Acta mathematica scientia, Series B >
REPRESENTATIONS OF AFFINE HECKE ALGEBRAS OF TYPE G2
Received date: 2008-08-05
Online published: 2009-05-20
Supported by
Xi was partially supported by Natural Sciences Foundation of China (10671193)
Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affine Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affine Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affine Weyl groups, but that is the theme of another paper.
Key words： representation; affine Hecke algebra; affine Weyl groups; Hecke algebra
XI Na-Hua . REPRESENTATIONS OF AFFINE HECKE ALGEBRAS OF TYPE G2[J]. Acta mathematica scientia, Series B, 2009 , 29(3) : 515 -526 . DOI: 10.1016/S0252-9602(09)60050-3
[1] Grojnowski I. Representations of affine Hecke algebras and affine quantum GLn at roots of unity. Inter Math Res Notices, 1994, 5: 213--216
[2] Kazhdan D, Lusztig G. Representations of Coxeter groups and Hecke algebras. Invent Math, 1979, 53: 165--184
[3] Kazhdan D, Lusztig G. Proof of the Deligne-Langlands conjecture for Hecke algebras. Invent Math, 1987, 87: 153--215
[4] Lusztig G. Singularities, character formulas, and a q-analog of weight multiplicities. Ast\'erisque, 1983, 101/102: 208--227
[5] Lusztig G. Cells in affine Weyl groups//Algebraic Groups and Related Topics. Advanced Studies in Pure Math, Vol 6. Kinokuniya and North Holland, 1985: 255--287
[6] Lusztig G. Cells in affine Weyl groups, II. J Alg, 1987, 109: 536--548
[7] Lusztig G. Leading coefficients of character values of Hecke algebras. Proc Sympos Pure Math Vol 47, Part 2.
Providence R I: Amer Math Soc, 1987: 235--262
[8] Lusztig G. Cells in affine Weyl groups IV. J Fac Sci Univ Tokyo Sect IA Math, 1989, 36: 297--328
[9] Stuler U. Unipotente und nilpotente Klassen in einfach Gruppen und Lie algebren vom Typ G2. Indag Math, 1971, 33: 365--378.
[10] Xi N. The based ring of the lowest two-sided cell of an affine Weyl group. J Alg, 1990, 134: 356--368
[11] Xi N. Representations of Affine Hecke Algebras. Lecture Notes in Mathematics, 587. Berlin: Springer-Verlag, 1994
[12] Xi N. The based ring of the lowest two-sided cell of an affine Weyl group, II. Ann Sci Éc Norm Sup, 1994, 27: 47--61
[13] Xi N. Representations of affine Hecke algebras and based ring of affine Weyl groups. J Amer Math Soc, 2007, 20: 211--217
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