$ where

  • $\Lambda$ is finite,
  • each strata is smooth connected and locally closed, and
  • for any $\lambda \in \Lambda $, the closure $\overline{X} _\lambda$ is a union of $X _\mu$ for $\mu\in \Lambda$.

We write $j_ \lambda : X_ \lambda\rightarrow X$.

There are various settings for sheaf theories:

  1. Analytic: $\mathbb{F}= \mathbb{C}$, $\mathbb{k}$ arbitrary, then $D(X,\mathbb{k})$ is constructible derived category of $\mathbb{k}$-sheaves with respect to analytic topology.
  2. Étale: $\mathbb{F}$ arbitrary, $\mathbb{k}$ is a finite filed with $\textrm{char}(k)\neq \textrm{char}(F)$, or a finite extension of $\mathbb{Q}_\ell$ with $\ell\neq \textrm{char}(F)$. Then $D(X,\mathbb{k})$ is the constructible dereived category of étale sheaves.
  3. Equivariant analytic setting.
  4. Equivariant étale setting.

In these settings, we have derived functors $j _\lambda ^\ast$, $j _\lambda ^!$, and $(j _{\lambda}) _{\ast}$, $(j _\lambda) _!$.

Given a local system $Z _\lambda$ on $X _\lambda$ for every $\lambda\in \Lambda$, such that $\mathrm{End} _{D(X,\mathbb{k})} (Z _\lambda)=\mathbb{k}$, and $\mathrm{Hom}(Z _\lambda, Z _\lambda[2n+1])=0$ for any $n\geq\mathbb{Z} _{\geq 0}$. Then we set

  • standard object $\triangle _\lambda = (j _\lambda) _! Z _\lambda[\textrm{dim}(X _\lambda)]$.
  • costandard object $\nabla _\lambda = (j _\lambda) _\ast Z _\lambda[\textrm{dim}(X _\lambda)]$.

We will also assume that for any $\lambda,\mu$, we have that $$j _\mu ^\ast \nabla _\lambda\in \langle Z _\mu \rangle _\triangle.$$

Definition 7. Let $D_\Lambda(X,\mathbb{k})$ be the triangulated subcategory of $D(X,\mathbb{k})$ consisting of objects $\mathscr{F}$ such that $\mathscr{F} | _{X _\mu}$ belongs to $ \langle Z _\mu \rangle _\triangle$ for any $\mu\in\Lambda$.

We have Verdier duality $$ \mathbb{D}: D_\Lambda(X,\mathbb{k})\xrightarrow{\cong} D_{\Lambda^{\textrm{dual}}}(X,\mathbb{k})$$ and we will only consider the case where $\Lambda$ is self-dual.

Parity complexes

Definition 8 (parity sheaves). Let $\mathscr{F}\in D _\Lambda (X, \mathbb{k})$,

  1. we call $\mathscr{F}$ $\ast$-even, if for any $\lambda \in \Lambda$, we have $H^n (j_\lambda^{\ast} \mathscr{F})=0$ unless $n$ is even.
  2. we call $\mathscr{F}$ $!$-even, if for any $\lambda\in\Lambda$, we have $H^n(j_\lambda^! \mathscr{F})=0$ unless $n$ is even.
  3. We call $\mathscr{F}$ is even, if it is $\ast$-even and $!$-even.
  4. We call $\mathscr{F}$ is a parity complex, if it is isomorphic to a direct sum of an even and an odd object.

Lemma 9. Let $\mathscr{F}\in D_\Lambda(X,\mathbb{k})$.

  1. If $|\Lambda|=1$, then the following are equivalent:
    1. $\mathscr{F}$ is $\ast$-even,
    2. $\mathscr{F}$ is $!$-even,
    3. $\mathscr{F}$ is even.
    4. $\mathscr{F}$ is a direct sum of $Z_\lambda[n]$ for $n$ even. Furthermore, if $\mathscr{F}$ and $\mathscr{G}$ are even, then any $n\in\mathbb{Z}$, we have $\mathrm{Hom}(\mathscr{F},\mathscr{G})=0$ unless $n$ is even.
  2. $\mathscr{F}$ is $!$-even iff $\mathbb{D}(F)$ is $\ast$-even. So $\mathscr{F}$ is parity iff $\mathbb{D}(\mathscr{F})$ is parity.
  3. $\mathscr{F}$ is even iff $\mathscr{F}[1]$ is odd. So partiy sheaves are stable under shifts $[1]$.
  4. $\mathscr{F}$ is even iff $H^n(\mathscr{F})=0$ and $H^n(\mathbb{\mathscr{F}})=0$ for all $n$ odd.

Proposition 10. Let $\mathscr{F}, \mathscr{G} \in D_{\Lambda}(X, \mathbb{k})$. If $\mathscr{F}$ is a direct sum of a $\ast$-even and $\ast$-odd object, and $\mathscr{G}$ is a direct sum of a $!$-even and $!$-odd object, then $\mathrm{Hom} (\mathscr{F},\mathscr{G}) \cong \bigoplus_{\lambda \in \Lambda} \mathrm{Hom} (j _\lambda^\ast \mathscr{F}, j _\lambda ^! \mathscr{G}).$

Proof.

Proof is given by induction on $|\Lambda|$. Let $Y$ be the support of $\mathscr{F}$ and $X_\mu\subset Y$ be an ipen stratrum. Let $j: X\backslash (Y\backslash X_\mu)\hookrightarrow X$ be the open embedding, and $i$ the complementary closed immersion. Then consider the long exact sequence associated to the exision distinguished triangle and use induction.

Corollary 11. Let $\mathscr{F}, \mathscr{G}\in D_{\Lambda}(X,\mathbb{k})$.

  1. If $\mathscr{F}$ is $\ast$-even, $\mathscr{G}$ is $!$-odd, then $\mathrm{Hom}(\mathscr{F},\mathscr{G})=0.$
  2. If $\mathscr{F}$ and $\mathscr{G}$ are parity sheaves. Let $U\subset X$ be an open union of strata, $j:U\hookrightarrow X$, then $$ \mathrm{Hom}(\mathscr{F},\mathscr{G}) \rightarrow \mathrm{Hom} (j^\ast\mathscr{F},j^\ast\mathscr{G}) $$ is surjective.
  3. Let $\mathscr{F}$ be indecomposible, and parity. Then for $j: U\hookrightarrow$ as above, $j^\ast\mathscr{F}$ is either indecomposible or $0$.

Theorem 12. For each $\lambda \in \Lambda$, there exsits at most one indecomposible partity sheaf $\mathscr{E} _\lambda$ supported on $\overline{X} _\lambda$ and such that $$ \mathscr{E} _\lambda| _{X _\lambda} \cong \mathscr{L} _\lambda [\mathrm{dim}(X _\lambda)].$$ Moreover, any indecomposible parity complex is isomorphic to $\mathscr{E} _\lambda[n]$ for some $\lambda$ and $n\in\mathbb{Z}$.