p-adic analog of the Kazhdan-Lusztig Hypothesis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 15 April 2014

Notes and References

This is an excerpt of the paper p-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, , +, and 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+ with finite support by M(S). The elements of M(S) 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(s) is the multiplicity of occurrence of the element sS in the multiset a. Let us set |a|=sSa(s); in particular, if a is a set, then |a| is the number of its elements. A mapping λ:{1,2,,|a|}S such that |λ-1(s)|=a(s) for sS will be called an order of the multiset a. We will sometimes write the order λ in the form {λ(1),λ(2),,λ(|a|)}.

Example. The elements of the set M(N) are called partitions; if λM(N) and kk·λ(k)=n, then we say that λ is a partition of the number n.

If S is a disjoint union αASα, then the semigroup M(S) is the direct sum αAM(Sα).

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

Let Δ, Δ𝒥. We say that Δ precedes Δ (in symbols, Δ Δ) if b( Δ)+1b( Δ)e( Δ)+1e( Δ). We say that Δ and Δ are connected if either Δ Δ or Δ Δ.

1.3. Graded Partitions. The elements of the set M(𝒥) will be called graded partitions. Thus, a graded partition a is a collection of those segments in in which each segment Δ occurs with multiplicity a( Δ).

Let us define a mapping M(𝒥)M() by setting (a)(i)= Δia( Δ). For each φM() let us set M(𝒥)φ={aM(𝒥)|s(a)=φ}; we call the elements of M(𝒥)φ graded partitions of the multiset φ. Obviously, M(𝒥)=qM()M(𝒥)φ, and all the sets M(𝒥)φ are finite.

1.4. Representations of the Groups Gn (See [Zel1980,Zel1977,Zel1977-2]). Let F be a non-Archimedean local field. Let us set Gn=GL(n,F) (n0; we assume that G0={e}). Let us denote the category of the algebraic Gn-modules of finite length by 𝒜n, its Grothendieck group by n, the set of equivalence classes of irreducible Gn-modules by Ωn, and the set of equivalence classes of the cuspidal representations by 𝒞n Ωn. We set =n0n, Ω=n0 Ωn, and 𝒞=n0𝒞n; thus, is a graded free Abelian group with a basis Ω. For each π𝒜n we denote its image in by the same letter π and the Jordan-Hölder series of π by JHπM( Ω) [in other words, we have π=ω ΩJHπ(ω)·ω in ].

For all k,l,n+ with k+l=n we define the induction functor 𝒜k×𝒜l𝒜n ((π,τ)π·τ); see [Zel1980, Sec. 1], With the help of these functors, becomes a graded ring. It is commutative and associative (see [Zel1980, Sec. 1]).

1.5. Classification of Irreducible Representations. For each φM(𝒞) we set πφ=ρ𝒞 ρφ(ρ). Let us denote the set of all irreducible components of πφ by Ωφ Ω [i.e., Ωφ={ω Ω|JHπφ(ω)>0}], and let us set φ=ω Ωφ·ω.

1.5.1. Proposition (see [Zel1980, 1.10]). The ring is graded with the help of the semigroup M(𝒞), i.e., =φM(𝒞)φ (this means that Ω=φM(𝒞) Ωφ) and φ·φφ+φ.

For all k and n+ we define the character νk of the group Gn by the equality νk(g)=|detg|k, where || is the standard norm in the field F. The group acts freely on 𝒞 by the formula (k,ρ)νkρ. The orbits of this action are called lines in 𝒞; let us denote the set of all lines in 𝒞 by , so that 𝒞= Π Π. We have M(𝒞)= ΠM( Π) (see Sec. 1.1). For each Π we set Ω Π=φM( Π) Ωφ and Π=φM( Π)φ. By Proposition 1.5.1, Π is a subring of .

1.5.2. Proposition (see [Zel1980, 8.6]). The ring is the tensor product of its subrings Π ( Π). More precisely, let φM(𝒞) be represented in the form φ=φ1+φ2++φp, where φiM( Πi), and suppose that Πi,, Πp are different lines in 𝒞. Then the multiplication of representations defines a bijection of Ωφ1× Ωφ2×× Ωφp with Ωφ.

This proposition reduces the classification of irreducible representations of the groups Gn, i.e., the description of the set Ω, to the description of each Ω Π. We fix a line Π in 𝒞. The choice of the point ρ Π enables us to identify Π with (kρk=νkρ). By the same token, M( Π) is identified with M(); the notions from Secs. 1.2 and 1.3 are transferred from to Π by means of transfer of the structure. Obviously, the property of segments in Π of being connected or to precede one another does not depend on the choice of the point ρ.

Now we fix ρ Π. The classification theorem (see [Zel1980], 6.1) asserts that there exists a bijection aa=a(ρ) between M(𝒥) and Ω Π that transforms the set M(𝒥)φ into Ωφ for each φM()=M( Π).

1.6. Construction of the Bijection aa. Step 1. Let Δ=[i,j]𝒥. By virtue of [Zel1980, Sec. 2.10]), the module ρi·ρi+1··ρj has a unique irreducible submodule; let us denote it by Δ= Δ(ρ).

Example (see [Zel1980, 3.2]). If ρ is the unique representation of the group Gi, then Δ(ρ)=ν(i+j)/2 Ωj-i+1.

Step 2. Let aM(𝒥). We choose an order { Δ1,, Δr} of the multiset a such that for i<j the segment Δi does not precede Δj (see Secs. 1.1 and 1.2). By virtue of [Zel1980, Sec. 6.1], the representation Δ1· Δ2·· Δr has a unique irreducible submodule, and it does not depend on the arbitrariness in the choice of the order of a. This is the module a=a(ρ).

1.7. Multiplicity mb,a. For all ρ𝒞 and aM(𝒮) we set πa(ρ)= Δ𝒮 ( Δ(ρ))a( Δ) . Obviously, each irreducible component of πa(ρ) has the form b(ρ), where bM(𝒥) and s(b)=s(a) (see Sec. 1.3). Let us set mb,a(ρ)=JHπa(ρ)(b(ρ)), i.e., mb,a(ρ) is the multiplicity of occurrence of b(ρ) in the Jordan-Hölder series of πa(ρ).

Hypothesis. The multiplicity mb,a(ρ) does not depend on ρ, i.e., it is determined by the graded partitions a and b.

Thus, if we accept this hypothesis, then the computation of the multiplicities reduces to the computation of the matrix (mb,a),a,bM(𝒥)φ for each φM().

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 Xa. Let us consider a finite-dimensional graded vector space V=nVn over (many of the following results are valid over an arbitrary field, but we will not dwell on this). We say that V has type φM() if dimVi=φ(i) for each i.

Let V have type φ. Let E(V) or Eφ denote the set of the operators T:VV of degree +1, i.e., of operators T such that T(Vi)Vi+1 for i; this is an affine space of dimension iφ(i)·φ(i+1). Obviously, the operators from E(V) are nilpotent. The group AutV=iGL(Vi) acts naturally on E(V).

Proposition-Definition. Each operator TE(V) has a Jordan basis that consists of homogeneous elements of V. The orbits of the group AutV on E(V) are parametrized by the set M(𝒥)φ; to each graded partition a there corresponds the orbit Xa that consists of the operators having exactly a( Δ) Jordan cells of the form vb( Δ)vb( Δ)+1ve( Δ)0, where viVi, for each Δ𝒮.

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 Δ 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 bM(𝒥)φ we denote the closure of Xb in Eφ (say, in the complex topology) by Xb It is clear that Xb is the union of certain Xa, aM(𝒥)φ.

Hypothesis. The sheaves i(Xb) (see Introduction) are equal to 0 for odd i. The coefficient mb,a(ρ) is nonzero for all ρ𝒞 if and only if XaXb (this will be proved in Sec. 2), and mb,a(ρ)= i0dim 2i (Xb)xa, where xa is an arbitrary point of Xa.

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