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

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 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. ${\stackrel{‾}{X}}_{b}=\underset{a\ge b}{⨆}{X}_{a}$ for each $b\in M\left(𝒥\right)\text{.}$ In other words, $\text{"}b\le a\text{"}$ and $\text{"}{X}_{a}\subset {\stackrel{‾}{X}}_{b}\text{"}$ are equivalent.

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

2.3. Proof of the Implication $b\le a⇒{X}_{a}\subset {\stackrel{‾}{X}}_{b}\text{.}$ 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 $ℤ\text{.}$ 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 $T\in {\stackrel{‾}{X}}_{b},$ which was required to be proved.

2.4. Correspondence $a\to {d}_{a}\text{.}$ With each graded partition $a$ we associate a finite function ${d}_{a}:𝒥\to {ℤ}^{+}$ 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 $\mathrm{\Delta }=\left[i,j\right]\in 𝒥$ we set ${\mathrm{\Delta }}^{+}=\left[i,j+1\right],$ ${}^{+}\mathrm{\Delta }=\left[i-1,j\right],$ and ${}^{+}{\mathrm{\Delta }}^{+}=\left[i-1,j+1\right]\text{.}$ In this notation we have $a(Δ)=da (Δ)-da (Δ+)- da(+Δ) +da(+Δ+) ,Δ∈𝒥.$

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 $\underset{T}{\mathrm{\Delta }}=\underset{T}{\left[i,j\right]}\in \underset{T}{𝒥},$ 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 ${d}_{a}\left(\mathrm{\Delta }\right)=\text{rk} {T}^{\mathrm{\Delta }}$ for all $\mathrm{\Delta }\in 𝒥$ if [in particular, ${d}_{a}\left(\left\{i\right\}\right)=\text{dim} {V}_{i}=\phi \left(i\right)$ for $i\in ℤ\text{].}$

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 ${X}_{a}\subset {\stackrel{‾}{X}}_{b}⇒b\le a\text{.}$ By virtue of the preceding proposition, the inclusion ${X}_{a}\subset {\stackrel{‾}{X}}_{b}$ 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 $i\in ℤ\text{.}$

2.6. Proof of the Combinatorial Proposition. In fact, the implication (1) $⇒$ (2) has already been proved [we have shown that (1) $⇒{X}_{a}\subset {\stackrel{‾}{X}}_{b}⇒$ (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 ${d}_{b}={d}_{a}+{\chi }_{\left({}^{+}{{\mathrm{\Delta }}^{\cap }}^{+}\subset \mathrm{\Delta }\subset {\mathrm{\Delta }}^{\cup }\right)}$ (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 $i\in ℤ,$ and suppose that the function $c:𝒥\to ℤ$ 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 ${}^{+}{\left({\mathrm{\Delta }}_{1}\cap {\mathrm{\Delta }}_{2}\right)}^{+}\subset \mathrm{\Delta }\subset {\mathrm{\Delta }}_{1}\cup {\mathrm{\Delta }}_{2}\text{.}$

We deduce the implication (2) $⇒$ (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 $d\left(\left[i,i+1\right]\right)>0$ for a certain $i\in ℤ\text{.}$ If $d\left(\left[i,i+1\right]\right)=0$ for all $i,$ then the function ${d}^{-}:𝒥\to {ℤ}^{+},$ 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 ${d}^{-}\left(\left[i,i+1\right]\right)=0$ for all $i\in ℤ,$ then we consider the function ${\left({d}^{-}\right)}^{-},$ etc. Since $d$ is nonzero, we finally arrive at a function $d\prime$ such that $d\prime \left(\left[i,i+1\right]\right)>0$ for a certain $i,$ and it is sufficient to prove the assertion of the lemma for $d\prime \text{.}$ Thus, we can assume that $d\left(\left[i,i+1\right]\right)>0$ for a certain $i\text{;}$ without loss of generality, we will assume that $d\left(\left[0,1\right]\right)>0\text{.}$

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

$\text{(}*\text{)}$ If ${\mathrm{\Delta }}_{0}$ and $\stackrel{‾}{\mathrm{\Delta }}$ are two segments of $ℤ$ such that ${\mathrm{\Delta }}_{0}=\left[{i}_{0},{j}_{0}\right]\subset \stackrel{‾}{\mathrm{\Delta }}=\left[\stackrel{‾}{i},\stackrel{‾}{j}\right],$ then $∑Δ0⊂Δ⊂Δ‾ c(Δ)=d(Δ0) -d([i0,j‾+1]) -d([i‾-1,j0]) +d(+Δ‾+).$

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

In the same manner, there exists an $l\ge 1$ such that $c\left(\left[1,j\right]\right)\ge 0$ for $1\le j and $c\left(\left[1,l\right]\right)<0\text{;}$ in addition, $d([0,j])≥ d([0,1])>0 for1≤j≤l. (3)$

Step 3. We consider the following statement for $j=1,2,\dots ,l\text{:}$ $d(Δ)>0 for[0,1]⊂ Δ⊂[k,j]. (Aj)$ 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 $d\left(\left[i,j+1\right]\right)>0$ 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 ${\mathrm{\Delta }}_{0}=\left[0,1\right]$ and $\stackrel{‾}{\mathrm{\Delta }}=\left[i+1,j\right],$ and rewrite the obtained equation in the form $d([i,j+1])= ∑Δ0⊂Δ⊂Δ‾ c(Δ)+ { d([0,j+1])- d([0,1])+ d([i,1]) } . (4)$ 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 ${\mathrm{\Delta }}_{0}\subset \mathrm{\Delta }\subset \stackrel{‾}{\mathrm{\Delta }}\text{.}$ Let us suppose that this is not so, i.e., $c\left(\mathrm{\Delta }\right)<0$ for a certain $\mathrm{\Delta }$ such that ${\mathrm{\Delta }}_{0}\subset \mathrm{\Delta }\subset \stackrel{‾}{\mathrm{\Delta }}\text{.}$ Taking $\left({A}_{j}\right)$ into account, we see that the segments ${\mathrm{\Delta }}_{1}=\left[k,0\right]$ 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 $j\in \left[1,l\right]\text{;}$ in particular, $\left({A}_{l}\right)$ is valid. But this means that the segments ${\mathrm{\Delta }}_{1}=\left[k,0\right]$ and ${\mathrm{\Delta }}_{2}=\left[1,l\right]$ 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 $\left(n×n\right)\text{-matrices}$ 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 $\left\{k,l\right\}$ with $k\ge l>0$ by the pair $\left\{k+1,l-1\right\}$ 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 ${d}_{a}\left(\mathrm{\Delta }\right)=\underset{n\in \mathrm{\Delta }}{\text{min}} \phi \left(n\right)\text{.}$