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.

4. Duality

4.1. Involution ωωt. Let ρ𝒞 and Π be the line in 𝒞 that contains ρ (see Sees. 1.4 and 1.5). It follows from Theorem 7.1 of [Zel1980] (see also Sec. 2.1) that Π is the ring of polynomials over in the variables Δ(ρ), Δ𝒥. Suppose that ωωt is an endomorphism of the ring Π that transforms [i,j](ρ) into the representation {i},{i+1},,{j}(ρ), corresponding to the collection of one-point segments (obviously, this endomorphism does not depend on the choice of ρ). It is proved in [Zel1980, Sec, 9] that it is an involutive automorphism of Π, 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 ρ𝒞 the equality (a(ρ))t=ai(ρ) defines an involution aat on the set of graded partitions; obviously, it transforms each of the sets M(𝒥)φ into itself.

Hypothesis. This involution does not depend on ρ.

This hypothesis follows from Hypothesis 1.7 and the results of [Zel1980, Sec. 9] (and, namely, from the fact that the expansion of (πa(ρ))t with respect to the basis {πb(ρ)} does not depend on ρ; see [Zel1980], 9.14).

4.2. Example. We fix a φM(). Let amin and amax be the minimum and the maximum elements of M(𝒥)φ [amin is defined in Sec. 2.8, b) or in [Zel1980], Sec. 9.10; amax consists of one-point segments, i.e., amax( Δ)=0 for Δ𝒥, | Δ|2]. Then amint=amax, amaxt=amin (see [Zel1980, 4.2 and 9.7]).

Corollary. mamin,a(ρ)=1 for aM(𝒥)φ.

Proof. It is sufficient to verify that mamin,amax(ρ)=1. Let us apply the automorphism ωωt to the equation πamax(ρ)=ama,amax(ρ)·a(ρ). Since (πamax(ρ))t=πamax(ρ), we see that ma,amax(ρ)=mat,amax(ρ) for all aM(𝒮)φ. In particular, mamin,amax(ρ)=mamax,amax(ρ)=1, which was required to be proved.

4.3. Involution on the Cells Xa. We will use the following theorem of V. S. Pyasetskii.

Theorem (See [Pya1975]). Let E be a finite-dimensional vector space over , G be a connected complex Lie group, σ:GGL(E) be an analytic representation, and σ*:GGL(E*) be the adjoint representation. Suppose that the action σ of the group G on E has a finite number of orbits. Then the action σ* of the group G* on E* also has a finite number of orbits and there exists a natural bijection between the G-orbits on E and on E*. More precisely, suppose that 𝔤 is the Lie algebra of G, the algebraic set M(σ)E×E* is defined by the equality M(σ)= { (T,T)E×E* |dσ(X)·T,T =0for allX𝔤 } and π1:M(σ)E and π2:M(σ)E* are projections. Then for each G-orbit X on E there exists exactly one G-orbit X* on E* such that π1-1(X)=π2-1(X*) (the bar denotes the Zariski closure); the correspondence XX* is a bijection between the G-orbits on E and on E*.

Let us apply this theorem in the following situation: E=E(V)=Eφ, G=AutV, and σ(g)T=gTg-1. Let E(V) be the space of the operators T:VV of degree (-1). We define a pairing between E(V) and E(V) by the equality T,T=tr(TT). It is easily seen that it is nondegenerate, i.e., it identifies E* with E(V) and, in addition, the action σ* of the group G on E(V) is given by the equality σ*(g)T=gTg-1. As in Sec. 1.8, it is proved that the G-orbits in E(V) can be enumerated by the elements of M(𝒮)φ: With an element a is associated the orbit Xa that consists of the operators having exactly a( Δ) Jordan cells of the form ve( Δ)ve( Δ)-1vb( Δ)0 (viVi) for each Δ𝒮. We can verify this in another way by passage to the adjoint operators: We have V*=nVn*, and the mapping T(T)* obviously defines an isomorphism E(V)E(V*)=Eφ that transforms each Xa into Xa.

Thus, Pyasetskii's construction leads to the bijection aa* of the set M(𝒮)φ onto itself, defined by the equality (Xa)*=Xa*. It is easily seen that it is an involution, i.e., (a*)*=a for all aM(𝒮); This follows from the fact that the definition of M(σ) is symmetric with respect to E and E*: M(σ*)={(T,T)|(T,T)M(σ)}.

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

Proof. Obviously, 𝔤=EndV=nEndVn and dσ(X)T=XT-TX for X𝔤 and TE(V). Hence dσ(X)T,T=tr((XT-TX)T)=tr(X(TT-TT)). Since the form X,Y=tr(XY) is nondegenerate on 𝔤, the proposition is proved.

The following practically convenient method of computing the involution aa* follows from this proposition. For TE(V) we set Z(T)={TE(V)|Tcommutes withT}. If TXa, then a* is characterized by the condition that Xa*Z(T) is Zariski-open in Z(T).

4.5. Hypothesis. at=a* for all aM(𝒮).

We leave the verification of this hypothesis in all the cases, where at is known to us, to the reader (see [Zel1980, Sec. 11]).

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