A glance at the Kazhdan-Lusztig theory

Let $G$ be a split reductive group over a finite field $k = \mathbb{F}_q$. Let $T \subset B \subset G$ be a choice of maximal and Borel. One of the fundamental object in representation theory is the Hecke algebra

$$\mathcal{H}(k) = \operatorname{Fun}_{B(k) \times B(k)}(G(k),\mathbb{C})$$ consisting of complex valued functions on $G(k)$ invariant on both sides by $B(k)$.  This is an algebra with the convolution product

$$(f_1 \star f_2)(g) = \frac{1}{\left| B(k) \right|} \sum_{h_1,h_2 \in G(k), h_1h_2=g} f_1(h_1)f_2(h_2).$$ It is a routine check that character functions on double cosets $B(k)wB(k)$ with $w \in W$ the Weyl group, form a $\mathbb{Z}$-basis of the Hecke algebra. 

Let $S \subset W$ be the set of simple reflections, then Iwahori showed that $\mathcal{H}(k)$ admits a canonical basis $t_s$ (with $s \in S$) such that

$$\begin{align*} t_s^2 & = (q-1)t_s + q t_1 \\ \underset{m_{su}}{\underbrace{t_s t_u \cdots}} & = \underset{m_{su}}{\underbrace{t_u t_s \cdots}} \end{align*}$$ where $s \neq u$ in the second relation and $m_{su}$ denotes the order of $su$ in $W$. This relation is universal in the sense that it is applicable to any Coxeter group and hence we can turn $q$ into a formal variable and obtain an algebra $\mathcal{H}$, which specializes to $\mathcal{H}(k)$ by $q \longmapsto \left| \mathbb{F}_q \right|$, hence justifying the notation. 

More concretely, let $q$ be a formal variable and $\mathcal{H}$ is the $\mathbb{Z}[q^{\pm 1/2}]$ generated by symbols $T_s$ ($s \in S$) subjecting to realtions 

$$\begin{align*} T_s^2 & = (q-1)T_s + q T_1 \\ \underset{m_{su}}{\underbrace{T_s T_u \cdots}} & = \underset{m_{su}}{\underbrace{T_u T_s \cdots}} \end{align*}.$$ There is an involution operator

$$D \colon \mathcal{H} \longrightarrow \mathcal{H}$$ given by

$$\begin{align*} D(q^{1/2}) & = q^{1/2} \\ D(T_s) & = q^{-1}T_s + q^{-1}-1 \end{align*}$$ (here one can check that $D(T_s)$ is invert of $T_s$. More generally, all $T_w$ are invertible). Kazdan-Lusztig later found that there is a unique basis $\left \{H_w \mid w \in W \right \}$ of $\mathcal{H}$ such that they are convolution-preserved, i.e., $D(H_w) = H_w$ and if we write

$$H_w = q^{-\ell(w)/2}\sum_{u \in W}P_{u,w}T_u$$ then the coefficients $P_{u,w} \in \mathbb{Z}[q^{\pm 1/2}]$ satisfy:

• $P_{u,w}=0$ whenver $u > w$ in the Bruhat order.
• $P_{w,w}=1$. 
• If $ u < w$, then $P_{u,w} \in \mathbb{Z}[q]$ and has degree $\leq \frac{1}{2}(\ell(w)-\ell(u)-1)$.

 (where $\ell \colon W \longrightarrow \mathbb{Z}_{\geq 0}$ denotes the length function). The polynomials $P_{u,w} \in \mathbb{Z}[q]$ are called Kazhdan-Lusztig polynomials. One of the long-standing conjecture in representation theory is the Kazhdan-Lusztig conjectures, relating $P_{u,w}(1)$ with representations of complex semisimple Lie groups and Lie algebras that I prefer not to state here. The solution to this problem requires the following key fact, which is one of the earlies application of the theory of perverse sheaves to representation theory:

Theorem. For $u,w \in W$, one has that

$$P_{u,w} = q^{\ell(w)/2} \sum_{i \in \mathbb{Z}} \operatorname{rank} \mathrm{H}^i(\mathbf{IC}_w(\mathbb{Q})_{\mid \mathcal{B}_u} ) q^{i/2}.$$

 Let us clarify the notation: here if one considers the flag variety $ \mathcal{B} = G/B$ then one has the Bruhat decomposition

$$G/B = \coprod_{w \in W} B\dot{w}B/B,$$ where $\mathcal{B}_w =  B\cdot{w}B/B \simeq \mathbb{A}^{\ell(w)}$ are Schubert cells. Those $\mathbf{IC}_w(\mathbb{Q})$ are intersection complexes supported on $\overline{\mathcal{B}}_w$, the Schubert varieties. 

One might call this the categofication theorem. To really understand what's going on beneath the surface, we need to pass to the geometric-derived worl instead of just staying with algebras! 

Grothendieck-Verdier's philosophy tells us that even if you only care about cohomology, it is better to work with derived categories. The obvious categorified candidate for the Hecke algebras is the (cohomologically bounded constructible) derived category 

$$\mathbf{D}^b_{B \times B}(G,\mathbb{Q})$$ of $B \times B$-"equivariant complexes". Equivalently, one can think this category as the $B$-equivariant category on $G/B$

$$\mathbf{D}^b_B(G/B,\mathbb{Q})$$ or the derived category of the double quotient stack $B \setminus G/ B$. The equivariant derived categories are developed by Bernstein-Lunts and the derived categories for algebraic stacks are studied by Laszlo-Olsson. Both approaches either require technical assumptions or topological natures and more importantly, stay in the realm of triangulated categories, which often lack many functorialities. The modern approach is the one of Liu-Zheng, using Lurie's theory of $(\infty,1)$-categories. In this new language, the category $\mathbf{D}^b_B(G/B,\mathbb{Q})$ is a one-line definition

$$\mathbf{D}^b_B(\mathcal{B},\mathbb{Q})  = \operatorname{lim}_{f^*} \ \mathbf{D}^b_{ct}(\underset{n-\text{factors}}{\mathcal{B} \times_B \cdots \times_B \mathcal{B}},\mathbb{Q}),$$ in which we view $\mathbf{D}^b(\mathcal{B}^{n/B})$ as $\infty$-categories and take their $\infty$-limit with respect to the $f^*$-functoriality. In general, for any Artin stack $\mathfrak{X}$ with an atlas $U \longrightarrow \mathfrak{X}$ where $U$ is a scheme, one can define

$$\mathbf{D}^b_{ct}(\mathfrak{X},\mathbb{Q}) =  \operatorname{lim}_{f^*} \ \mathbf{D}^b_{ct}(\underset{n-\text{factors}}{U \times_{\mathfrak{X}} \cdots \times_{\mathfrak{X}} U},\mathbb{Q}),$$

and one still has six-functor formalism as in classical theory. For quotient stacks, one has that if $G$ acts on a variety $X$ so that $X/G$ exists as a scheme then the quotient map $p \colon X \longrightarrow X/G$ induces an equivalence 

$$p^* \colon \mathbf{D}^b_{ct}(X/G,\mathbb{Q}) \simeq \mathbf{D}^b_B(X,\mathbb{Q}).$$

This is just to convince that one (after forgetting these technicalities) can perform six operations as if one is working with schemes. Now we have a "categorified Hecke algebra" $\mathbf{D}^b_B(G/B,\mathbb{Q})$. One might want to turn it into an "algebra", which is a tensor structure in this case. 

The Grothendieck's sheaf-function dictionary tells us how to do this. I omit the details here and go straight into the constructions. There is a convolution diagram

$$G/B \times G/B \overset{p}{\longleftarrow} G \times G/B \overset{q}{\longrightarrow} G \times^B G/B \longrightarrow G/B,$$ where 

• $p,q$ are quotient maps and $m$ is the multiplcation.
• Let $B \times B$ act on $G/B \times G/B$ by $(b_1,b_2) \cdot ([g_1],[g_2])=([b_1g_1],[b_2g_2])$.
• Let $B \times B$ act on $G \times G/B$ by $(b_1,b_2) \cdot (g_1,[g_2]) = (b_1g_1b_2^{-1},[b_2g_2])$. 

With this, $p$ is $B \times B$-equivariant and $q$ is the quotient morphism by the second copy and lastly $m$ is $B$-equivariant.

Let $M,N \in \mathbf{D}^b_B(G/B,\mathbb{Q})$, then one has the external tensor product

$$M \boxtimes N = \mathrm{pr}_1^*(M) \otimes \mathrm{pr}_2^*(N) \in \mathbf{D}^b_{B \times B}G/B \times G/B,\mathbb{Q}).$$ By the observation above, $q$ induces an equivalence

$$\mathbf{D}^b_{B \times B}(G/B \times G/B,\mathbb{Q}) \simeq \mathbf{D}^b_B(G \times^B G/B,\mathbb{Q})$$ and via this equivalence, $M \boxtimes N$ corresponds to something we call the twisted external tensor product

$$M \widetilde{\boxtimes} N \in   \mathbf{D}^b_B(G \times^B G/B,\mathbb{Q}).$$

The convolution product is defined by

$$M \star N = m_*(M  \widetilde{\boxtimes} N) \in \mathbf{D}^b_B(G/B,\mathbb{Q}).$$ Some formal, functorial arguments show the following properties of the convolution products:

• The category $\mathbf{D}^b_B(G/B,\mathbb{Q})$ is a unital monoidal category (even in a $\infty$-categorical sense) with unit the intersection cohomology complex $\mathbf{IC}_e(\mathbb{Q})$ supported on $\overline{\mathcal{B}}_e$. 
• The Verdier duality $\mathbb{D}$ (playing the role of the involution in the Hecke algebra) commutes with the convolution product.
• The category $\operatorname{Semis}_B(\mathcal{B}) \subset \mathbf{D}^b_B(G/B,\mathbb{Q})$ consisting of semisimple objects is closed under the convolution product. In particular, its split Grothendieck group $K_{\oplus}(\operatorname{Semis}_B(\mathcal{B}))$ is naturally a ring. 

Now we can have a closer look at the categorified theorem.

Theorem. The morphism  

$$\begin{align*} \operatorname{ch} \colon K_{\oplus}(\operatorname{Semis}_B(\mathcal{B})) & \longrightarrow \mathcal{H} \\ [M] & \longmapsto \sum_{w \in W,i \in \mathbb{Z}} (\operatorname{rank} \mathrm{H}^i(M_{\mid \mathcal{B}_w})) q^{i/2} T_w \end{align*}$$ is an isomorphism of rings. Moreover,

$$ \operatorname{ch}([M[1]]) = q^{-1/2}\operatorname{ch}([M]) \ \ \ \ \operatorname{ch}([\mathbf{D}(M)]) = D(\operatorname{ch}([M])) \ \ \ \ \operatorname{ch}([\mathbf{IC}_w(\mathbb{Q})]) = H_w.$$ In particular, one can derive the formula expressing Kazhdan-Lusztig polynomials in terms of intersection complexes. The proof of this theorem replies on the theory of parity sheaves that I omit here. There are variants of the theorem above as well, including its $\ell$-adic version where the weight formalism enters the play. Some other closely related topics are the comparison between parity sheaves with Soergel bimodules and Koszul duality.