$p\text{-adic}$ 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.

2. Contiguity Theorem

2.1. A Partial Order on $M\left(𝒥\right)$ (see [Zel1980, Sec. 7]). Let $a,b\in M\left(𝒥\right)$ be graded partitions. We say that $b$ is obtained from $a$ by an elementary operation if $b$ is obtained from $a$ by the deletion of a pair of connected segments ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ and addition in its place of the segments ${\mathrm{\Delta }}^{\cup }={\mathrm{\Delta }}_1\cup {\mathrm{\Delta }}_2$ and ${\mathrm{\Delta }}^{\cap }={\mathrm{\Delta }}_1\cap {\mathrm{\Delta }}_2$ (if ${\mathrm{\Delta }}_1\cap {\mathrm{\Delta }}_2=\varnothing ,$ then we add only the segment ${\mathrm{\Delta }}^{\cup }\text{);}$ more formally, $b=a-{\chi }_{{\mathrm{\Delta }}_1}-{\chi }_{{\mathrm{\Delta }}_2}+{\chi }_{{\mathrm{\Delta }}^{\cup }}+{\chi }_{{\mathrm{\Delta }}^{\cap }},$ where ${\chi }_{\mathrm{\Delta }}\in M\left(𝒥\right)$ is the characteristic function of the singleton $\left\{\mathrm{\Delta }\right\}\subset 𝒥\text{.}$ We write $b<a$ if $b$ can be obtained from $a$ by a series of elementary operations. It is easily seen that this definition turns $M\left(𝒥\right)$ into a partially ordered set, where the elements from different $M{\left(𝒥\right)}_{\phi }$ are incomparable with each other.

Theorem 7.1 of [Zel1980] asserts that the multiplicity $m_{b,a}^{\left(\rho \right)}$ is nonzero if and only if $b\le a\text{;}$ in addition, $m_{a,a}^{\left(\rho \right)}=1\text{.}$

2.2. Contiguity Theorem. ${\bar{X}}_b=\underset{a\ge b}{⨆}{X}_a$ for each $b\in M\left(𝒥\right)\text{.}$ In other words, $\text{"}b\le a\text{"}$ and $\text{"}{X}_a\subset {\bar{X}}_b\text{"}$ are equivalent.

The remaining part of Sec. 2 is devoted to the proof of this theorem.

2.3. Proof of the Implication $b\le a\Rightarrow X_a\subset {\bar{X}}_b\text{.}$ We can assume that $b$ can be obtained from $a$ by an elementary operation. By the same token, the whole problem reduces to the case where $a={\chi }_{{\mathrm{\Delta }}_1}+{\chi }_{{\mathrm{\Delta }}_2}$ and $b={\chi }_{{\mathrm{\Delta }}_1\cup {\mathrm{\Delta }}_2}+{\chi }_{{\mathrm{\Delta }}_1\cap {\mathrm{\Delta }}_2},$ where ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are connected segments in $\mathbb{Z}\text{.}$ Let $T\in X_a\text{.}$ This means that $V$ has a basis that consists of the vectors ${\xi }_i\in V_i$ $\text{(}i\in {\mathrm{\Delta }}_1\text{)}$ and ${\eta }_i\in V_i$ $\text{(}i\in {\mathrm{\Delta }}_2\text{)}$ such that $T{\xi }_i={\xi }_{i+1}$ and $T{\eta }_i={\eta }_{i+1}$ (we assume that ${\xi }_{e\left({\mathrm{\Delta }}_1\right)+1}=0$ and ${\eta }_{e\left({\mathrm{\Delta }}_2\right)+1}=0\text{).}$ Let ${\mathrm{\Delta }}_1\to {\mathrm{\Delta }}_2$ (see Sec. 1.2). Let us consider the operator $T_0\in E\left(V\right)$ such that $T_0{\xi }_i={\eta }_{i+1}$ for $b\left({\mathrm{\Delta }}_2\right)-1\le i\le e\left({\mathrm{\Delta }}_1\right)$ and $T_0$ is equal to $0$ on the remaining basis vectors. It is easily verified that the operator $T+\epsilon ·T_0$ belongs to $X_b$ for $\epsilon \ne 0\text{.}$ Therefore $T\in {\bar{X}}_b,$ which was required to be proved.

2.4. Correspondence $a\to d_a\text{.}$ With each graded partition $a$ we associate a finite function $d_a:𝒥\to {\mathbb{Z}}^{+}$ by setting $d_a\left(\mathrm{\Delta }\right)=\sum _{\mathrm{\Delta }\prime \supset \mathrm{\Delta }}a\left(\mathrm{\Delta }\prime \right)$ for $\mathrm{\Delta }\in 𝒮\text{.}$ Obviously, $a$ is uniquely regenerated by $d_a\text{.}$ Namely, for each $\mathrm{\Delta }=[i,j]\in 𝒥$ we set ${\mathrm{\Delta }}^{+}=[i,j+1],$ ${}^{+}\mathrm{\Delta }=[i-1,j],$ and ${}^{+}{\mathrm{\Delta }}^{+}=[i-1,j+1]\text{.}$ In this notation we have $$a\left(\mathrm{\Delta }\right)=d_a\left(\mathrm{\Delta }\right)-d_a\left({\mathrm{\Delta }}^{+}\right)-d_a\left({}^{+}\mathrm{\Delta }\right)+d_a\left({}^{+}{\mathrm{\Delta }}^{+}\right),\mathrm{\Delta }\in 𝒥\text{.}$$

The following proposition shows the origin of the function $d_a\text{.}$

Proposition. For each $T\in E\left(V\right)=E_{\phi }$ and each segment $\underset{T}{\mathrm{\Delta }}=\underset{T}{[i,j]}\in \underset{T}{𝒥},$ define the operator $T^{\mathrm{\Delta }}:V_i\to V_j$ as the composition $V_i\to V_{i+1}\to \dots \to V_j\text{.}$ If $T\in X_a,$ then $d_a\left(\mathrm{\Delta }\right)=\text{rk}\hspace{0.17em}{T}^{\mathrm{\Delta }}$ for all $\mathrm{\Delta }\in 𝒥$ if [in particular, $d_a\left(\left\{i\right\}\right)=\text{dim}\hspace{0.17em}{V}_i=\phi \left(i\right)$ for $i\in \mathbb{Z}\text{].}$

The proof is obvious. Let us observe that this proposition gives an elementary proof of the fact that $a\in M\left(𝒥\right)$ is uniquely determined by the operator $T$ (see Sec. 1.8).

2.5. Proof of the Implication $X_a\subset {\bar{X}}_b\Rightarrow b\le a\text{.}$ By virtue of the preceding proposition, the inclusion $X_a\subset {\bar{X}}_b$ implies that $d_a\left(\mathrm{\Delta }\right)\le d_b\left(\mathrm{\Delta }\right)$ for all $\mathrm{\Delta }\in 𝒥\text{.}$ Therefore, it remains to prove the following proposition.

Combinatorial Proposition. Let $a,b\in M\left(𝒥\right)$ be graded partitions. The following conditions are equivalent:

(1)	$b\le a\text{;}$
(2)	$d_a\left(\mathrm{\Delta }\right)\le d_b\left(\mathrm{\Delta }\right)$ for all $\mathrm{\Delta }\in 𝒥,$ moreover, $d_a\left(\left\{i\right\}\right)=d_b\left(\left\{i\right\}\right)$ for $i\in \mathbb{Z}\text{.}$

2.6. Proof of the Combinatorial Proposition. In fact, the implication (1) $\Rightarrow$ (2) has already been proved [we have shown that (1) $\Rightarrow X_a\subset {\bar{X}}_b\Rightarrow$ (2)]. It can also be easily proved directly: If $b=a-{\chi }_{{\mathrm{\Delta }}_1}-{\chi }_{{\mathrm{\Delta }}_2}+{\chi }_{{\mathrm{\Delta }}^{\cup }}+{\chi }_{{\mathrm{\Delta }}^{\cap }}$ (see Sec. 2.1), then we easily see that $d_b=d_a+{\chi }_{({}^{+}{{\mathrm{\Delta }}^{\cap }}^{+}\subset \mathrm{\Delta }\subset {\mathrm{\Delta }}^{\cup })}$ (if ${\mathrm{\Delta }}^{\cap }=\varnothing ,$ i.e., say, $b\left({\mathrm{\Delta }}_2\right)=e\left({\mathrm{\Delta }}_1\right)+1,$ then ${}^{+}{{\mathrm{\Delta }}^{\cap }}^{+}$ is defined as the segment $\text{[}e\left({\mathrm{\Delta }}_1\right),b\left({\mathrm{\Delta }}_2\right)\text{]).}$

Fundamental Lemma. Let $d:𝒥\to L^{+}$ be a nonzero finite function such that $d\left(\left\{i\right\}\right)=0$ for all $i\in \mathbb{Z},$ and suppose that the function $c:𝒥\to \mathbb{Z}$ is defined by the equality $c\left(\mathrm{\Delta }\right)=d\left(\mathrm{\Delta }\right)-d\left({}^{+}\mathrm{\Delta }\right)-d\left({\mathrm{\Delta }}^{+}\right)+d\left({}^{+}{\mathrm{\Delta }}^{+}\right)$ $\text{(}\mathrm{\Delta }\in 𝒮\text{).}$ Then there exist connected segments ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ such that $c\left({\mathrm{\Delta }}_1\right)<0,$ $c\left({\mathrm{\Delta }}_2\right)<0$ and $d\left(\mathrm{\Delta }\right)>0$ for any segment $\mathrm{\Delta }$ such that ${}^{+}{({\mathrm{\Delta }}_1\cap {\mathrm{\Delta }}_2)}^{+}\subset \mathrm{\Delta }\subset {\mathrm{\Delta }}_1\cup {\mathrm{\Delta }}_2\text{.}$

We deduce the implication (2) $\Rightarrow$ (1) from this lemma. Let $a$ and $b$ satisfy condition (2) and suppose that $a\ne b\text{.}$ Let us apply the lemma to $d=d_b-d_a$ (so that $c=b-a\text{).}$ Let ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ be the segments whose existence is asserted in the lemma. We have $a\left({\mathrm{\Delta }}_1\right)>0$ and $a\left({\mathrm{\Delta }}_2\right)>0,$ so that $a\prime =a-{\chi }_{{\mathrm{\Delta }}_1}-{\chi }_{{\mathrm{\Delta }}_2}+{\chi }_{{\mathrm{\Delta }}^{\cup }}+{\chi }_{{\mathrm{\Delta }}^{\cap }}\in M\left(𝒥\right)\text{.}$ Therefore, $a>a\prime ,$ and the partitions $a\prime$ and $b$ satisfy the condition (2). By means of obvious induction, we can assume that $a\prime \ge b,$ i.e., $a>b,$ which was required to be proved.

2.7. Proof of the Fundamental Lemma. Step 1. Reduction to the case where $d\left([i,i+1]\right)>0$ for a certain $i\in \mathbb{Z}\text{.}$ If $d\left([i,i+1]\right)=0$ for all $i,$ then the function $d^{-}:𝒥\to {\mathbb{Z}}^{+},$ defined by the equality $d^{-}\left(\mathrm{\Delta }\right)=d\left({\mathrm{\Delta }}^{+}\right),$ satisfies the conditions of the fundamental lemma. The validity of the statement of the lemma for $d$ follows obviously from its validity for $d^{-}\text{.}$ If $d^{-}\left([i,i+1]\right)=0$ for all $i\in \mathbb{Z},$ then we consider the function ${\left(d^{-}\right)}^{-},$ etc. Since $d$ is nonzero, we finally arrive at a function $d\prime$ such that $d\prime \left([i,i+1]\right)>0$ for a certain $i,$ and it is sufficient to prove the assertion of the lemma for $d\prime \text{.}$ Thus, we can assume that $d\left([i,i+1]\right)>0$ for a certain $i\text{;}$ without loss of generality, we will assume that $d\left([0,1]\right)>0\text{.}$

Step 2. We will often use the following identity, which can be directly verified:

$\text{(}*\text{)}$ If ${\mathrm{\Delta }}_0$ and $\bar{\mathrm{\Delta }}$ are two segments of $\mathbb{Z}$ such that ${\mathrm{\Delta }}_0=[i_0,j_0]\subset \bar{\mathrm{\Delta }}=[\bar{i},\bar{j}],$ then $$\sum _{{\mathrm{\Delta }}_0\subset \mathrm{\Delta }\subset \bar{\mathrm{\Delta }}}c\left(\mathrm{\Delta }\right)=d\left({\mathrm{\Delta }}_0\right)-d\left([i_0,\bar{j}+1]\right)-d\left([\bar{i}-1,j_0]\right)+d\left({}^{+}{\bar{\mathrm{\Delta }}}^{+}\right)\text{.}$$

First of all we apply $\text{(}*\text{)}$ to the segments ${\mathrm{\Delta }}_0=\left\{0\right\}$ and $\bar{\mathrm{\Delta }}=[i,0],$ where $i\le 0\text{.}$ We get $$\begin{array}{cc}\sum _{i\le i\prime \le 0}c\left([i\prime ,0]\right)=-d\left([0,1]\right)-d\left([i-1,0]\right)-d\left([i-1,1]\right)\text{.}& \text{(1)}\end{array}$$ Taking limit as $i\to -\infty ,$ we see that $\sum _{i\prime \le 0}c\left([i\prime ,0]\right)=-d\left([0,1]\right)<0\text{.}$ Hence there exists a $k\le 0$ such that $c\left([i,0]\right)\ge 0$ for $k<i\le 0,$ and $c\left([k,0]\right)<0\text{.}$ Again applying (1), we see that $$\begin{array}{cc}d\left([i,1]\right)\ge d\left([0,1]\right)>0\text{for}k\le i\le 0\text{.}& \text{(2)}\end{array}$$

In the same manner, there exists an $l\ge 1$ such that $c\left([1,j]\right)\ge 0$ for $1\le j<l,$ and $c\left([1,l]\right)<0\text{;}$ in addition, $$\begin{array}{cc}d\left([0,j]\right)\ge d\left([0,1]\right)>0\text{for}1\le j\le l\text{.}& \text{(3)}\end{array}$$

Step 3. We consider the following statement for $j=1,2,\dots ,l\text{:}$ $$\begin{array}{cc}d\left(\mathrm{\Delta }\right)>0\text{for}[0,1]\subset \mathrm{\Delta }\subset [k,j]\text{.}& \left(A_j\right)\end{array}$$ Let us observe that $\left(A_1\right)$ is valid by virtue of the inequality (2).

Let us now suppose that the assertion of the fundamental lemma is not fulfilled for our function $d\text{.}$ Starting from this supposition, we prove the validity of all $\left(A_j\right)$ by means of induction on $j\text{.}$

Let $\left(A_j\right)$ be valid. We prove $\left(A_{j+1}\right),$ i.e., that $d\left([i,j+1]\right)>0$ for $k\le i\le 0\text{.}$ By virtue of (3), we can assume that $i<0\text{.}$ Let us apply $\text{(}*\text{)}$ to the segments ${\mathrm{\Delta }}_0=[0,1]$ and $\bar{\mathrm{\Delta }}=[i+1,j],$ and rewrite the obtained equation in the form $$\begin{array}{cc}d\left([i,j+1]\right)=\sum _{{\mathrm{\Delta }}_0\subset \mathrm{\Delta }\subset \bar{\mathrm{\Delta }}}c\left(\mathrm{\Delta }\right)+\{d\left([0,j+1]\right)-d\left([0,1]\right)+d\left([i,1]\right)\}\text{.}& \text{(4)}\end{array}$$ By virtue of (2) and (3), the expression within the curly brackets is positive. It remains to verify that $c\left(\mathrm{\Delta }\right)\ge 0$ for ${\mathrm{\Delta }}_0\subset \mathrm{\Delta }\subset \bar{\mathrm{\Delta }}\text{.}$ Let us suppose that this is not so, i.e., $c\left(\mathrm{\Delta }\right)<0$ for a certain $\mathrm{\Delta }$ such that ${\mathrm{\Delta }}_0\subset \mathrm{\Delta }\subset \bar{\mathrm{\Delta }}\text{.}$ Taking $\left(A_j\right)$ into account, we see that the segments ${\mathrm{\Delta }}_1=[k,0]$ and ${\mathrm{\Delta }}_2=\mathrm{\Delta }$ satisfy the conclusion of the fundamental lemma, which contradicts our supposition.

Thus, we have proved the statement $\left(A_j\right)$ for all $j\in [1,l]\text{;}$ in particular, $\left(A_l\right)$ is valid. But this means that the segments ${\mathrm{\Delta }}_1=[k,0]$ and ${\mathrm{\Delta }}_2=[1,l]$ satisfy the conclusion of the lemma; the last statement is a contradiction!

The fundamental Lemma is proved and, with it, Proposition 2.5 and Theorem 2.2 are also proved.

2.8. Remarks. a) The results of this section have "nongraded" analogs. With each partition $\lambda$ of a number $n$ (see Sec. 1.1) we associate the variety $X_{\lambda }$ of the nilpotent $(n\times n)\text{-matrices}$ that have $\lambda \left(k\right)$ Jordan cells of dimension $k\text{.}$ The analog of Theorem 2.2 describes the contiguity of the cells $X_{\lambda }\text{.}$ The resulting partial order relation on the partitions is well known (see, e.g., [LVi1973]). The analog of the elementary operation (see Sec. 2.1) is the replacement of the pair $\{k,l\}$ with $k\ge l>0$ by the pair $\{k+1,l-1\}$ in the partition $\lambda ,$ and the analog of the function $d_a$ is defined by the equality $d_{\lambda }\left(k\right)=\sum _{m\ge k}\lambda \left(m\right)\text{.}$ The analog of Proposition 2.5 is well known; its proof is substantially simpler than that in the graded case.

b) The proofs of the combinatorial results about the relation $\le$ in [Zel1980] are greatly simplified with the help of Proposition 2.5. Let us observe, e.g., the following result (see [Zel1980, 9.10]): Each set $M{\left(𝒥\right)}_{\phi }$ has a unique minimal element $a=a_{\text{min}}\left(\phi \right)\text{.}$ Proof: The corresponding function $d_a$ is defined by the equality $d_a\left(\mathrm{\Delta }\right)=\underset{n\in \mathrm{\Delta }}{\text{min}}\hspace{0.17em}\phi \left(n\right)\text{.}$

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