$p\text{-adic}$ 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.

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 $\mathbb{Z}$ 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 ${⟨[i,j]⟩}^{\left(\rho \right)}$ into the representation ${⟨\left\{i\right\},\{i+1\},\dots ,\left\{j\right\}⟩}^{\left(\rho \right)},$ 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 $\phi \in M\left(\mathbb{Z}\right)\text{.}$ 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 $\mathrm{\Delta }\in 𝒥,$ $\left|\mathrm{\Delta }\right|\ge 2\text{].}$ 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 $M\left(\sigma \right)\subset E\times E^{*}$ is defined by the equality $$M\left(\sigma \right)=\{(T,T\prime )\in E\times E^{*}\hspace{0.17em}|\hspace{0.17em}⟨d\sigma \left(X\right)·T,T\prime ⟩=0\hspace{0.17em}\text{for all}\hspace{0.17em}X\in 𝔤\}$$ 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 $\overline{{\pi }_1^{-1}\left(X\right)}=\overline{{\pi }_2^{-1}\left(X^{*}\right)}$ (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: $E=E\left(V\right)=E_{\phi },$ $G=\text{Aut}\hspace{0.17em}V,$ 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 $⟨T,T\prime ⟩=\text{tr}(T\circ T\prime )\text{.}$ 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 $V^{*}=\underset{n\in \mathbb{Z}}{⨁}{V}_n^{*},$ 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 $E^{*}\text{:}$ $M\left({\sigma }^{*}\right)=\left\{(T\prime ,T)\hspace{0.17em}\right|\hspace{0.17em}(T,T\prime )\in M\left(\sigma \right)\}\text{.}$

4.4. Proposition. In the notation of Sec. 4.3 $$M\left(\sigma \right)=\{(T,T\prime )\in E\left(V\right)\times E\prime \left(V\right)\hspace{0.17em}|\hspace{0.17em}T\hspace{0.17em}\text{and}\hspace{0.17em}T\prime \hspace{0.17em}\text{commute with one another}\}\text{.}$$

Proof. Obviously, $𝔤=\text{End}\hspace{0.17em}V=\underset{n\in \mathbb{Z}}{⨁}\text{End}\hspace{0.17em}{V}_n$ 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 $⟨d\sigma \left(X\right)T,T\prime ⟩=\text{tr}((X\circ T-T\circ X)\circ T\prime )=\text{tr}(X\circ (T\circ T\prime -T\prime \circ T))\text{.}$ Since the form $⟨X,Y⟩=\text{tr}(X\circ Y)$ 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 $Z\left(T\right)=\{T\prime \in E\prime \left(V\right)\hspace{0.17em}|\hspace{0.17em}T\prime \hspace{0.17em}\text{commutes with}\hspace{0.17em}T\}\text{.}$ 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]).

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