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

## 1. Basic Definitions and Formulation of the Hypothesis

As usual, $ℤ,$ ${ℤ}^{+},$ and $ℕ$ denote the sets of the integers, the nonnegative integers, and the positive integers, respectively.

1.1. Multisets. For each set $S$ we denote the semigroup that consists of the functions $a:S\to {ℤ}^{+}$ with finite support by $M\left(S\right)\text{.}$ The elements of $M\left(S\right)$ are called finite multisets on $S$ (we will omit the word "finite" in the sequel). It is convenient to represent them as a collection of the elements of $S$ in which repetitions are allowed; the number $a\left(s\right)$ is the multiplicity of occurrence of the element $s\in S$ in the multiset $a\text{.}$ Let us set $|a|=\sum _{s\in S}a\left(s\right)\text{;}$ in particular, if $a$ is a set, then $|a|$ is the number of its elements. A mapping $\lambda :\left\{1,2,\dots ,|a|\right\}\to S$ such that $|{\lambda }^{-1}\left(s\right)|=a\left(s\right)$ for $s\in S$ will be called an order of the multiset $a\text{.}$ We will sometimes write the order $\lambda$ in the form $\left\{\lambda \left(1\right),\lambda \left(2\right),\dots ,\lambda \left(|a|\right)\right\}\text{.}$

Example. The elements of the set $M\left(N\right)$ are called partitions; if $\lambda \in M\left(N\right)$ and $\sum _{k\in ℕ}k·\lambda \left(k\right)=n,$ then we say that $\lambda$ is a partition of the number $n\text{.}$

If $S$ is a disjoint union $\underset{\alpha \in A}{⨆}{S}_{\alpha },$ then the semigroup $M\left(S\right)$ is the direct sum $\underset{\alpha \in A}{⨁}M\left({S}_{\alpha }\right)\text{.}$

1.2. Segments in $ℤ\text{.}$ A nonempty finite subset $\mathrm{\Delta }=\left\{i,i+1,\dots ,j\right\}$ of $ℤ$ that consists of several running subsequences of integers is called a segment in $ℤ\text{.}$ We write $\mathrm{\Delta }=\left[i,j\right]\text{;}$ the number $i$ is called the beginning of $\mathrm{\Delta }$ and is denoted by $b\left(\mathrm{\Delta }\right),$ and $j$ is called the end of $\mathrm{\Delta }$ and is denoted by $e\left(\mathrm{\Delta }\right)\text{.}$ Let us denote the set of all segments in $ℤ$ by $𝒥\text{.}$

Let $\mathrm{\Delta },\mathrm{\Delta }\prime \in 𝒥\text{.}$ We say that $\mathrm{\Delta }$ precedes $\mathrm{\Delta }\prime$ (in symbols, $\mathrm{\Delta }\to \mathrm{\Delta }\prime \text{)}$ if $b\left(\mathrm{\Delta }\right)+1\le b\left(\mathrm{\Delta }\prime \right)\le e\left(\mathrm{\Delta }\right)+1\le e\left(\mathrm{\Delta }\prime \right)\text{.}$ We say that $\mathrm{\Delta }$ and $\mathrm{\Delta }\prime$ are connected if either $\mathrm{\Delta }\to \mathrm{\Delta }\prime$ or $\mathrm{\Delta }\prime \to \mathrm{\Delta }\text{.}$

1.3. Graded Partitions. The elements of the set $M\left(𝒥\right)$ will be called graded partitions. Thus, a graded partition $a$ is a collection of those segments in $ℤ$ in which each segment $\mathrm{\Delta }$ occurs with multiplicity $a\left(\mathrm{\Delta }\right)\text{.}$

Let us define a mapping $M\left(𝒥\right)\to M\left(ℤ\right)$ by setting $\left(a\right)\left(i\right)=\sum _{\mathrm{\Delta }\ni i}a\left(\mathrm{\Delta }\right)\text{.}$ For each $\phi \in M\left(ℤ\right)$ let us set $M{\left(𝒥\right)}_{\phi }=\left\{a\in M\left(𝒥\right) | s\left(a\right)=\phi \right\}\text{;}$ we call the elements of $M{\left(𝒥\right)}_{\phi }$ graded partitions of the multiset $\phi \text{.}$ Obviously, $M\left(𝒥\right)=\underset{q\in M\left(ℤ\right)}{⨆}M{\left(𝒥\right)}_{\phi },$ and all the sets $M{\left(𝒥\right)}_{\phi }$ are finite.

1.4. Representations of the Groups ${G}_{n}$ (See [Zel1980,Zel1977,Zel1977-2]). Let $F$ be a non-Archimedean local field. Let us set ${G}_{n}=GL\left(n,F\right)$ $\text{(}n\ge 0\text{;}$ we assume that ${G}_{0}=\left\{e\right\}\text{).}$ Let us denote the category of the algebraic ${G}_{n}\text{-modules}$ of finite length by ${𝒜}_{n},$ its Grothendieck group by ${ℛ}_{n},$ the set of equivalence classes of irreducible ${G}_{n}\text{-modules}$ by ${\mathrm{\Omega }}_{n},$ and the set of equivalence classes of the cuspidal representations by ${𝒞}_{n}\subset {\mathrm{\Omega }}_{n}\text{.}$ We set $ℛ=\underset{n\ge 0}{⨁}{ℛ}_{n},$ $\mathrm{\Omega }=\underset{n\ge 0}{⨆}{\mathrm{\Omega }}_{n},$ and $𝒞=\underset{n\ge 0}{⨆}{𝒞}_{n}\text{;}$ thus, $ℛ$ is a graded free Abelian group with a basis $\mathrm{\Omega }\text{.}$ For each $\pi \in {𝒜}_{n}$ we denote its image in $ℛ$ by the same letter $\pi$ and the Jordan-Hölder series of $\pi$ by $J{H}_{\pi }\in M\left(\mathrm{\Omega }\right)$ [in other words, we have $\pi =\sum _{\omega \in \mathrm{\Omega }}J{H}_{\pi }\left(\omega \right)·\omega$ in $ℛ\text{].}$

For all $k,l,n\in {ℤ}^{+}$ with $k+l=n$ we define the induction functor ${𝒜}_{k}×{𝒜}_{l}\to {𝒜}_{n}$ $\text{(}\left(\pi ,\tau \right)↦\pi ·\tau \text{);}$ see [Zel1980, Sec. 1], With the help of these functors, $ℛ$ becomes a graded ring. It is commutative and associative (see [Zel1980, Sec. 1]).

1.5. Classification of Irreducible Representations. For each $\phi \in M\left(𝒞\right)$ we set $πφ=∏ρ∈𝒞 ρφ(ρ)∈ℛ.$ Let us denote the set of all irreducible components of ${\pi }_{\phi }$ by ${\mathrm{\Omega }}_{\phi }\subset \mathrm{\Omega }$ [i.e., ${\mathrm{\Omega }}_{\phi }=\left\{\omega \in \mathrm{\Omega } | J{H}_{{\pi }_{\phi }}\left(\omega \right)>0\right\}\text{]},$ and let us set ${ℛ}_{\phi }=\underset{\omega \in {\mathrm{\Omega }}_{\phi }}{⨁}ℤ·\omega \subset ℛ\text{.}$

1.5.1. Proposition (see [Zel1980, 1.10]). The ring $ℛ$ is graded with the help of the semigroup $M\left(𝒞\right),$ i.e., $ℛ=\underset{\phi \in M\left(𝒞\right)}{⨁}{ℛ}_{\phi }$ (this means that $\mathrm{\Omega }=\underset{\phi \in M\left(𝒞\right)}{⨆}{\mathrm{\Omega }}_{\phi }\text{)}$ and ${ℛ}_{\phi }·{ℛ}_{\phi \prime }\subset {ℛ}_{\phi +\phi \prime }\text{.}$

For all $k\in ℝ$ and $n\in {ℤ}^{+}$ we define the character ${\nu }^{k}$ of the group ${G}_{n}$ by the equality ${\nu }^{k}\left(g\right)={|\text{det} g|}^{k},$ where $| |$ is the standard norm in the field $F\text{.}$ The group $ℤ$ acts freely on $𝒞$ by the formula $\left(k,\rho \right)↦{\nu }^{k}\otimes \rho \text{.}$ The orbits of this action are called lines in $𝒞\text{;}$ let us denote the set of all lines in $𝒞$ by $ℒ,$ so that $𝒞=\underset{\mathrm{\Pi }\in ℒ}{⨆}\mathrm{\Pi }\text{.}$ We have $M\left(𝒞\right)=\underset{\mathrm{\Pi }\in ℒ}{⨁}M\left(\mathrm{\Pi }\right)$ (see Sec. 1.1). For each $\mathrm{\Pi }\in ℒ$ we set ${\mathrm{\Omega }}_{\mathrm{\Pi }}=\underset{\phi \in M\left(\mathrm{\Pi }\right)}{⨆}{\mathrm{\Omega }}_{\phi }$ and ${ℛ}_{\mathrm{\Pi }}=\underset{\phi \in M\left(\mathrm{\Pi }\right)}{⨁}{ℛ}_{\phi }\text{.}$ By Proposition 1.5.1, ${ℛ}_{\mathrm{\Pi }}$ is a subring of $ℛ\text{.}$

1.5.2. Proposition (see [Zel1980, 8.6]). The ring $ℛ$ is the tensor product of its subrings ${ℛ}_{\mathrm{\Pi }}$ $\text{(}\mathrm{\Pi }\in ℒ\text{).}$ More precisely, let $\phi \in M\left(𝒞\right)$ be represented in the form $\phi ={\phi }_{1}+{\phi }_{2}+\dots +{\phi }_{p},$ where ${\phi }_{i}\in M\left({\mathrm{\Pi }}_{i}\right),$ and suppose that ${\mathrm{\Pi }}_{i},\dots ,{\mathrm{\Pi }}_{p}$ are different lines in $𝒞\text{.}$ Then the multiplication of representations defines a bijection of ${\mathrm{\Omega }}_{{\phi }_{1}}×{\mathrm{\Omega }}_{{\phi }_{2}}×\dots ×{\mathrm{\Omega }}_{{\phi }_{p}}$ with ${\mathrm{\Omega }}_{\phi }\text{.}$

This proposition reduces the classification of irreducible representations of the groups ${G}_{n},$ i.e., the description of the set $\mathrm{\Omega },$ to the description of each ${\mathrm{\Omega }}_{\mathrm{\Pi }}\text{.}$ We fix a line $\mathrm{\Pi }$ in $𝒞\text{.}$ The choice of the point $\rho \in \mathrm{\Pi }$ enables us to identify $\mathrm{\Pi }$ with $ℤ$ $\text{(}k↦{\rho }_{k}={\nu }^{k}\otimes \rho \text{).}$ By the same token, $M\left(\mathrm{\Pi }\right)$ is identified with $M\left(ℤ\right)\text{;}$ the notions from Secs. 1.2 and 1.3 are transferred from $ℤ$ to $\mathrm{\Pi }$ by means of transfer of the structure. Obviously, the property of segments in $\mathrm{\Pi }$ of being connected or to precede one another does not depend on the choice of the point $\rho \text{.}$

Now we fix $\rho \in \mathrm{\Pi }\text{.}$ The classification theorem (see [Zel1980], 6.1) asserts that there exists a bijection $a↦⟨a⟩={⟨a⟩}^{\left(\rho \right)}$ between $M\left(𝒥\right)$ and ${\mathrm{\Omega }}_{\mathrm{\Pi }}$ that transforms the set $M{\left(𝒥\right)}_{\phi }$ into ${\mathrm{\Omega }}_{\phi }$ for each $\phi \in M\left(ℤ\right)=M\left(\mathrm{\Pi }\right)\text{.}$

1.6. Construction of the Bijection $a↦⟨a⟩\text{.}$ Step 1. Let $\mathrm{\Delta }=\left[i,j\right]\in 𝒥\text{.}$ By virtue of [Zel1980, Sec. 2.10]), the module ${\rho }_{i}·{\rho }_{i+1}·\dots ·{\rho }_{j}$ has a unique irreducible submodule; let us denote it by $⟨\mathrm{\Delta }⟩={⟨\mathrm{\Delta }⟩}^{\left(\rho \right)}\text{.}$

Example (see [Zel1980, 3.2]). If $\rho$ is the unique representation of the group ${G}_{i},$ then ${⟨\mathrm{\Delta }⟩}^{\left(\rho \right)}={\nu }^{\left(i+j\right)/2}\in {\mathrm{\Omega }}_{j-i+1}\text{.}$

Step 2. Let $a\in M\left(𝒥\right)\text{.}$ We choose an order $\left\{{\mathrm{\Delta }}_{1},\dots ,{\mathrm{\Delta }}_{r}\right\}$ of the multiset $a$ such that for $i the segment ${\mathrm{\Delta }}_{i}$ does not precede ${\mathrm{\Delta }}_{j}$ (see Secs. 1.1 and 1.2). By virtue of [Zel1980, Sec. 6.1], the representation $⟨{\mathrm{\Delta }}_{1}⟩·⟨{\mathrm{\Delta }}_{2}⟩·\dots ·⟨{\mathrm{\Delta }}_{r}⟩$ has a unique irreducible submodule, and it does not depend on the arbitrariness in the choice of the order of $a\text{.}$ This is the module $⟨a⟩={⟨a⟩}^{\left(\rho \right)}\text{.}$

1.7. Multiplicity ${m}_{b,a}\text{.}$ For all $\rho \in 𝒞$ and $a\in M\left(𝒮\right)$ we set $πa(ρ)= ∏ Δ∈𝒮 (⟨ Δ⟩(ρ))a( Δ) ∈ℛ.$ Obviously, each irreducible component of ${\pi }_{a}^{\left(\rho \right)}$ has the form ${⟨b⟩}^{\left(\rho \right)},$ where $b\in M\left(𝒥\right)$ and $s\left(b\right)=s\left(a\right)$ (see Sec. 1.3). Let us set ${m}_{b,a}^{\left(\rho \right)}=J{H}_{{\pi }_{a}^{\left(\rho \right)}}\left({⟨b⟩}^{\left(\rho \right)}\right),$ i.e., ${m}_{b,a}^{\left(\rho \right)}$ is the multiplicity of occurrence of ${⟨b⟩}^{\left(\rho \right)}$ in the Jordan-Hölder series of ${\pi }_{a}^{\left(\rho \right)}\text{.}$

Hypothesis. The multiplicity ${m}_{b,a}^{\left(\rho \right)}$ does not depend on $\rho ,$ i.e., it is determined by the graded partitions $a$ and $b\text{.}$

Thus, if we accept this hypothesis, then the computation of the multiplicities reduces to the computation of the matrix $\left({m}_{b,a}\right),a,b\in M{\left(𝒥\right)}_{\phi }$ for each $\phi \in M\left(ℤ\right)\text{.}$

Thus, we have explained the left-hand side of our hypothetical formula (see Introduction). We pass to the description of the right-hand side.

1.8. Varieties ${X}_{a}\text{.}$ Let us consider a finite-dimensional graded vector space $V=\underset{n\in ℤ}{⨁}{V}_{n}$ over $ℂ$ (many of the following results are valid over an arbitrary field, but we will not dwell on this). We say that $V$ has type $\phi \in M\left(ℤ\right)$ if $\text{dim} {V}_{i}=\phi \left(i\right)$ for each $i\in ℤ\text{.}$

Let $V$ have type $\phi \text{.}$ Let $E\left(V\right)$ or ${E}_{\phi }$ denote the set of the operators $T:V\to V$ of degree $+1,$ i.e., of operators $T$ such that $T\left({V}_{i}\right)\subset {V}_{i+1}$ for $i\in ℤ\text{;}$ this is an affine space of dimension $\sum _{i}\phi \left(i\right)·\phi \left(i+1\right)\text{.}$ Obviously, the operators from $E\left(V\right)$ are nilpotent. The group $\text{Aut} V=\prod _{i\in ℤ}GL\left({V}_{i}\right)$ acts naturally on $E\left(V\right)\text{.}$

Proposition-Definition. Each operator $T\in E\left(V\right)$ has a Jordan basis that consists of homogeneous elements of $V\text{.}$ The orbits of the group $\text{Aut} V$ on $E\left(V\right)$ are parametrized by the set $M{\left(𝒥\right)}_{\phi }\text{;}$ to each graded partition $a$ there corresponds the orbit ${X}_{a}$ that consists of the operators having exactly $a\left(\mathrm{\Delta }\right)$ Jordan cells of the form ${v}_{b\left(\mathrm{\Delta }\right)}\to {v}_{b\left(\mathrm{\Delta }\right)+1}\to \dots \to {v}_{e\left(\mathrm{\Delta }\right)}\to 0,$ where ${v}_{i}\in {V}_{i},$ for each $\mathrm{\Delta }\in 𝒮\text{.}$

Proof. The first statement follows from any proof of the theorem on the Jordan normal form of an operator. The fact that the number of the Jordan cells over each $\mathrm{\Delta }$ is uniquely determined by the type of the operator follows, e.g., from the Krull-Remak-Schmidt theorem; an elementary proof will be given below in Sec.2.

1.9. Formulation of the Hypothesis. For each $b\in M{\left(𝒥\right)}_{\phi }$ we denote the closure of ${X}_{b}$ in ${E}_{\phi }$ (say, in the complex topology) by ${\stackrel{‾}{X}}_{b}$ It is clear that ${X}_{b}$ is the union of certain ${X}_{a},$ $a\in M{\left(𝒥\right)}_{\phi }\text{.}$

Hypothesis. The sheaves ${ℋ}^{i}\left({\stackrel{‾}{X}}_{b}\right)$ (see Introduction) are equal to $0$ for odd $i\text{.}$ The coefficient ${m}_{b,a}^{\left(\rho \right)}$ is nonzero for all $\rho \in 𝒞$ if and only if ${X}_{a}\subset {\stackrel{‾}{X}}_{b}$ (this will be proved in Sec. 2), and $mb,a(ρ)= ∑i≥0dim ℋ2i (X‾b)xa,$ where ${x}_{a}$ is an arbitrary point of ${X}_{a}\text{.}$