Hecke Algebras With Unequal Parameters

By G. Lusztig. 2002

413 citations recorded by the source corpus.

Introduction Coxeter groups Partial order on $W$ The algebra ${\mathcal H}$ The bar operator The elements $c_w$ Left or right multiplication by $c_s$ Dihedral groups Cells Cosets of parabolic subgroups Inversion The longest element for a finite $W$ Examples of elements $D_w$ The function $\mathbf{a}$ Conjectures Example: The split case Example: The quasisplit case Example: The infinite dihedral case The ring $J$ Algebras with trace form The function ${\mathbf{a}}_E$ Study of a left cell Constructible representations Two-sided cells Virtual cells Relative Coxeter groups Representations A new realization of Hecke algebras Bibliography Other titles in this series.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a6992cc7b649ce7dcbb9e9c40fb4342c2cfcf3ce

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Citations in the corpus

Character sheaves on disconnected groups, X (2003, methodology): “(This plays a key role in the argument in 35.22.) In §34 we study the algebra Hn of 31.2 (or rather an extension H D n of it) in the spirit of our earlier st udy [L12] of a usual Iwahori-Hecke algebra by means of the asymptotic Hecke algebra.”
Affine flag varieties and quantum symmetric pairs (2016, neutral): “ The case for i“ 0 follows from a similar argument to that of the case for i“ r. Note that when r“ 0, one uses the non-degenerate symplectic form on L1{L0 “ ε´1L# 0 {L0, inherited from that of V (see [Lu03]). 5.3. The comultiplication ∆c Recall Λn,d from (2.2.2), and we denote Λc n,d“ ! paiqiPZ ˇ ˇ ”
Left cells in type $B_n$ with unequal parameters (2003, result): “For this purpose, it is convenient to work in the general setting described by Lusztig (see [13] and [17]).”
An exotic Deligne-Langlands correspondence for symplectic groups (2006, methodology): “r result to multi-parameter cases. (It is called the unequal parameter case.) Lusztig realized the “graded version” of H(with unequal parameters) via several geometric means [Lu88, Lu89, Lu95b] (c.f. [Lu03]) and classiﬁed their representations in certain cases. Unfortunately, his geometries admit es”
Hecke algebras of finite type are cellular (2006, methodology): “Kazhdan–Lusztig theory and the asymptotic ring References for this section are the books [14] and [25].”
On fusion categories (2002, neutral): “and the Galois group of L over Q is abelian.”
On the Induction of Kazhdan–Lusztig Cells (2003, background): “This case is studied in detail in Lusztig’s lecture notes [8].”
On the classification of irreducible representations of affine Hecke algebras with unequal parameters (2010, background): “s can change however. It is reasonable to expect that something similar holds for general aﬃne Hecke algebras, probably that would follow from the existence of an appropriate asymptotic Hecke algebra [Lus8]. 98 q = 1 1 q q = 1 0 2 0 q = q = 1 0 2 q = q = q 2 1 q = q = q 0 1/2 2 1 0 q q = q = 1 2 1 q”
On Domino Insertion and Kazhdan–Lusztig Cells in Type Bn (2006, methodology): “onjectures have been veriﬁed for n 66 by explicit computation using CHEVIE[18] and the program Coxeterdeveloped by du Cloux [7]. For the basic deﬁnitions concerning Kazhdan–Lusztig cells, see Lusztig [30]. Conjecture A. Assume that Γ = Z, a = 2 and b = 2r + 1 where r > 0. Then the following hold.”
Gröbner–Shirshov bases and their calculation (2013, background): “Two K -bases for H are known; one is natural, and the other is the Kazhdan–Lusztig canonical basis [155].”
Kazhdan–Lusztig Cells and the Murphy Basis (2005, background): “In [ 17 ], Lusztig writes cw instead of C ′ w , but we shall adhere to the older notation here.”
Categorification of (induced) cell modules and the rough structure of generalized Verma modules (2007, neutral): “This filtration is based on the Gelfand–Kirillov dimension which is closely related to Lusztig’s a-function (see [50]) and should be compared with the appearance of the a-function in [12] and [28].”