## $p\text{-adic}$ 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.

## 4. Duality

4.1. Involution $\omega \to {\omega }^{t}\text{.}$ Let $\rho \in 𝒞$ and $\mathrm{\Pi }$ be the line in $𝒞$ that contains $\rho$ (see Sees. 1.4 and 1.5). It follows from Theorem 7.1 of [Zel1980] (see also Sec. 2.1) that ${ℛ}_{\mathrm{\Pi }}$ is the ring of polynomials over $ℤ$ in the variables ${⟨\mathrm{\Delta }⟩}^{\left(\rho \right)},$ $\mathrm{\Delta }\in 𝒥\text{.}$ Suppose that $\omega \to {\omega }^{t}$ is an endomorphism of the ring ${ℛ}_{\mathrm{\Pi }}$ that transforms ${⟨\left[i,j\right]⟩}^{\left(\rho \right)}$ into the representation ${⟨\left\{i\right\},\left\{i+1\right\},\dots ,\left\{j\right\}⟩}^{\left(\rho \right)},$ corresponding to the collection of one-point segments (obviously, this endomorphism does not depend on the choice of $\rho \text{).}$ It is proved in [Zel1980, Sec, 9] that it is an involutive automorphism of ${ℛ}_{\mathrm{\Pi }},$ and the hypothesis that it transforms irreducible representations into irreducible representations is put forth. This hypothesis has recently been proved by I. N. Bernshtein. Thus, for each $\rho \in 𝒞$ the equality ${\left({⟨a⟩}^{\left(\rho \right)}\right)}^{t}={⟨{a}^{i}⟩}^{\left(\rho \right)}$ defines an involution $a\to {a}^{t}$ on the set of graded partitions; obviously, it transforms each of the sets $M{\left(𝒥\right)}_{\phi }$ into itself.

Hypothesis. This involution does not depend on $\rho \text{.}$

This hypothesis follows from Hypothesis 1.7 and the results of [Zel1980, Sec. 9] (and, namely, from the fact that the expansion of ${\left({\pi }_{a}^{\left(\rho \right)}\right)}^{t}$ with respect to the basis $\left\{{\pi }_{b}^{\left(\rho \right)}\right\}$ does not depend on $\rho \text{;}$ see [Zel1980], 9.14).

4.2. Example. We fix a $\phi \in M\left(ℤ\right)\text{.}$ Let ${a}_{\text{min}}$ and ${a}_{\text{max}}$ be the minimum and the maximum elements of $M{\left(𝒥\right)}_{\phi }$ $\text{[}{a}_{\text{min}}$ is defined in Sec. 2.8, b) or in [Zel1980], Sec. 9.10; ${a}_{\text{max}}$ consists of one-point segments, i.e., ${a}_{\text{max}}\left(\mathrm{\Delta }\right)=0$ for $\mathrm{\Delta }\in 𝒥,$ $|\mathrm{\Delta }|\ge 2\text{].}$ Then ${a}_{\text{min}}^{t}={a}_{\text{max}},$ ${a}_{\text{max}}^{t}={a}_{\text{min}}$ (see [Zel1980, 4.2 and 9.7]).

Corollary. ${m}_{{a}_{\text{min}},a}^{\left(\rho \right)}=1$ for $a\in M{\left(𝒥\right)}_{\phi }\text{.}$

Proof. It is sufficient to verify that ${m}_{{a}_{\text{min}},{a}_{\text{max}}}^{\left(\rho \right)}=1\text{.}$ Let us apply the automorphism $\omega \to {\omega }^{t}$ to the equation ${\pi }_{{a}_{\text{max}}}^{\left(\rho \right)}=\sum _{a}{m}_{a,{a}_{\text{max}}}^{\left(\rho \right)}·{⟨a⟩}^{\left(\rho \right)}\text{.}$ Since ${\left({\pi }_{{a}_{\text{max}}}^{\left(\rho \right)}\right)}^{t}={\pi }_{{a}_{\text{max}}}^{\left(\rho \right)},$ we see that ${m}_{a,{a}_{\text{max}}}^{\left(\rho \right)}={m}_{{a}^{t},{a}_{\text{max}}}^{\left(\rho \right)}$ for all $a\in M{\left(𝒮\right)}_{\phi }\text{.}$ In particular, ${m}_{{a}_{\text{min}},{a}_{\text{max}}}^{\left(\rho \right)}={m}_{{a}_{\text{max}},{a}_{\text{max}}}^{\left(\rho \right)}=1,$ which was required to be proved.

4.3. Involution on the Cells ${X}_{a}\text{.}$ We will use the following theorem of V. S. Pyasetskii.

Theorem (See [Pya1975]). Let $E$ be a finite-dimensional vector space over $ℂ,$ $G$ be a connected complex Lie group, $\sigma :G\to GL\left(E\right)$ be an analytic representation, and ${\sigma }^{*}:G\to GL\left({E}^{*}\right)$ be the adjoint representation. Suppose that the action $\sigma$ of the group $G$ on $E$ has a finite number of orbits. Then the action ${\sigma }^{*}$ of the group ${G}^{*}$ on ${E}^{*}$ also has a finite number of orbits and there exists a natural bijection between the $G\text{-orbits}$ on $E$ and on ${E}^{*}\text{.}$ More precisely, suppose that $𝔤$ is the Lie algebra of $G,$ the algebraic set $M\left(\sigma \right)\subset E×{E}^{*}$ is defined by the equality $M(σ)= { (T,T′)∈E×E* | ⟨dσ(X)·T,T′⟩ =0 for all X∈𝔤 }$ and ${\pi }_{1}:M\left(\sigma \right)\to E$ and ${\pi }_{2}:M\left(\sigma \right)\to {E}^{*}$ are projections. Then for each $G\text{-orbit}$ $X$ on $E$ there exists exactly one $G\text{-orbit}$ ${X}^{*}$ on ${E}^{*}$ such that $\stackrel{‾}{{\pi }_{1}^{-1}\left(X\right)}=\stackrel{‾}{{\pi }_{2}^{-1}\left({X}^{*}\right)}$ (the bar denotes the Zariski closure); the correspondence $X\to {X}^{*}$ is a bijection between the $G\text{-orbits}$ on $E$ and on ${E}^{*}\text{.}$

Let us apply this theorem in the following situation: $E=E\left(V\right)={E}_{\phi },$ $G=\text{Aut} V,$ and $\sigma \left(g\right)T=g\circ T\circ {g}^{-1}\text{.}$ Let $E\prime \left(V\right)$ be the space of the operators $T\prime :V\to V$ of degree $\text{(}-1\text{)}\text{.}$ We define a pairing between $E\left(V\right)$ and $E\prime \left(V\right)$ by the equality $⟨T,T\prime ⟩=\text{tr}\left(T\circ T\prime \right)\text{.}$ It is easily seen that it is nondegenerate, i.e., it identifies ${E}^{*}$ with $E\prime \left(V\right)$ and, in addition, the action ${\sigma }^{*}$ of the group $G$ on $E\prime \left(V\right)$ is given by the equality ${\sigma }^{*}\left(g\right)T\prime =g\circ T\prime \circ {g}^{-1}\text{.}$ As in Sec. 1.8, it is proved that the $G\text{-orbits}$ in $E\prime \left(V\right)$ can be enumerated by the elements of $M{\left(𝒮\right)}_{\phi }\text{:}$ With an element $a$ is associated the orbit ${X}_{a}^{\prime }$ that consists of the operators having exactly $a\left(\mathrm{\Delta }\right)$ Jordan cells of the form ${v}_{e\left(\mathrm{\Delta }\right)}\to {v}_{e\left(\mathrm{\Delta }\right)-1}\cdots \to {v}_{b\left(\mathrm{\Delta }\right)}\to 0$ $\text{(}{v}_{i}\in {V}_{i}\text{)}$ for each $\mathrm{\Delta }\in 𝒮\text{.}$ We can verify this in another way by passage to the adjoint operators: We have ${V}^{*}=\underset{n\in ℤ}{⨁}{V}_{n}^{*},$ and the mapping $T\prime \to {\left(T\prime \right)}^{*}$ obviously defines an isomorphism $E\prime \left(V\right)\to E\left({V}^{*}\right)={E}_{\phi }$ that transforms each ${X}_{a}^{\prime }$ into ${X}_{a}\text{.}$

Thus, Pyasetskii's construction leads to the bijection $a\to {a}^{*}$ of the set $M{\left(𝒮\right)}_{\phi }$ onto itself, defined by the equality ${\left({X}_{a}\right)}^{*}={X}_{{a}^{*}}^{\prime }\text{.}$ It is easily seen that it is an involution, i.e., ${\left({a}^{*}\right)}^{*}=a$ for all $a\in M\left(𝒮\right)\text{;}$ This follows from the fact that the definition of $M\left(\sigma \right)$ is symmetric with respect to $E$ and ${E}^{*}\text{:}$ $M\left({\sigma }^{*}\right)=\left\{\left(T\prime ,T\right) | \left(T,T\prime \right)\in M\left(\sigma \right)\right\}\text{.}$

4.4. Proposition. In the notation of Sec. 4.3 $M(σ)= { (T,T′)∈E(V) ×E′(V) | T and T′ commute with one another } .$

Proof. Obviously, $𝔤=\text{End} V=\underset{n\in ℤ}{⨁}\text{End} {V}_{n}$ and $d\sigma \left(X\right)T=X\circ T-T\circ X$ for $X\in 𝔤$ and $T\in E\left(V\right)\text{.}$ Hence $⟨d\sigma \left(X\right)T,T\prime ⟩=\text{tr}\left(\left(X\circ T-T\circ X\right)\circ T\prime \right)=\text{tr}\left(X\circ \left(T\circ T\prime -T\prime \circ T\right)\right)\text{.}$ Since the form $⟨X,Y⟩=\text{tr}\left(X\circ Y\right)$ is nondegenerate on $𝔤,$ the proposition is proved.

The following practically convenient method of computing the involution $a\to {a}^{*}$ follows from this proposition. For $T\in E\left(V\right)$ we set $Z\left(T\right)=\left\{T\prime \in E\prime \left(V\right) | T\prime \text{commutes with} T\right\}\text{.}$ If $T\in {X}_{a},$ then ${a}^{*}$ is characterized by the condition that ${X}_{{a}^{*}}\cap Z\left(T\right)$ is Zariski-open in $Z\left(T\right)\text{.}$

4.5. Hypothesis. ${a}^{t}={a}^{*}$ for all $a\in M\left(𝒮\right)\text{.}$

We leave the verification of this hypothesis in all the cases, where ${a}^{t}$ is known to us, to the reader (see [Zel1980, Sec. 11]).