An analogue of the character formula for Hecke algebras

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

Last update: 23 April 2014

Notes and References

This is an html version of the paper An analogue of the character formula for Hecke algebras by I.V. Cherednik.

M.V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 21, No. 2, pp. 94-95, April-June, 1987. Original article submitted March 19, 1986.

An analogue of the character formula for Hecke algebras

In this note the classical character formula of Frobenius [JKe1981] for the symmetric group S is generalized to affine Hekke algebras. In the spirit of [BGG1975-2], resolutions realizing these formulas are immediately constructed. The construction was motivated by [Zel1987-2] and inspired by discussions with A. V. Zelevinskii, to whom the author expresses his deep gratitude. The author is thankful to I. M. Gel'fand for his attention to this work.

1. Suppose a C-algebra Hn is generated by elements T1,,Tn-1 for which [Ti,Tj]=0 for ij±1, TiTi+1Ti=Ti+1TiTi+1, (Ti-q)(Ti+1)=0. Henceforth, q is a power of a prime (as in [Zel1980, Rog1985]) or q is taken in some defective neighborhood of 1 in C (as in [Che1986]). Adding pairwise commuting x1,,xn, with relations [xi,Tj]=0 for ij, j+1, xiTi-Tixi+1=(q-1)xi=Tixi-xi+1Ti, we obtain an affine Hecke algebra n. For an arbitrary family u=(u1,,un) we extend the left action of Hn on itself to an action of n on Hn putting xk(1)=quk. The obtained n-module is denoted by Iu. Next, (w) is the length of the reduced decomposition of wSn relative to si=(i,i+1), wwdefl(w)=l(ww-1)+l(w), (id)=0, (w)n(n-1)/2. On a function f(λ1,,λn) the permutations wSn act by the formula (wf)(λ1,,λn)=f(w-1(λ1,,λn)), λiC

For a,bC we will write abab++, otherwise, a<b. We associate with a sequence of pairs μ=({ii}), 1ir, the family uμ=(uk), ikn=defi=1r(li-li) of all numbers u(i,j) of the form iu(i,j)=i+ji+1, enumerated by the rule uk=u(ik,jk), k<mik<im or jk-jm<0=ik-im. We denote by wμSn the permutation of indices uk preserving ik and corresponding to the transformation i+ji-j+1.

Lemma 1 [Rog1985]. The family of functions φsi=i+(i-qλ2-λ1)(i-q)-1Ti is uniquely extended to a family {φw(λ1,,λn),wSn} by the cocyclic relations φxy=y-1φxφy for xyy, x,ySn. 2) The submodule Iμ=Hnφwμ(uμ) is an n-submodule of Iuμ and contains, for each wwμ the leading coefficient φw(uμ) of the decomposition, relative to λ0, of the function φw(uν), ν=({i+iλ,i+iλ}).

2. Next, suppose that lj<li, lj<li for all j<i. We associate to each permutation σSrμσ=({i,σ(i)}) and the n-module Iσ=Iμσ. If for some i, i-σ(i)<0, then μσ=def, Iσ=def0. Put wσ=wμσ, uσ=uμσ. We will write στ if (σ)=(τ)+1 and τ is obtained by dropping some σi from the reduced decomposition of σ relative to σi=(i,i+1)Sr.

Lemma 2. 1) If στ, then there exists a unique permutation Snwσ,τwτ for which wσ,τ(uτ)=wσ(uσ). 2) The condition φwσ(uσ)φwσ,τ(uτ) uniquely determines an embedding of n-modules ρσ,τ:IσIτ; ρσ,τ(Iσ)Iτ, if Iτ0. 3) Conversely, if (σ)=(τ)+1 and Iτ0, then there exists a nonzero n-homomorphism between Iσ and Iτ, ρστ, ρ=cρσ,τ, c*.

A family σστ, σττ, στ will be called a square. Each triple σττ, στ can be extended to a square.

Proposition 3. 1) For each square ρσ,τρσ,σ(Iσ)=ρτ,τρσ,τ(Iσ). 2) The image Iσ of each Iσ (σSr) in Iμ does not depend on the choice of a chain σσ0 and the corresponding sequence of embeddings.

Let σ=k=1lσik=σilσi1 be some reduced decomposition σp=k=1pσikSr. We associate to each σip the element σipS permuting the subfamilies (i+1,,i+1) and (i+1,,i+1+1) in wτ(uτ) for τ=σp-1, μ=μτ=({i,i}). Put σ=k=1lσik. We can verify that σ does not depend on the choice of σ and σ=defσwμwμ.

Corollary 4. If one puts ωσ=wσ-1σ, then σωσ, Iσ=Hnφσ(uμ) for each σSr. The isomorphism of Iσ and Iσ mapping φwσ(uσ) into φσ(uμ) is induced by multiplication of Hn on the right by φωσ(uμ). If Iσ0, then IσIτστ for some chain.

Proposition 5 [BGG1975-2]. On the set of pairs στ there exists a function ε(σ,τ)=±1 for which ε(σ,τ)ε(σ,σ)=-ε(τ,τ)ε(σ,τ) on each square.

Put Vp=(σ)=pIσ. Let νσ,τ:VpVp-1 be the homomorphism, defined for στ, (σ)=p, inducing the natural embedding IσIτIr and mapping each IσVp for σσ into zero. Put dp=στε(σ,τ)νσ,τ, l(σ)=p. We denote by d0 the homomorphism of Iμ=V0 onto its only irreducible quotient module V-10 (cf. [Zel1980]) generalizing the representation of Sn associated with Jung's skew scheme corresponding to μ [JKe1981, Che1986, Che1986-2]. Let Vp=0 for p>r(r-1)/2, p<-1.

Theorem 6. The sequence {Vp,dp} is exact.

3. Remarks. We will give an example of the function ε. A reduced decomposition of σSr is said to be canonical if: a) for 1i<r, the decomposition remains reduced after eliminating all σ1,,σi; b) for adjacent σiσj in the decomposition we always have i>j when ji±1. The author's attention was attracted to such decompositions by A. N. Kirillov. Put ε(σ,τ)=(-1)k+π, where σik is the transposition eliminated from the canonical decomposition of σ in the passage to τ (k is its number in the decomposition), π is the number of permutations of adjacent pairs σiσjσjσi for ij±1 (the replacements σiσi±1σiσi±1σiσi±1 are not counted) used to transform the obtained reduced decomposition for τ to the canonical one.

2)To prove the theorem, using the results of [Che1986, Che1986-2] on branching of special bases in V-1, we impose an induction restriction on n-1n. Here, {V,d} splits into a direct sum of some sequences {Vi,di} for μi in which i are replaced by i-1. For an admissible μi (satisfying the same inequalities as μ), after eliminating all Iσ whose construction is not compatible with the passage ii-1, the sequence {Vi,di} coincides with the sequence of the theorem for n-1 instead of n and μi instead of μ. If μi is not admissible, then it turns out that the sequence {Vi,di,V-1i=def0} is exact. In the proof of this fact one verifies that for each σ one of the embeddings of the form νσ,τi or ντ,σi is an isomorphism for a suitable τ.

3) All constructions of this note, as well as the theorem, are extended verbatim to the degenerate algebra n (q1) (cf. [Che1986, Che1986-2, Dri1986-2]). For this algebra, Ti can be identified with si and xisi-sixi+1=1=sixi-xi+1si. Respectively, one has to put xk(1)=uk, φsi=1+(λ2-λ1)si. Then Iμ is isomorphic as an Sn-module to the representation induced from the identity representation of the subgroup i=1rSli-liSn; V-1 for integers {i} V-1 corresponds to Jung's skew scheme [JKe1981] represented by cells with the set of centers (i,j)Z2, 0ir-1, r-i+i<jr-i+i. Thus, the theorem, indeed, generalizes the character formula of Frobenius. A replacement of (uk) by (-uk) in all constructions results in a similar "antisymmetric" resolution for the n-module corresponding to the scheme obtained from the scheme μ by the reflection in the diagonal i=j. This is also true for q1. The statement and the proof of Theorem 6 are extended to representations of quantum R-algebras of the A series corresponding to μ [Che1986-2, Dri1986-2] (cf. [Che1986-3]).

Literature cited

[JKe1981] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.

[BGG1975-2] I.N. Bernstein, I.M. Gel'fand, and S.I. Gel'fand, Publ. of 1971 Summer School in Math., Budapest (1975), pp. 21-64.

[Zel1987-2] A.V. Zelevinskii, Funktsion. Anal. Prilozhen., 21, No. 2, 74-75 (1987).

[Zel1980] A. Zelevinsky, Induced representations of 𝔭-adic groups II: On irreducible representations of GL(n), Ann. Sci. École Norm. Sup. Ser. (4) 13 (1980) 165–210.

[Rog1985] J. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985) 443–465. MR 86j:22028

[Che1986] I.V. Cherednik, in: Group-Theoretic Methods in Physics. Proceedings of the 3rd Int. Sem. [in Russian], Nauka, Moscow (1986).

[Che1986-2] I.V. Cherednik, Funktsion. Anal. Prilozhen., 20, No. 1, 87-88 (1986).

[Dri1986-2] V.G. Drinfel'd, Funktsion. Anal. Prilozhen., 20, No. 1, 69-70 (1986).

[Che1986-3] I.V. Cherednik, Dokl. Akad. Nauk SSSR, 291, No. 1, 49-53 (1986).

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