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 affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representati...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 affne 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 affne 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 affne Weyl groups, but that is the theme of another paper.展开更多
For type B_(3), we show that Lusztig's conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp_(6)(C) with three equal Jordan blocks needs modification.
Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig poly...Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig polynomial P_(y,w)∈Z[q].We determine someμ(y,w)for y∈c_(0) and w∈c_(2),where c0 is the lowest two-sided cell of B_(2) and c_(2) is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.展开更多
基金partially supported by Natural Sciences Foundation of China (10671193)
文摘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 affne 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 affne 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 affne Weyl groups, but that is the theme of another paper.
基金supported by National Key R&D Program of China (Grant No. 2020YFA0712600)National Natural Science Foundation of China (Grant No. 11688101)the AMSS for hospitality and for financial supports。
文摘For type B_(3), we show that Lusztig's conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp_(6)(C) with three equal Jordan blocks needs modification.
基金Supported in part by the Natural Science Foundation of China(Grant no.11101233)Beijing Youth Top-notch Talent Support Program(Grant no.21351918007).
文摘Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig polynomial P_(y,w)∈Z[q].We determine someμ(y,w)for y∈c_(0) and w∈c_(2),where c0 is the lowest two-sided cell of B_(2) and c_(2) is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.
