\include{preamble}

\begin{document}

Speaker: Daniel Juteau

Title: Parity sheaves I, II, III

(joint work with Carl Mautner and Geordie Williamson)

February 19, 20, 22, 2012

\section{Introduction.}

\begin{tabular}{p{3in} | p{3in}}
Representation theory & Geometry / Topology \\ \hline
$\cA$ = category of representations ($k$-linear, Abelian) & Perv($X,k$) = category of perverse sheaves with $k$-coefficients; $X = \coprod_{\lambda \in \Lambda} X_\lambda$ stratified complex variety \\ \hline
$D^b(\cA)$ = bounded derived category & $D^b_\Lambda(X,k)$ = bounded $\Lambda$-constructible category\\ \hline
modules, complexes, characters & perverse sheaves, complexes, ``characters'' = graded dimensions of cohomology of the stalks\\ \hline
simple modules & simple perverse sheaves = IC (intersection cohomology complexes)\\ \hline
standard / co-standard objects & perverse extensions ${\rho_j}_*$ and perverse shriek extensions ${\rho_j}_!$ \\ \hline
tilting objects & parity sheaves
\end{tabular}

~

If ${\rm char}(k) = 0$, then there is a very powerful tool, the decomposition theorem, which gives a recursive computation of the IC stalks. For example, when $X = G/B$, this gives Kazhdan--Lusztig theory.

But if ${\rm char}(k) = p > 0$, this theorem ``fails''. In particular, the IC stalks are very hard to compute. Two kinds of problems: 
\begin{compactitem}
\item Problems with torsion (working over $\bZ$)
\item No parity vanishing (usually odd cohomology vanishes)
\end{compactitem}

However, in representation theory situtations, the varieties of interest seem to satisfy ``parity conditions''. This motivates the definition of ``parity sheaves''. In the case $G/B$, these are the special sheaves of Soergel, and used by Fiebig in the affine case.

\begin{compactitem}
\item In nice situations, we get the same classification as for IC.
\item Are they computable? The failure of the decomposition theorem is measured by the rank of ``intersection forms'' modulo $p$. This is explained in JMW, which is a modular version of an argument of de Cataldo--Migliorini.
\begin{compactitem}
\item They can be computed by moment graph techniques (Fiebig--Williamson (on arXiv), modular version of Braden--Mac Pherson)
\item Presentation by generators and relations for $\ext^*({\rm parities})$ for $G/B$ (Elias--Williamson, not available yet)
\end{compactitem}
\end{compactitem}

\section{Perverse sheaves review.}

Let $X$ be a complex variety with a stratification $X = \coprod_{\lambda \in \Lambda} X_\lambda$. Here each $X_\lambda$ is locally closed, connected, smooth, and $\ol{X_\lambda}$ is a union of other $X_\mu$ (in which case we write $\mu \le \lambda$). We denote the inclusion $i_\lambda \colon X_\lambda \subset X$ and set $d_\lambda = \dim_\bC(X_\lambda)$. 

Let $\cF$ be a sheaf on $X$ with $k$-coefficients ($k$ is any commutative ring). We say that $\cF$ is {\bf $\Lambda$-constructible} if for all $\lambda$, $i_\lambda^* \cF = \cF|_{X_\lambda}$ is a local system (i.e., locally isomorphic to a constant sheaf with stalks which are finitely generated over $k$).

Let $D^b_\Lambda(X,k)$ be the full subcategory of $D^b({\rm Sh}(X,k))$ (= bounded derived category of $k$-sheaves) whose objects are complexes $\bK_\bullet$ such that the cohomology sheaves $\sH^n(\bK_\bullet)$ are $\Lambda$-constructible. Also, we define $D^b_c(X,k)$ to be the full subcategory of complexes which belong to $D^b_\Lambda(X,k)$ for some stratification $\Lambda$.

\subsection{Grothendieck's formalism of 6 operations.}

\begin{compactitem}
\item Two internal bifunctors: $\rR\sH om$ and $\stackrel{\rL}{\otimes}$
\item Given $f \colon X \to Y$, we have morphisms $f_*, f_! \colon D^b_c(X,k) \to D^b_c(Y,k)$ and $f^*, f^! \colon D^b_c(Y,k) \to D^b_c(X,k)$. Here $f_*$ and $f_!$ are right derived functors of functors between categories of sheaves (but we drop R from the notation). $f^*$ is already exact on sheaves but $f^!$ is in general not a derived functor.
\end{compactitem}

We have adjunctions $(f^*, f_*)$ and $(f_!, f^!)$. 

In the case when $Y$ is a point, then $f_* = \cR \Gamma(X,-)$ gives $\rH^*(X,-)$, and $f_! = \cR \Gamma_c(X,-)$ gives cohomology with compact support. Also, $f^* \ul{k}_{\rm pt} = \ul{k}_X$ (constant sheaf) and $f^! \ul{k}_{\rm pt} = \omega_X$ is the dualizing complex. If $X$ is smooth, then $\omega_X \cong \ul{k}_X[2 \dim X]$.

The Grothendieck--Verdier duality is the functor $\bD_X = \cR\sH om(-,\omega_X)$. It satisfies $\bD^2 \simeq 1_X$ and $\bD f_* \simeq f_! \bD$ and $\bD f^* \simeq f^! \bD$. 

\begin{remark}
When $X$ is smooth and $Y$ is a point, then gives Poincar\'e duality between $\rH^*(X,k)$ and $\rH_c^*(X,k)$. 
\end{remark}

Now assume that $\Lambda$ is a Whitney stratification. When $k$ is a field, this condition implies that $D^b_\Lambda(X,k)$ is preserved by $\bD$, and also compatible with $i_\lambda^?$ and ${i_\lambda}_?$.

\begin{remark} When $k = \bZ$, have two categories ${\rm Perv}_\Lambda(X,\bZ)$ and ${\rm Perv}_\Lambda(X,\bZ)^+$ which are exchanged under $\bD$ (``dual perverse sheaves'').
\end{remark}

\begin{example}
When $X$ is a point, $\bD(\bZ/n) = \bD(\bZ[-1] \xrightarrow{n} \bZ[0]) = \bD(\bZ[0] \xrightarrow{n} \bZ[1]) \cong (\bZ/n)[-1]$.
\end{example}

\subsection{Perverse sheaves.}

We define ${\rm Perv}_\Lambda(X,k) = \leftexp{p}{D}^{\le 0} \cap \leftexp{p}{D}^{\ge 0}$ where
\begin{align*}
\leftexp{p}{D}^{\le 0} &= \{ \bK \in D^b_\Lambda(X,k) \mid i_\lambda^* \bK \in D^{\le -d_\lambda} \}\\
\leftexp{p}{D}^{\ge 0} &= \{ \bK \in D^b_\Lambda(X,k) \mid i_\lambda^! \bK \in D^{\ge -d_\lambda} \}.
\end{align*}
Here $D^{\le d}$ means complexes with cohomology concentrated in degrees $\le d$ (and similarly for $\ge d$). This is an Abelian category, and we have ``perverse cohomology sheaves'' functors $\leftexp{p}{\cH}^i \colon D^b_\Lambda(X,k) \to {\rm Perv}_\Lambda(X,k)$ and truncations $\leftexp{p}{\tau}_{\le i}, \leftexp{p}{\tau}_{\ge i}$.

\begin{remark}
${\rm Perv}_\Lambda(X,k)$ is Abelian because it is obtained by the ``gluing'' procedure (recollement). Here we are gluing the Abelian categories ${\rm Loc}(X_\lambda, k)[d_\lambda] \subset D^b_\Lambda(X,k)$ (this shift implies self-duality). Since $X_\lambda$ is smooth, this implies that for $\cL$ a local system, we get $\bD(\cL[d_\lambda]) \simeq \cL^\vee[d_\lambda]$.
\end{remark}

\subsection{Intersection cohomology complexes.}

Given $\lambda \in \Lambda$ and $\cL \in {\rm Loc}(X_\lambda, k)$, there is a unique $\bK =: {\rm IC}(\ol{X_\lambda}, \cL)$ in $D^b_\Lambda(X,k)$ such that
\begin{compactitem}
\item $\bK$ is supported by $\ol{X_\lambda}$,
\item $\bK|_{X_\lambda} \simeq \cL[d_\lambda]$,
\item For all $\mu < \lambda$, $i_\mu^* \bK \in D^{< -d_\mu}$ and $i_\mu^! \bK \in D^{> -d_\mu}$.
\end{compactitem}

\begin{theorem}[Beilinson--Bernstein--Deligne]
Assume $k$ is a field. Then ${\rm Perv}_\Lambda(X,k)$ is Noetherian and Artinian. The simple objects are ${\rm IC}(\ol{X_\lambda}, \cL)$ where $\cL$ is a simple object in ${\rm Loc}(X_\lambda, k)$.
\end{theorem}

\subsection{Decomposition theorem.}

\begin{theorem} 
Suppose $k$ is a field of characteristic $0$. Let $\pi \colon Y \to X$ be a proper morphism. Let $\cL$ be a local system on an open dense smooth $U \subset Y$ with finite monodromy (i.e., the monodromy representation of the fundamental group factors through a finite group). Then $\bK = \pi_*{\rm IC}(Y, \cL)$ is semisimple.
\end{theorem}

\steven{It was remarked that maybe the finite monodromy condition can be dropped, but not sure.}

\begin{remark} Semisimplicity of $\bK$ means
\begin{compactitem}
\item $\bK \cong \bigoplus \leftexp{p}{\cH}^i \bK[-i]$
\item $\leftexp{p}{\cH}^i \bK$ is a semisimple perverse sheaf, i.e.,
$\leftexp{p}{\cH}^i \bK = \bigoplus_\lambda {\rm IC}(X, \cL_\lambda)$ where each $\cL_\lambda$ is a semisimple local system. \qedhere
\end{compactitem}
\end{remark}

%February 20, 2012

\subsection{Deligne's construction of IC complexes.}

Let $X_{\lambda_{\rm max}}$ be the open dense stratum in $X$, and let $\cL$ be a local system on $X_{\lambda_{\rm max}}$. We will give an alternate construction of ${\rm IC}(X, \cL)$ (to get the smaller strata, just apply replace $X$ with $X \setminus X_{\lambda_{\rm max}}$. 

Let $S_i$ be the union of the strata of dimension $i$, and $U_i$ be the union of strata of dimensions $i, \dots, d = \dim X$. We have inclusions $U_d \subset U_{d-1} \subset \cdots \subset U_1 \subset U_0 = X$. Denote the inclusion $j_i \colon U_i \subset U_{i-1}$ and set $j = X_{\lambda_{\rm max}} \subset U_0$.

Set ${\rm IC}(X, \cL) = {j_!}_*(\cL[d])$ where ${j_!}_*$ is the intermediate extension functor defined by
\[
{j_!}_* = (\tau_{\le -1} {j_1}_*) \cdots (\tau_{\le -d+1} {j_{d-1}}_*) (\tau_{\le -d} {j_d}_*)
\]
This gives a condition on the stalks and costalks:
\[
\begin{array}{l|llllll}
& -d & -d+1 & -d+2 & \cdots & 0 \\ \hline
X_{\lambda_{\rm max}} & \cL & 0 & 0 & \cdots \\
S_{d-1} & * & 0 & 0 & \cdots \\
S_{d-2} & * & * & 0 & \cdots \\
\vdots \\
S_0 & * & * & * & \cdots & 0
\end{array}
\]

\begin{example}
Let $X$ be the nilpotent cone of $\fsl_2$, which is $\left\{ \begin{pmatrix} a & b \\ c & -a \end{pmatrix} \mid (a,b,c) \in \bC^3, \ a^2 + bc = 0\right\}$. We can also think of this as the simple surface singularity of type ${\rm A}_1$, $\bC^2 / \{\pm 1\} = \Spec (\bC[uv,u^2,-v^2])$. We have two strata given by the regular orbit $U$ and the zero orbit $\{0\}$.

Let $\tilde{X} = \{(x,\ell) \mid x \in X,\ L \supset \im(x)\} \cong \rT^* \bP^1$ be the Springer resolution and let $\pi \colon \tilde{X} \to X$ be the first projection. This is an isomorphism over $U$ and $\pi^{-1}(0) = \bP^1$. Then we have $\cH^m(\pi_* \ul{k}_{\tilde{X}}[2])_x = \rH^{m+2}(\pi^{-1}(x), k)$. We encode this with a table
\[
\begin{array}{l|lll}
\pi_* (\ul{k}_{\tilde{X}}[2]) & -2 & -1 & 0 \\ \hline
U & k & 0 & 0 \\
\{0\} & k & 0 & k 
\end{array}
\]
If ${\rm char}(k) = 0$, we can use the decomposition theorem. This will give us that $\pi_*(\ul{k}_{\tilde{X}}[2]) = \ul{k}_X[2] \oplus \ul{k}_0$ (and we have $\ul{k}_X[2] = {\rm IC}(X,k)$).

Denote $j \colon U \subset X$ and $i \colon \{0\} \subset X$. We will use Deligne's construction. Then 
\[
\cH^m(j_*\ul{k}_X)_0 = \lim_{V \ni 0} \rH^m(j^{-1}(V), k) = \rH^m(U,k).
\]
Also, $U = (\bC^2 \setminus \{0\}) / \{\pm 1\}$, which is homotopy equivalent to $\rS^3 / \{\pm 1\} = \bR\bP^3$. Also, $\rH^*(\bR\bP^3, \bZ) = \bZ[0] \oplus \bZ/2[2] \oplus \bZ[3]$. If $k$ is a field of characteristic different from 2, we have $\rH^*(\bR\bP^3, k) = k[0] \oplus k[3]$ and also we have $\rH^*(\bR\bP^3, \bF_2) = \bF_2[0] \oplus \bF_2[1] \oplus \bF_2[2] \oplus\bF_2[3]$.

Hence for $k$ of characteristic 2, we can describe the stalks of the cohomology of the IC complexes
\[
\begin{array}{l|lll}
b = {\rm IC}(X,k) & -2 & -1 & 0 \\ \hline
U & k & 0 & 0 \\
\{0\} & k & k & 0 
\end{array}
\quad 
\begin{array}{l|lll}
a = {\rm IC}(\{0\},k) & -2 & -1 & 0 \\ \hline
U & 0 & 0 & 0 \\
\{0\} & 0 & 0 & k
\end{array}
\]
We can use this to conclude that $[\pi_*(\ul{k}_{\tilde{X}}[2])] = [b] + 2[a]$ in the Grothendieck group.

We remark that the fiber $\bP^1 = \pi^{-1})(0) \subset \rT^* \bP^1$ has self-intersection $-2$, and in general the fiber will have an intersection form which will be useful later.
\end{example}

\section{Parity sheaves.}

\begin{definition}
\begin{compactenum}
\item A {\bf pariversity} is a function $\dagger \colon \Lambda \to \bZ/2$. Fix $? \in \{*, !\}$. 
\item A complex $\cF \in D^b_\Lambda(X,k)$ is {\bf ?-even} if $\cH^m(i_\lambda^? \cF)$ vanishes for $m \not\equiv \dagger(\lambda) \pmod 2$ (and the $\cH^m(i_\lambda^? \cF)$ are torsionfree if $k$ is not a field). $\cF$ is {\bf ?-odd} if $\cF[i]$ is $(\dagger,?)$-even.
\item $\cF$ is {\bf $\dagger$-even} if it is both $(\dagger, *)$ and $(\dagger, !)$-even.
\item $\cF$ is a {\bf $\dagger$-parity sheaf} if $\cF = \cF_{\rm even} \oplus \cF_{\rm odd}$ where $\cF_{\rm even}$ and $\cF_{\rm odd}[1]$ are $\dagger$-even. \qedhere
\end{compactenum}
\end{definition}

\begin{example}
The constant pariversity is denoted $\natural(\lambda) = \ol{0}$ and we also have the function $\diamond(\lambda) = d_\lambda \pmod 2$.
\end{example}

We now make an assumption which is usually satisfied in representation-theoretic contexts: For all $\lambda \in \Lambda$ and $\cL \in {\rm Loc}_{k\text{-free}}(X_\lambda,k)$, we have $\rH^{\rm odd}(X_\lambda, \cL) = 0$.

\begin{remark} This assumption holds for the nilpotent cone of $\fg = {\rm Lie}(G)$ in $G$-equivariant case, if $k$ is a field of characteristic $p$, where $p$ is not a torsion prime for $G$.
\end{remark}

\begin{theorem}[Uniqueness, JMW]
Assume that $k$ is a field or a complete local PID. Let $\cF$ be an indecomposable parity complex. Then
\begin{compactenum}[\rm 1.]
\item There is $\lambda \in \Lambda$ such that $\supp \cF = \ol{X_\lambda}$.
\item $\cF|_{X_\lambda} \simeq \cL[d]$ for some indecomposable $\cL \in {\rm Loc}(X_\lambda, k)$ and some $d \in \bZ$.
\item Any indecomposable parity complex $\cF'$ satisfying the above two conditions with the same $\lambda, \cL, d$ is isomorphic to $\cF$.
\end{compactenum}
\end{theorem}

If $\cF$ is as above, with $d = d_\lambda$, we call it the parity sheaf $\cE(\lambda, \cL)$.

\begin{proposition}
$\bD(\cE(\lambda, \cL)) = \cE(\lambda, \cL^\vee)$.
\end{proposition}

On existence: 
\begin{compactitem}
\item One can use proper ``even'' maps (in particular, ``even'' resolutions) (this means that the fibers do not have nonzero cohomology in odd degrees).
\item There is an analogue of Deligne's construction. Given $j \colon U \subset X$ and $i \colon F \subset X$, the sequence $A \to i^*j_* F \to B$, where $A = \leftexp{p}{\tau}_{\le -1}$, gives a construction of ${j_!}_*$.

If it is possible to put $i^*j_* F$ in a triangle with $A$ in even degrees and $B$ in odd degrees (or the other way around), then we can do one step more in constructing a parity extension (for either extension of the pariversity). This works in both directions ($i^* j_* F = \text{even part} \oplus \text{odd part}$) if $F$ is contractible (in the non-equivariant setting).
\end{compactitem}

\begin{example}
\begin{compactenum}
\item $X = G/B = \coprod_{x \in W} BxB/B$. 

For $\dagger = \natural$, we get $\cE^\natural(x,\bQ) = {\rm IC}(x,\bQ)$. 

For $\dagger = \diamond$, we get that $\cE^\diamond(x,k)$ is a tilting sheaf (as described in ``Tilting exercises'' by Beilinson--Bezrukavnikov--Mirkovic).

In positive characteristic, $\cE^\natural(x,k)$ gives the ``special sheaves'' of Soergel.
\item $X = \cG r_G = \coprod_\lambda \cG r_\lambda$ is the affine Grassmannian (the stratification is given by the $G[\![t]\!]$-orbits). Since all of the orbits are even dimensional, we get $\natural = \diamond$. If the characteristic $p$ is not too small, then $\cE^\natural(\lambda, k) = \cE^\diamond(\lambda,k)$ is perverse and by geometric Satake, it corresponds to $T(\lambda)$, which is a tilting module for $G^\vee$. \qedhere
\end{compactenum}
\end{example}

\section{The $p$-canonical basis.}

Fix $X = G/B = \coprod_{w \in W} BwB/B$ where $W$ is the Weyl group. Let $\ell \colon W \to \bN$ be the length function and $\le$ be the Bruhat order on $W$. Then $BwB/B \cong \bC^{\ell(w)}$, so they are contractible. In particular, they are simply-connected, so we only have the constant local systems and have vanishing of odd cohomology, so we get uniqueness of parity sheaves. Furthermore, parity sheaves exist for any pariversity (via the analogue of Deligne's construction).

From now on, fix $\dagger = \natural$ and a field $k$.

Let $\cH$ be the Hecke algebra with generators $H_s$ for each simple reflection $s \in S$, and relations $H_s^2 = (v^{-1} - v) H_s + 1$ and the braid relations. Given $\cF \in D^b_W(G/B, k)$. Set 
\[
{\rm ch}(\cF) = \sum_{i,x} \dim \cH^i(\cF_x) v^{\ell(x) - i} H_x.
\]

\begin{example} 
${\rm ch}({\rm IC}(x,\bQ)) = \ul{H}_x$ is the Kazhdan--Lusztig basis.
\end{example}

Define $\leftexp{p}{\ul{H}}_x = {\rm ch}(\cE(x, \bF_p))$. Then
\begin{compactenum}[(a)]
\item $(\leftexp{p}{\ul{H}}_x)_{x \in W}$ is a self-dual basis for $\cH$, which we call the {\bf $p$-canonical basis}.
\item $\leftexp{p}{\ul{H}}_x \leftexp{p}{\ul{H}}_y = \sum_z \leftexp{p}{\mu}_{x,y}^z \leftexp{p}{\ul{H}}_z$ with $\leftexp{p}{\mu}_{x,y}^z \in \bN[v,v^{-1}]$ since the convolution product sends parity sheaves to parity sheaves.
\item $\leftexp{p}{\ul{H}}_y = \sum_{x \le y} \leftexp{p}{m}_{x,y} \ul{H}_x$ with $\leftexp{p}{m}_{x,y} \in \bN[v,v^{-1}]$ because ${\rm ch}(\cE(y,\bF_p)) = {\rm ch}(\cE(y,\bZ_p)) = {\rm ch}(\bQ_p \otimes \cE(y,\bZ_p)) = {\rm ch}(\bigoplus {\rm IC}[])$.
\end{compactenum}

\subsection{Basic problems.}

\begin{compactenum}[(BP1)]
\item When is $\leftexp{p}{\ul{H}}_x = \ul{H}_x$? This is true if $p$ is large enough with respect to $x$.
\item Determine the equivalence classes generated by $x \sim_P y$ if $\leftexp{p}{m}_{x,y} \ne 0$.
\item Determine $\leftexp{p}{m}_{x,y}$ in general.
\end{compactenum}

\begin{remark} 
\begin{compactenum}[1.]
\item By (Soergel 2000, ``On the relation...''), if the answer to (BP1) for flag varieties $G/B$ is ``always'' for $p$ bigger than the Coxeter number, then ``Lusztig's conjecture around the Steinberg weight'' is true.
\item Fiebig: the same thing for closed subsets $Z$ of the affine flag manifold implies all of Lusztig's conjecture.
\item (BP2) for $\cG r_G$ (which is hard) is equivalent, via geometric Satake, to the linkage principle (which is known).
\item (BP3) for $\cG r_G$ is equivalent, via geometric Satake, to computing characters of tilting modules for $G^\vee_{\bF_p}$. \qedhere
\end{compactenum}
\end{remark}

\subsection{Examples.}

Williamson ``Modular IC complexes on flag varieties'' defines ``separated elements'' $W_\sigma \subset W$, and gives a sufficient condition to have $\leftexp{p}{\ul{H}}_x = \ul{H}_x$ (for all $p$ and all $x \in W_\sigma$)
using certain positivity constraints.

Using computer calculations for low ranks, we have $W_\sigma = W$ for ${\rm A}_n$ with $n \le 6$. In particular, Lusztig's conjecture around St holds for $\SL_n$ with $n \le 7$. The case ${\rm A}_7$ is interesting. In an appendix by Tom Braden, it is shown that there is 2-torsion in IC for the hexagon permutation in ${\rm A}_7$. We have $\leftexp{p}{\ul{H}}_x = \ul{H}_x$ for all $p$ except for 38 elements in $S_8$, which fall into 3 families where $\leftexp{2}{\ul{H}}_x \ne \ul{H}_x$. In particular, for $x = 62845173$ and $y = 21645387$, we have $\leftexp{2}{\ul{H}}_x = \ul{H}_x + \ul{H}_y$ (related to the ``Kashiwara--Saito singularity''.)

In type ${\rm B}_2$, this is okay except $\leftexp{2}{\ul{H}}_{sts} = \ul{H}_{sts} + \ul{H}_s$ (related to ${\rm A}_1$ singularity). Here $s$ is the short root.

In type ${\rm G}_2$, there are some problems for $p=2,3$.

For $\tilde{\rm A}_1$, $\leftexp{p}{\ul{H}}_x$ is equivalent to characters of tilting modules for $\SL_2$ (which are known).

Recently, Polo found $n$-torsion for $\SL_{4n}$.

\subsection{The characteristic variety.}

Let $\pi \colon T^* X \to X$ be the projection of the cotangent bundle. Let $M$ be a $\rD_X$-module. A good filtration for $M$ gives an associated graded ${\rm gr}(M)$ which is a $\pi_*\cO_{T^* X}$-module. Its support is denoted ${\rm CC}(M)$ and called the {\bf characteristic cycle} in $T^*X$.

If $\cF$ is a constructible complex on $X$, then we can define ${\rm CC}(\cF)$ using a Morse function $f$ and the functor $i^!$, where $i \colon \{x \mid f(x) \ge 0\} \to X$. 

The $\rD_X$-module gives rise to a constructible complex, and the two constructions of CC agree under the Riemann--Hilbert correspondence.

\begin{conjecture}[``Finkelberg's conjecture'', Isle of Skye 2010, (Kazhdan)]
$\leftexp{p}{\ul{H}}_x \ne \ul{H}_x$ implies that ${\rm CC}({\rm IC}(x, \bQ))$ is reducible.
\end{conjecture}

Observation (Vilonen--Williamson): ${\rm CC}(-)$ commutes with extension of scalars (take Euler characteristic). So
\[
{\rm CC}({\rm IC}(x, \bQ)) = {\rm CC}({\rm IC}(x,\bZ) \otimes_\bZ^{\rL} \bF_p).
\]
So if ${\rm CC}({\rm IC}(x, \bQ))$ is irreducible, then ${\rm IC}(x,\bZ) \otimes_\bZ^{\rL} \bF_p$ is simple for all $p$. In particular, ${\rm CC}({\rm IC}(x,\bQ))$ gives bounds for decomposition numbers (independent of $p$).

\subsection{Intersection forms and failure of the decomposition theorem.}

Recall the definition of Borel--Moore homology: $\rH^{\rm BM}_i(X) = \rH^{-i}(X; \omega_X) = \hom(\ul{k}_X, \omega_X[-i])$. If $Y$ is smooth and connected of dimension $n$, then we have an isomorphism $\mu_Y \colon \ul{k}_Y \cong \omega_Y[-2n]$ which gives the fundamental class $\mu_Y \in \rH^{\rm BM}_{2n}(Y)$. If $Y$ is singular, we can still define fundamental classes, one for each irreducible component of maximal dimension $n$. 

Given a closed embedding $i \colon F \subset Y$ with $F$ compact and $Y$ smooth, then we can also define $\rH^{\rm BM}_m(F) = \rH^{2n-m}(Y, Y \setminus F)$. In particular, $\rH^{\rm BM}_\bullet(F)$ inherits an intersection form $\cup \colon \rH^{\rm BM}_p(F) \otimes \rH^{\rm BM}_q(F) \to \rH^{\rm BM}_{p+q-2n}(F)$. If $p+q=2n$, then we get this gives a pairing 
\[
\rH^{\rm BM}_p(F) \otimes \rH^{\rm BM}_q(F) \to \rH^{\rm BM}_0(F) \to \rH^{\rm BM}_0({\rm pt}) = k.
\]
So we get bilinear forms $B^m_F \colon \rH^{\rm BM}_{n+m}(F) \otimes \rH^{\rm BM}_{n-m}(F) \to k$. If $F$ is half-dimensional, there is only one of them.

\begin{theorem}[Decomposing parity complexes]
Let $f \colon \tilde{X} \to X$ be a proper even map where $X$ is stratified. We assume that $X$ satisfies the parity conditions for the strata (so we have uniqueness of parity sheaves). Then 
\[
f_* \ul{k}_{\tilde{X}}[d_X] = \bigoplus_{n, \lambda, \cL} \cE(\lambda, \cL)[-n]^{\oplus m_n(\lambda, \cL)}
\]
where $m_n(\lambda, \cL)$ are the ranks of the intersection forms on local systems of the Borel--Moore homology of the fibers.
\end{theorem}

\end{document}