$p\text{-adic}$ analog of the Kazhdan-Lusztig Hypothesis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 15 April 2014

Notes and References

This is an excerpt of the paper $p\text{-adic}$ analog of the Kazhdan-Lusztig Hypothesis by A. V. Zelevinskii. Terrestrial Physics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 15, No. 2, pp. 9-21, April-June, 1981. Original article submitted November 27, 1980.

3. Examples

3.1. Sheaves $\pi$ and ${\mathscr{H}}^i\left(X\right)\text{.}$ Let us recall definitions from [KLu1980]. Let $X$ be an $n\text{-dimensional}$ algebraic variety over $ℂ\text{.}$ Then there exists on $X$ a complex $\pi$ of sheaves of vector spaces over $ℂ,$ uniquely defined as an object of the corresponding derived category (see Verdier's article in [Del1977]) with the following properties:

(1)	The cohomological sheaves ${\mathscr{H}}^i\left(X\right)$ of the complex $\pi$ are constructive and are equal to zero for $i<0,$ and $\text{dim}\hspace{0.17em}\text{Supp}\hspace{0.17em}{\mathscr{H}}^i\left(X\right)\le n-i-1$ for $i>0\text{.}$
(2)	$\pi$ is self-dual in the derived category.
(3)	The restriction of $\pi$ to the smooth part of $X$ is equivalent to a complex that reduces to the constant sheaf $ℂ$ in a component of degree 0.

All further computations are based on the following proposition.

Proposition. Let $p:\stackrel{\sim }{X}\to X$ be a resolution of singularities of $X,$ i.e., $\stackrel{\sim }{X}$ is a nonsingular variety and $p$ is a proper morphism that is an isomorphism over the smooth part of $X\text{.}$ Suppose that the following condition is fulfilled:

(1')	$\text{dim}\hspace{0.17em}\{x\in X\hspace{0.17em}|\hspace{0.17em}\text{dim}\hspace{0.17em}{p}^{-1}\left(x\right)\ge i\}\le n-2i-1$ for $i>0\text{.}$

Let $1_{\stackrel{\sim }{X}}$ be the complex of the sheaves on $\stackrel{\sim }{X}$ that reduce to the constant sheaf $ℂ$ in a component of degree 0. Then the complex $\pi ={\text{Rp}}_{*}\left(1_{\stackrel{\sim }{X}}\right)$ satisfies the conditions (1)-(3) and ${\mathscr{H}}^i{\left(X\right)}_x=H^i\left(p^{-1}\left(x\right)\right)$ for $i\ge 0$ and $x\in X$ (on the right are the usual cohomologies).

This proposition follows directly from the definitions.

3.2. Nonsingular Cases. For all graded partitions $a$ and $b$ such that $b\le a$ we set $${\mu }_{b,a}=\sum _{i\ge 0}\text{dim}\hspace{0.17em}{\mathscr{H}}^{2i}{\left({\bar{X}}_b\right)}_{x_a},\text{where}{x}_a\in X_a\subset {\bar{X}}_b\text{;}$$ thus, Hypothesis 1.9 asserts that $m_{b,a}^{\left(\rho \right)}={\mu }_{b,a}\text{.}$

Proposition. The number ${\mu }_{b,a}$ is equal to $1$ if at least one of the following conditions is fulfilled:

(1)	$b=a\text{;}$
(2)	$a,b\in M{\left(𝒥\right)}_{\phi },$ where $\phi \left(i\right)\le 1$ for all $i\in \mathbb{Z}\text{;}$
(3)	$b=a_{\text{min}}\left(\phi \right)$ for a certain $\phi \in M\left(\mathbb{Z}\right)$ [see Sec. 2.8, b)].

Proof. By virtue of Sec. 3.1, if $x$ is a nonsingular point of $X,$ then ${\mathscr{H}}^0{\left(X\right)}_x=ℂ$ and ${\mathscr{H}}^i{\left(X\right)}_x=0$ for $i>0\text{.}$ It remains to prove that $x_a$ is a nonsingular point of ${\bar{X}}_b$ in each of our cases.

(1)	It is clear that $x_a$ is a nonsingular point of ${\bar{X}}_a,$ since $X_a$ is an open orbit of the group $\text{Aut}\hspace{0.17em}V$ in ${\bar{X}}_a\text{.}$
(2)	The condition means that $\text{dim}\hspace{0.17em}{V}_i\le 1$ for each $i\in \mathbb{Z}\text{.}$ It is easily verified that in this case $${\bar{X}}_b=\{T\in E_{\phi }\hspace{0.17em}|\hspace{0.17em}{T}^{[i,i+1]}=0,\hspace{0.17em}\text{if}\hspace{0.17em}{d}_b\left([i,i+1]\right)=0\}$$ (see Sec. 2.4). Therefore, ${\bar{X}}_b$ is an affine space, i.e., it is a nonsingular variety.
(3)	If $b=a_{\text{min}}\left(\phi \right),$ then ${\bar{X}}_b=E_{\phi }$ is a nonsingular variety.

3.3. Determinant Varieties. Let us analyze the case in which $a,b\in M{\left(𝒥\right)}_{\phi },$ where $\text{supp}\hspace{0.17em}\phi$ consists of two adjacent points (for definiteness, let $\text{supp}\hspace{0.17em}\phi =\{0,1\}\text{).}$ In other words, two spaces $V_0$ and $V_1$ with $\text{dim}\hspace{0.17em}{V}_i=\phi \left(i\right)$ are given, and $E_{\phi }$ consists of all linear operators $T_0:V_0\to V_1\text{.}$ Let $\text{min}\hspace{0.17em}(\phi \left(0\right),\phi \left(1\right))=n\text{.}$ It is clear that $M{\left(𝒥\right)}_{\phi }$ consists of $(n+1)$ elements $a_0,a_1,\dots ,a_n,$ where $a_r$ is determined by the condition $d_{a_r}\left([0,1]\right)=r\text{.}$ We set $X_r=X_{a_r}\text{.}$ It is clear that $X_r=\left\{T\hspace{0.17em}\right|\hspace{0.17em}\text{rk}\hspace{0.17em}T=r\}$ and ${\bar{X}}_r=\left\{T\hspace{0.17em}\right|\hspace{0.17em}\text{rk}\hspace{0.17em}T\le r\}\text{.}$

Proposition. Suppose that $0\le r_0\le r\le n$ and $x\in X_{r_0}\subset {\bar{X}}_r\text{.}$

a)	${\mathscr{H}}^i{\left({\bar{X}}_r\right)}_x=H^i\left(G_{r-r_0}\left({ℂ}^{n-r_0}\right)\right)$ for $i\ge 0,$ where $G_k\left({ℂ}^m\right)$ is the Grassmanian of the $k\text{-dimensional}$ planes in ${ℂ}^m\text{.}$ In particular, ${\mathscr{H}}^i\left({\bar{X}}_r\right)=0$ for odd $i\text{.}$
b)	${\mu }_{a_r,a_{r_0}}=\left(\genfrac{}{}{0ex}{}{n-r_0}{r-r_0}\right)\text{.}$

Proof. a) By virtue of Proposition 3.1, it is sufficient to construct a resolution of singularities $p:\stackrel{\sim }{X}\to {\bar{X}}_r$ whose fiber over each point $x\in X_{r_0}$ is equal to $G_{r-r_0}\left({ℂ}^{n-r_0}\right)$ [easy computations show that the condition (1') of Sec. 3.1 is fulfilled here]. Let us set $$\begin{array}{c}{\stackrel{\sim }{X}}_I=\{(T,I)\in E_{\phi }\times G_r\left(V_1\right)\hspace{0.17em}|\hspace{0.17em}\text{Im}\hspace{0.17em}T\subset I\},\\ {\stackrel{\sim }{X}}_K=\{(T,K)\in E_{\phi }\times G_{\phi \left(0\right)-r}\left(V_0\right)\hspace{0.17em}|\hspace{0.17em}\text{Ker}\hspace{0.17em}T\supset K\}\text{;}\end{array}$$ suppose that $p_I:{\stackrel{\sim }{X}}_I\to {\bar{X}}_r$ and $p_K:{\stackrel{\sim }{X}}_K\to {\bar{X}}_r$ are the projections on the first factor. It is easily verified that each of the mappings $p_I$ and $p_K$ is a resolution of singularities of the variety ${\bar{X}}_r\text{.}$ One of them is the desired one: If $\phi \left(1\right)\le \phi \left(0\right),$ then we can choose $p_I$ as $p,$ and if $\phi \left(0\right)\le \phi \left(1\right),$ then we can take $p_K$ as $p\text{.}$

b) By virtue of a), ${\mu }_{a_r,a_{r_0}}=\sum _{i\ge 0}\text{dim}\hspace{0.17em}{H}^i\left(G_{r-r_0}\left({ℂ}^{n-r_0}\right)\right)\text{.}$ The Grassmanian $G_{r-r_0}\left({ℂ}^{n-r_0}\right)$ has a triangulation into (complex) Schubert cells; therefore ${\mu }_{a_r,a_{r_0}}$ is equal to the number of these cells. Let us recall that the Schubert cells on $G_k\left({ℂ}^m\right)$ are enumerated by collections of the numbers $1\le n_1<n_2<\dots <n_k\le m\text{;}$ the closure of the cell corresponding to the collection $\left\{n_i\right\}$ is equal to $\{U\in G_k\left({ℂ}^m\right)\hspace{0.17em}|\hspace{0.17em}\text{dim}\hspace{0.17em}(U\cap U_{n_i})\ge i\hspace{0.17em}\text{for}\hspace{0.17em}i=1,2,\dots ,k\},$ where $U_1\subset U_2\subset \dots \subset U_m={ℂ}^m$ is a fixed complete flag. Hence the number of these cells is equal to $\left(\genfrac{}{}{0ex}{}{m}{k}\right),$ which was required to be proved.

3.4. Case $\phi \left(0\right)=\phi \left(2\right)=1,$ $\phi \left(1\right)=2,$ and $\phi \left(i\right)=0$ for $i\notin [0,2]\text{.}$ In other words, three spaces $V_0,$ $V_1,$ and $V_2$ with $\text{dim}\hspace{0.17em}{V}_0=\text{dim}\hspace{0.17em}{V}_2=1,$ $\text{dim}\hspace{0.17em}{V}_1=2$ are given; the variety $E_{\phi }$ consists of the pairs $T=(T^{01}:V_0\to V_1,T^{12}:V_1\to V_2)\text{.}$ The set $M{\left(𝒥\right)}_{\phi }$ consists of five elements; the corresponding varieties ${\bar{X}}_b$ are equal to $\left\{0\right\},$ $\left\{T\right|T^{01}=0\},$ $\left\{T\right|T^{12}=0\},$ $\left\{T\right|T^{12}\circ T^{01}=0\},$ and $E_{\phi }\text{.}$ Only one of these varieties is singular, and, viz., $X=\left\{T\right|T^{12}\circ T^{01}=0\}\text{.}$ Let us set $\stackrel{\sim }{X}=\{(T,L)\in E_{\phi }\times G_1\left(V_1\right)\hspace{0.17em}|\hspace{0.17em}\text{Im}\hspace{0.17em}{T}^{01}\subset L\subset \text{Ker}\hspace{0.17em}{T}^{12}\},$ and let $p:\stackrel{\sim }{X}\to X$ be the projection on the first factor. It is easily seen that $p:\stackrel{\sim }{X}\to X$ is a resolution of singularities of $X$ that is an isomorphism over $X\backslash \left\{0\right\},$ and $p^{-1}\left(0\right)=ℂ{\mathbb{P}}^1\text{.}$ Hence ${\mu }_{b,a}=2$ if $X_a=\left\{0\right\}$ and ${\bar{X}}_b=X\text{.}$ All the remaining coefficients ${\mu }_{b,a}$ are equal to 1 for $a,b\in M{\left(𝒥\right)}_{\phi }$ such that $b\le a\text{.}$

3.5. Remarks. a) In all the cases analyzed above, except 3.2(3), the coefficients $m_{b,a}^{\left(\rho \right)}$ have been computed in [Zel1980, Sec. 11] and Hypothesis 1.9 is verified. The equality $m_{b,a}^{\left(\rho \right)}=1$ for $b=a_{\text{min}}\left(\phi \right)\le a$ will be proved below in Sec. 4.2.

b) For each $b\in M\left(𝒥\right),$ we can easily construct a resolution of singularities of the variety ${\bar{X}}_b$ (and not one only!). For example, let $I_b$ consist of the collections $\{I_{ij}\in G_{d_b\left([i,j]\right)}\left(V_j\right)\},$ $[i,j]\in 𝒥,$ such that $I_{i\prime ,j}\subset I_{ij}$ for $i\prime \le i\text{.}$ Let us set $${\stackrel{\sim }{X}}_I\left(b\right)=\{(T,\left\{I_{ij}\right\})\in E_{\phi }\times I_b\hspace{0.17em}|\hspace{0.17em}{T}^{[j,j+1]}\left(I_{ij}\right)\subset I_{i,j+1}\hspace{0.17em}\text{for}\hspace{0.17em}i\le j\},$$ and let $p:{\stackrel{\sim }{X}}_I\left(b\right)\to {\bar{X}}_b$ be the projection of the first factor. It is easily seen that $p$ is a resolution of singularities of ${\bar{X}}_b\text{.}$ In the same manner we can construct a resolution of ${\bar{X}}_b$ that is connected with the kernels of the operators $T^{[i,i+1]}$ (cf. Sec. 3.3); moreover, there exist many "mixed" resolutions, using kernels as well as images (cf. Sec. 3.4). The author does not know whether we can always select one of these resolutions so that it satisfies the condition (1') of Sec. 3.1, i.e., it would be suitable for the computation of the sheaves ${\mathscr{H}}^i\left({\bar{X}}_b\right)\text{.}$

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