$p\text{-adic}$ 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.

1. Basic Definitions and Formulation of the Hypothesis

As usual, $\mathbb{Z},$ ${\mathbb{Z}}^{+},$ and $\mathbb{N}$ 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 $a:S\to {\mathbb{Z}}^{+}$ 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 $\left|a\right|=\sum _{s\in S}a\left(s\right)\text{;}$ in particular, if $a$ is a set, then $\left|a\right|$ is the number of its elements. A mapping $\lambda :\{1,2,\dots ,\left|a\right|\}\to S$ such that $\left|{\lambda }^{-1}\left(s\right)\right|=a\left(s\right)$ for $s\in S$ will be called an order of the multiset $a\text{.}$ We will sometimes write the order $\lambda$ in the form $\{\lambda \left(1\right),\lambda \left(2\right),\dots ,\lambda \left(\left|a\right|\right)\}\text{.}$

Example. The elements of the set $M\left(N\right)$ are called partitions; if $\lambda \in M\left(N\right)$ and $\sum _{k\in \mathbb{N}}k·\lambda \left(k\right)=n,$ 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 $\mathbb{Z}\text{.}$ A nonempty finite subset $\mathrm{\Delta }=\{i,i+1,\dots ,j\}$ of $\mathbb{Z}$ that consists of several running subsequences of integers is called a segment in $\mathbb{Z}\text{.}$ We write $\mathrm{\Delta }=[i,j]\text{;}$ 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 $\mathbb{Z}$ 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 $\mathbb{Z}$ in which each segment $\mathrm{\Delta }$ occurs with multiplicity $a\left(\mathrm{\Delta }\right)\text{.}$

Let us define a mapping $M\left(𝒥\right)\to M\left(\mathbb{Z}\right)$ by setting $\left(a\right)\left(i\right)=\sum _{\mathrm{\Delta }\ni i}a\left(\mathrm{\Delta }\right)\text{.}$ For each $\phi \in M\left(\mathbb{Z}\right)$ let us set $M{\left(𝒥\right)}_{\phi }=\{a\in M\left(𝒥\right)\hspace{0.17em}|\hspace{0.17em}s\left(a\right)=\phi \}\text{;}$ we call the elements of $M{\left(𝒥\right)}_{\phi }$ graded partitions of the multiset $\phi \text{.}$ Obviously, $M\left(𝒥\right)=\underset{q\in M\left(\mathbb{Z}\right)}{⨆}M{\left(𝒥\right)}_{\phi },$ 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 $G_n=GL(n,F)$ $\text{(}n\ge 0\text{;}$ 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 $k,l,n\in {\mathbb{Z}}^{+}$ with $k+l=n$ we define the induction functor ${𝒜}_k\times {𝒜}_l\to {𝒜}_n$ $\text{(}(\pi ,\tau )\mapsto \pi ·\tau \text{);}$ 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 $${\pi }_{\phi }=\prod _{\rho \in 𝒞}{\rho }^{\phi \left(\rho \right)}\in ℛ\text{.}$$ Let us denote the set of all irreducible components of ${\pi }_{\phi }$ by ${\mathrm{\Omega }}_{\phi }\subset \mathrm{\Omega }$ [i.e., ${\mathrm{\Omega }}_{\phi }=\{\omega \in \mathrm{\Omega }\hspace{0.17em}|\hspace{0.17em}J{H}_{{\pi }_{\phi }}\left(\omega \right)>0\}\text{]},$ and let us set ${ℛ}_{\phi }=\underset{\omega \in {\mathrm{\Omega }}_{\phi }}{⨁}\mathbb{Z}·\omega \subset ℛ\text{.}$

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 $n\in {\mathbb{Z}}^{+}$ we define the character ${\nu }^k$ of the group $G_n$ by the equality ${\nu }^k\left(g\right)={\left|\text{det}\hspace{0.17em}g\right|}^k,$ where $\left|\hspace{0.17em}\right|$ is the standard norm in the field $F\text{.}$ The group $\mathbb{Z}$ acts freely on $𝒞$ by the formula $(k,\rho )\mapsto {\nu }^k\otimes \rho \text{.}$ 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 ${\mathrm{\Omega }}_{{\phi }_1}\times {\mathrm{\Omega }}_{{\phi }_2}\times \dots \times {\mathrm{\Omega }}_{{\phi }_p}$ 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 $\mathbb{Z}$ $\text{(}k\mapsto {\rho }_k={\nu }^k\otimes \rho \text{).}$ By the same token, $M\left(\mathrm{\Pi }\right)$ is identified with $M\left(\mathbb{Z}\right)\text{;}$ the notions from Secs. 1.2 and 1.3 are transferred from $\mathbb{Z}$ 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 $a\mapsto ⟨a⟩={⟨a⟩}^{\left(\rho \right)}$ 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 $\phi \in M\left(\mathbb{Z}\right)=M\left(\mathrm{\Pi }\right)\text{.}$

1.6. Construction of the Bijection $a\mapsto ⟨a⟩\text{.}$ Step 1. Let $\mathrm{\Delta }=[i,j]\in 𝒥\text{.}$ 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 ${⟨\mathrm{\Delta }⟩}^{\left(\rho \right)}={\nu }^{(i+j)/2}\in {\mathrm{\Omega }}_{j-i+1}\text{.}$

Step 2. Let $a\in M\left(𝒥\right)\text{.}$ We choose an order $\{{\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_r\}$ of the multiset $a$ such that for $i<j$ 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 $${\pi }_a^{\left(\rho \right)}=\prod _{\mathrm{\Delta }\in 𝒮}{\left({⟨\mathrm{\Delta }⟩}^{\left(\rho \right)}\right)}^{a\left(\mathrm{\Delta }\right)}\in ℛ\text{.}$$ 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 $\phi \in M\left(\mathbb{Z}\right)\text{.}$

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 $V=\underset{n\in \mathbb{Z}}{⨁}{V}_n$ 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 $\phi \in M\left(\mathbb{Z}\right)$ if $\text{dim}\hspace{0.17em}{V}_i=\phi \left(i\right)$ for each $i\in \mathbb{Z}\text{.}$

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 $i\in \mathbb{Z}\text{;}$ this is an affine space of dimension $\sum _i\phi \left(i\right)·\phi (i+1)\text{.}$ Obviously, the operators from $E\left(V\right)$ are nilpotent. The group $\text{Aut}\hspace{0.17em}V=\prod _{i\in \mathbb{Z}}GL\left(V_i\right)$ 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 $\text{Aut}\hspace{0.17em}V$ 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 ${\bar{X}}_b$ It is clear that $X_b$ is the union of certain $X_a,$ $a\in M{\left(𝒥\right)}_{\phi }\text{.}$

Hypothesis. The sheaves ${\mathscr{H}}^i\left({\bar{X}}_b\right)$ (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 $X_a\subset {\bar{X}}_b$ (this will be proved in Sec. 2), and $$m_{b,a}^{\left(\rho \right)}=\sum _{i\ge 0}\text{dim}\hspace{0.17em}{\mathscr{H}}^{2i}{\left({\bar{X}}_b\right)}_{x_a},$$ where $x_a$ is an arbitrary point of $X_a\text{.}$

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