The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter gr...The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter group is found. The new base is described by using the notion of cell datum of Graham and Lehrer and the notion of norm.展开更多
Let A be a tame Hecke algebra of type A. A new minimal projective bimodule resolution for A is constructed and the dimensions of all the Hochschild homology groups and cyclic homology groups are calculated explicitly.
This paper proves that the restriction of a Specht module for a(degenerate or non-degenerate)cyclotomic Hecke algebra, or KLR(Khovanov-Lauda-Rouquier) algebra, of type A has a Specht filtration.
This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applic...This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary(the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.展开更多
We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalizatio...We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.展开更多
We introduce the doubled Hecke algebra,which is an infinite-dimensional algebra generated by two Hecke algebras.This concept originates from the degenerate doubled Hecke algebra in the theory of Schur-Weyl duality rel...We introduce the doubled Hecke algebra,which is an infinite-dimensional algebra generated by two Hecke algebras.This concept originates from the degenerate doubled Hecke algebra in the theory of Schur-Weyl duality related to enhanced reductive algebraic groups.We study the finite-dimensional natural representation of the doubled Hecke algebra on tensor space and prove that the doubled Hecke algebra forms a duality with the quantum group of Levi type.展开更多
This is a continuation of our previous work. We classify all the simple ?q(D n )-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf n(q) ≠ 0, this yields a classification of all the simple ? q (D...This is a continuation of our previous work. We classify all the simple ?q(D n )-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf n(q) ≠ 0, this yields a classification of all the simple ? q (D n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ(1) = λ(2),wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?q,1(B n )- module Dλ remains irreducible on restriction to ?q(D n ).展开更多
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
Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to giv...Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(г\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.展开更多
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach.For example,matrix coefficients of supercuspidal representations can be used as test ...Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach.For example,matrix coefficients of supercuspidal representations can be used as test functions for this.However,they kill a large class of interesting cuspidal automorphic representations.For the case of number fields,multipliers of the Schwartz algebra are used in the recent work(see Beuzart-Plessis et al.(2021))to isolate all the cuspidal spectrum.In particular,they are suitable for the comparison of orbital integrals.These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups(see Beuzart-Plessis et al.(2021,2022)).In this article,we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al.(2021)for the function field case.展开更多
文摘The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter group is found. The new base is described by using the notion of cell datum of Graham and Lehrer and the notion of norm.
文摘Let A be a tame Hecke algebra of type A. A new minimal projective bimodule resolution for A is constructed and the dimensions of all the Hochschild homology groups and cyclic homology groups are calculated explicitly.
文摘This paper proves that the restriction of a Specht module for a(degenerate or non-degenerate)cyclotomic Hecke algebra, or KLR(Khovanov-Lauda-Rouquier) algebra, of type A has a Specht filtration.
基金supported by a 2017 University of New South Wales Science Goldstar Grant(Jie Du)the Simons Foundation(Grant Nos. #359360(Brian Parshall) and #359363 (Leonard Scott))
文摘This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary(the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.
文摘We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.
基金supported by the National Natural Science Foundation of China(NSFC)Grant 12071136.
文摘We introduce the doubled Hecke algebra,which is an infinite-dimensional algebra generated by two Hecke algebras.This concept originates from the degenerate doubled Hecke algebra in the theory of Schur-Weyl duality related to enhanced reductive algebraic groups.We study the finite-dimensional natural representation of the doubled Hecke algebra on tensor space and prove that the doubled Hecke algebra forms a duality with the quantum group of Levi type.
基金the Tianyuan Math. Foundation of China (Grant No. TY10126011) the China Post-doctoral Science Foundation given to the first author.
文摘This is a continuation of our previous work. We classify all the simple ?q(D n )-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf n(q) ≠ 0, this yields a classification of all the simple ? q (D n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ(1) = λ(2),wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?q,1(B n )- module Dλ remains irreducible on restriction to ?q(D n ).
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.
文摘Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(г\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.
基金supported by National Natural Science Foundation of China(Grant No.11971254)supported by National Natural Science Foundation of China(Grant No.11501382)。
文摘Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach.For example,matrix coefficients of supercuspidal representations can be used as test functions for this.However,they kill a large class of interesting cuspidal automorphic representations.For the case of number fields,multipliers of the Schwartz algebra are used in the recent work(see Beuzart-Plessis et al.(2021))to isolate all the cuspidal spectrum.In particular,they are suitable for the comparison of orbital integrals.These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups(see Beuzart-Plessis et al.(2021,2022)).In this article,we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al.(2021)for the function field case.
