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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.
In this note the classical character formula of Frobenius [JKe1981] for the symmetric group 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 is generated by elements for which for Henceforth, is a power of a prime (as in [Zel1980, Rog1985]) or is taken in some defective neighborhood of in (as in [Che1986]). Adding pairwise commuting with relations for we obtain an affine Hecke algebra For an arbitrary family we extend the left action of on itself to an action of on putting The obtained is denoted by Next, is the length of the reduced decomposition of relative to On a function the permutations act by the formula
For we will write otherwise, We associate with a sequence of pairs the family of all numbers of the form enumerated by the rule or We denote by the permutation of indices preserving and corresponding to the transformation
Lemma 1 [Rog1985]. The family of functions is uniquely extended to a family by the cocyclic relations for 2) The submodule is an of and contains, for each the leading coefficient of the decomposition, relative to of the function
2. Next, suppose that for all We associate to each permutation and the If for some then Put We will write if and is obtained by dropping some from the reduced decomposition of relative to
Lemma 2. 1) If then there exists a unique permutation for which 2) The condition uniquely determines an embedding of if 3) Conversely, if and then there exists a nonzero between and
A family will be called a square. Each triple can be extended to a square.
Proposition 3. 1) For each square 2) The image of each in does not depend on the choice of a chain and the corresponding sequence of embeddings.
Let be some reduced decomposition We associate to each the element permuting the subfamilies and in for Put We can verify that does not depend on the choice of and
Corollary 4. If one puts then for each The isomorphism of and mapping into is induced by multiplication of on the right by If then for some chain.
Proposition 5 [BGG1975-2]. On the set of pairs there exists a function for which on each square.
Put Let be the homomorphism, defined for inducing the natural embedding and mapping each for into zero. Put We denote by the homomorphism of onto its only irreducible quotient module (cf. [Zel1980]) generalizing the representation of associated with Jung's skew scheme corresponding to [JKe1981, Che1986, Che1986-2]. Let for
Theorem 6. The sequence is exact.
3. Remarks. We will give an example of the function A reduced decomposition of is said to be canonical if: a) for the decomposition remains reduced after eliminating all b) for adjacent in the decomposition we always have when The author's attention was attracted to such decompositions by A. N. Kirillov. Put where is the transposition eliminated from the canonical decomposition of in the passage to is its number in the decomposition), is the number of permutations of adjacent pairs for (the replacements are not counted) used to transform the obtained reduced decomposition for to the canonical one.
2)To prove the theorem, using the results of [Che1986, Che1986-2] on branching of special bases in we impose an induction restriction on Here, splits into a direct sum of some sequences for in which are replaced by For an admissible (satisfying the same inequalities as after eliminating all whose construction is not compatible with the passage the sequence coincides with the sequence of the theorem for instead of and instead of If is not admissible, then it turns out that the sequence is exact. In the proof of this fact one verifies that for each one of the embeddings of the form or is an isomorphism for a suitable
3) All constructions of this note, as well as the theorem, are extended verbatim to the degenerate algebra (cf. [Che1986, Che1986-2, Dri1986-2]). For this algebra, can be identified with and Respectively, one has to put Then is isomorphic as an to the representation induced from the identity representation of the subgroup for integers corresponds to Jung's skew scheme [JKe1981] represented by cells with the set of centers Thus, the theorem, indeed, generalizes the character formula of Frobenius. A replacement of by in all constructions results in a similar "antisymmetric" resolution for the corresponding to the scheme obtained from the scheme by the reflection in the diagonal This is also true for The statement and the proof of Theorem 6 are extended to representations of quantum of the A series corresponding to [Che1986-2, Dri1986-2] (cf. [Che1986-3]).
[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 groups II: On irreducible representations of , Ann. Sci. École Norm. Sup. Ser. (4) 13 (1980) 165–210.
[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).