Centralizers of Iwahori-Hecke Algebras II: The General Case

This paper is a sequel to [4]. We establish the minimal basis theory for the centralizers of parabolic subalgebras of Iwahori-Hecke algebras associated to finite Coxeter groups of any type, generalizing the approach introduced in [3] from centres to the centralizer case. As a pro-requisite, we prove a reducibility property in the twisted J-conjugacy classes in finite Coxeter groups, which is a generalization of results in [7] and [4].2000 Mathematics Subject Classification: 20C08, 20F55     

Centralizers of Iwahori-Hecke algebras

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories.

This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type and , giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.

     

The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.     

Hecke Algebras of Classical Groups over p-adic Fields II

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (J , ) of an open compact subgroup J and its irreducible representation which is constructed from given data = (, P₀, ). Here, is a semisimple element in the Lie algebra of G, P₀ is a parahoric subgroup in the centralizer of in G, and is a cuspidal representation on the finite reductive quotient of P₀. In this paper, we explicitly describe those Hecke algebras when P₀ is a minimal parahoric subgroup, is trivial and is a character.     

强半单的n-李代数的表示

本文主要研究了强半单的n-李代数的表示,证明了强半单的n-李代数的表示(V,ρ)可转化为一个约化李代数Lρ(V)的表示,并证明了不变线性形等其它相关性质．     

Weyl groups,the Dirac inequality,and isolated unitary unramified representations

I present several applications of the Dirac inequality to the determination of isolated unitary representations and associated “spectral gaps” in the case of unramified principal series. The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. The results could be viewed as a $p$-adic analogue of Salamanca-Riba’s classification of irreducible unitary $(\mathfrak{g},K)$-modules with strongly regular infinitesimal character.     

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.     

Finite-Dimensional Representations of Twisted Hyper-Loop Algebras

We investigate the category of finite-dimensional representations of twisted hyper-loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper-loop algebras are isomorphic to appropriate simple and Weyl modules for the nontwisted hyper-loop algebras, respectively, via restriction of the action.     

The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

Hecke Algebras and Automorphic Forms

The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G ₂(F₅) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL₂ to PGSp₄.     

On the Left Cell Representations of Iwahori-Hecke Algebras of Finite Coxeter Groups

In this paper we investigate the left cell representations ofthe Iwahori-Hecke algebras associated to a finite Coxeter groupW. Our main result shows that , where w₀ is the element of longest length in W, acts essentiallyas an involution upon the canonical bases of a cell representation.We describe some properties of this involution, use it to furtherdescribe the left cells, and finally show how to realize eachcell representation as a submodule of . Our results rely uponcertain positivity properties of the structure constants ofthe Kazhdan-Lusztig bases of the Hecke algebra and so have notyet been shown to apply to all finite Coxeter groups.     

Row and Column Removal Theorems for Homomorphisms of Specht Modules and Weyl Modules

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author [16]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the q-Schur algebra.This research was supported by ARC grant DP0343023. The first author was also supported by a Sesqui Research Fellowship at the University of Sydney.     

Inductive Algebras for Finite Heisenberg Groups

A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups.     

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of with weight basis . The supporting graph P of is defined to be the directed graph whose vertices are the elements of and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, ) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, ) and the irreducible one-dimensional weight space representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, ), the exceptional Lie algebra G ₂, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.     

Generalized Graded Hecke Algebras of Types B and D

For a root system of type B we study an algebra similar to a graded Hecke algebra, isomorphic to a subalgebra of the rational Cherednik algebra. We introduce principal series modules over it and prove an irreducibility criterion for these modules. We deduce similar results for an algebra associated to a root system of type D.     

Weyl代数的表示

设Ｋ是一个域．证明了，若ｃｈＫ＝０，那么ｎ－ｔｈＷｅｙｌ代数Ａ_ｎ（ｋ）没有有限维表示．还给出了Ａ_ｎ（ｋ）的不可约Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模的分类．当Ｋ是一个特征非零的代数闭域时，给出了有限维不可约Ａ_ｎ（Ｋ）－模的分类．     

Hecke group algebras as quotients of affine Hecke algebras at level 0

The Hecke group algebra of a finite Coxeter group , as introduced by the first and last authors, is obtained from by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, is the natural quotient of the affine Hecke algebra H(W)(q) through its level 0 representation.The proof relies on the following core combinatorial result: at level 0 the 0-Hecke algebra H(W)(0) acts transitively on . Equivalently, in type A, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level 0 representation is a calibrated principal series representation M(t) for a suitable choice of character t, so that the quotient factors (non-trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the 0-Hecke algebra and that of the affine Hecke algebra H(W)(q) at this specialization.     

Murphy Operators and the Centre of the Iwahori-Hecke Algebras of Type A

In this paper we introduce a family of polynomials indexed by pairs of partitions and show that if these polynomials are self-orthogonal then the centre of the Iwahori-Hecke algebra of the symmetric group is precisely the set of symmetric polynomials in the Murphy operators.