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On Representations of Affine Hecke Algebras of Type B
Abstract
For the affine Hecke algebras of type A, Grojnowski has developed a combinatorial labelling of irreducible modules using formal characters and crystal operators, also obtaining certain special branching rules.
In this thesis, the same approach is applied to affine Hecke algebras of type B, where it does not lead to full results in all cases. For certain eigenvalues of lattice operators only partial results can be achieved.
One main result is the irreducibility of certain modules that are induced from type A to type B, giving a one-to-one corresondance of irreducible objects in certain full subcategories of the module categories in type A and type B.
This yields an analogous combinatorial description in type B as in type A, including branching rules.
... Plus précisément, le lien entre algèbres KLR et bases canoniques est donné par un isomorphisme explicite d'algèbres, due à Varagnolo-Vasserot [VV09b], entre les algèbres KLR et les algèbres d'extension de certains complexes de faisceaux constructibles qui interviennent dans la construction de Kazhdan-Lusztig. Les règles de branchement pour les algèbres de Hecke affines de type B ont été étudiées très récemment par [Eno09], [EK06,EK08a,EK08b], [Mie08] et [VV09a]. En particulier, dans [Eno09], [EK06,EK08a,EK08b] un analogue de la construction d'Ariki ont été conjecturé et étudié. ...
... The branching rules for affine Hecke algebras of type B have been investigated quite recently, see [Eno09], [EK06,EK08a,EK08b], [Mie08] and [VV09a]. In particular, in [Eno09], [EK06,EK08a,EK08b] an analogue of Ariki's construction is conjectured and studied for affine Hecke algebras of type B. ...
This thesis consists of three chapters. In Chapter I, we define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. Using these functors, we construct a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space. Chapter II is a joint work with Michela Varagnolo and Eric Vasserot. We prove a conjecture of Miemietz and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type D. In Chapter III, we prove a conjecture of Leclerc and Thibon on the graded multiplicities associated with the Jantzen filtration of Weyl modules over v-Schur algebras.
... Proof. Using the Mackey Lemma for affine Hecke algebras (see [Mi,Section 2] and [Kl]), ...
Let G be a general linear group over a p-adic field. It is well known that Bernstein components of the category of smooth representations of G are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of G by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
... is the analogue of [BDK,Lemma 5.4] and [Fl,Lemma 2.1]. See also [Mi,Theorem 1]. ...
In this paper, we study the relation between the cocenter and the
representation theory of affine Hecke algebras. The approach is based on the
interaction between the rigid cocenter, an important subspace of the cocenter,
and the dual object in representation theory, the rigid quotient of the
Grothendieck group of finite dimensional representations.
... The inductive approach to the representation theory of the usual, type A, Hecke algebras, that is, the algebras H(1, 1, n), heavily uses the representation theory of the affine Hecke algebra of type A. One could expect that, say, for the representation theory of the algebras H(2, 1, n), there will be a necessity to use representations of the affine Hecke algebra of type B (see [10,22] for definitions). But -and it is maybe surprising -the representation theory of the Hecke algebras H(m, 1, n) in the inductive approach requires, for all m, the study of representations of the same affine Hecke algebra of type A. ...
An inductive approach to the representation theory of cyclotomic Hecke
algebras, inspired by Okounkov and Vershik, is developed. We study the common
spectrum of the Jucys-Murphy elements using representations of the simplest
affine Hecke algebra. Representations are constructed with the help of a new
associative algebra whose underlying vector space is the tensor product of the
cyclotomic Hecke algebra with the free associative algebra generated by
standard m-tableaux.
Let G be a general linear group over a p-adic field. It is well known that Bernstein components of the category of smooth representations of G are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of G by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
Let G be a finite group of Lie type and let k be a field of characteristic distinct from the defining characteristic of G. In studying the non-describing rep- resentation theory of G, the endomorphism algebra S(G;k) = EndkG( L J ind G PJ k) plays an increasingly important role. In type A, by work of Dipper and James, S(G;k) identifies with a q-Schur algebra and so serves as a link between the rep- resentation theories of the finite general linear groups and certain quantum groups. This paper presents the first systematic study of the structure and homological alge- bra of these algebras for G of arbitrary type. Because S(G;k) has a reinterpretation as a Hecke endomorphism algebra, it may be analyzed using the theory of Hecke algebras. Its structure turns out to involve new applications of Kazhdan-Lusztig cell theory. In the course of this work, we prove two stratification conjectures about Coxeter group representations made in (CPS4) and we formulate a new conjecture about the structure of S(G;k). We verify this conjecture here in all rank 2 examples.
We determine a basis of the (twisted) cocenter of graded affine Hecke
algebras with arbitrary parameters. In this setting, we prove that the kernel
of the (twisted) trace map is the commutator subspace (Density theorem) and
that the image is the space of good forms (trace Paley-Wiener theorem).
Cette thèse se compose de trois chapitres. Dans le chapitre I, nous définissons les foncteurs de i-restriction et i-induction sur la catégorie O des algèbres de Hecke doublement affine rationnelles cyclotomiques. En utilisant ces foncteurs, nous construisons un cristal sur l'ensemble des classes d'isomorphisme des modules simples, qui est isomorphe au cristal de l'espace de Fock. Le chapitre II est un travail en collaboration avec Michela Varagnolo et Eric Vasserot. Nous démontrons une conjecture de Kashiwara et Miemietz sur bases canoniques et règles de branchement pour les algèbres de Hecke affines de type D. Dans le chapitre III, nous démontrons une conjecture de Leclerc et Thibon sur les multiplicités graduées associées à la filtration de Jantzen de modules de Weyl sur algèbres de v-Schur.
We relate the filtration by the support on the Grothendieck group [ O ] [\mathcal {O}] of the category O \mathcal {O} of cyclotomic rational double affine Hecke algebras to a representation-theoretic grading on [ O ] [\mathcal {O}] , defined using the action of an affine Lie algebra and of a Heisenberg algebra on the Fock space. This implies a recent conjecture of Etingof. The proof uses a categorification of the Heisenberg action, which is new, and a categorification of the affine Lie algebra action, which was introduced by the first author in an earlier paper.
We prove a conjecture of Kashiwara and Miemietz on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case. Comment: 24 pages
We prove a series of conjectures of Enomoto and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type B. The main ingredient of the proof is a new graded Ext-algebra associated with quiver with involutions that we compute explicitly. Comment: 82 pages
We give a simple description of the natural bijection between the set of FLOTW bipartitions and the set of Uglov bipartitions (which generalizes the set of Kleshchev bipartitions). These bipartitions, which label the crystal graphs of irreducible
-modules of level two, naturally appear in the context of the modular representation theory of Hecke algebras of type B
n
.
The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki,
asserts that the K-group of the representations of the affine Hecke algebras of
type A is isomorphic to the algebra of functions on the maximal unipotent
subgroup of the group associated with the Lie algebra or the affine
Lie algebra , and the irreducible representations correspond to
the upper global bases. In this note, we formulate analogous conjectures for
certain classes of irreducible representations of affine Hecke algebras of type
B. We corrected typos.
In the previous paper "Symmetric Crystals and Affine Hecke Algebras of Type B", we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for \gl_\infty. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type A. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type B.
This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.
We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r; p; n) and the corresponding cyclotomic Hecke algebras H r;p;n , can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to ane Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.
Ian Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (l+1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_l^(1). The present article is devoted to extending Grojnowski's machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov, and the Kac-Moody algebra A_l^(1) with the twisted algebra A_2l^(2). In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford superalgebras of the plus part U_\Z^+ of the enveloping algebra, as well as of Kashiwara's highest weight crystals B(\infty) and B(\la) for each dominant weight \la. The results of the article have applications to the modular representation theory of the double covers of the symmeric groups, as was predicted originally by Leclerc and Thibon. In particular, the parametrization of irreducibles, classification of blocks and analogues of the modular branching rules of the symmetric group for the double covers over fields of odd characteristic follow from the special case \lambda = \Lambda_0 of our main results. These matters are discussed in the final section of the paper.
We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r,p,n) and the corresponding cyclotomic Hecke algebras Hr,p,n, can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.
Hecke algebras.- Affine Weyl groups and affine Hecke algebras.- A generalized two-sided cell of an affine Weyl group.- qs-analogue of weight multiplicity.- Kazhdan-Lusztig classification on simple modules of affine Hecke algebras.- An equivalence relation in T x ?*.- The lowest two-sided cell.- Principal series representations and induced modules.- Isogenous affine Hecke algebras.- Quotient algebras.- The based rings of cells in affine Weyl groups of type .- Simple modules attached to c 1.
This is the third, substantially revised edition of this important
monograph by a giant in the field of mathematics. The book is concerned
with Kac-Moody algebras, a particular class of infinite-dimensional Lie
algebras, and their representations. Each chapter begins with a
motivating discussion and ends with a collection of exercises with hints
to the more challenging problems. The theory has applications in many
areas of mathematics, and Lie algebras have been significant in the
study of fundamental particles, including string theory, so this book
should appeal to mathematical physicists, as well as mathematicians.
In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincare polynomial. This verifies a conjecture of Lusztig proposed in 1989.
The volume under review is a much awaited contribution to the literature on representation theory of symmetric groups and associated groups and algebras. Related topics are presented in several recent surveys and monographs, such as [S. Ariki, Representations of quantum algebras and combinatorics of Young tableaux, Am. Math. Soc. (2002; Zbl 1003.17008); M. Cabanes and M. Enguehard, Representation theory of finite reductive groups, Cambridge University Press (2004; Zbl 1069.20032); M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford: Clarendon Press (2000; Zbl 0996.20004); A. Mathas, Iwahori-Hecke algebras and the Schur algebras of the symmetric group, Am. Math. Soc. (1999; Zbl 0940.20018)]. This book is a comprehensive presentation of the progress in the modular representation theory achieved especially by the work of A. Lascoux, B. Leclerc, J.-Y. Thibon, S. Ariki, J. Grojnowski, J. Brundan and of the author. The book is divided into two parts. Part I is devoted to the linear representations, and consists of 11 chapters. The first two chapters are introductory and present the approach of Okounkov and Vershik to the representation theory in characteristic zero. Chapter 3 contains the definition of the degenerate affine Hecke algebras ℋ n and their basic properties, including a discussion of the induction and restriction functors. Chapter 4 discusses central characters and blocks, and the so-called Kato module. Chapter 5 continues the study of ℋ n -modules by introducing the functors e a , the crystal operators e ˜ and f ˜, and proving several branching rules. In Chapter 6 methods for calculating characters of certain irreducible ℋ n -modules are developed. Chapter 7 presents the basic results concerning integral representations and cyclotomic Hecke algebras, and in the next chapter cyclotomic analogues of the functors e i are studied. Chapter 9 establishes a connection between the Grothendieck group K(∞) of integral representations of all affine Hecke algebras ℋ n and the affine Kac-Moody algebras. An important result here constructs an explicit isomorphism of graded Hopf algebras between the graded dual K(∞) * and the Kostant-Tits ℤ-form U ℤ + of the Kac-Moody algebra of type A p-1 (1) . As a consequence, the blocks of the cyclotomic Hecke algebra are classified. Chapters 10 and 11 study the crystal graph of the irreducible highest weight module and the socle branching graph of the cyclotomic Hecke algebras. The second part of the book is devoted to projective representations. Their study is done by regarding the twisted group algebra T n of S n as a superalgebra, and to work with the category of T n -supermodules. It turns out that it is even more convenient to work with the Sergeev superalgebra, which is “almost Morita equivalent” to T n . This gives a parallelism between the linear representation theory and the spin representation theory which is exposed in the remaining Chapters 14-22. Most of the material appears for the first time in book form, but at the same time, the theory is developed from scratch. Therefore, this volume will be useful not only to researchers, but also to the graduate students who want to understand this fascinating and dynamic subject, which is relevant not only for group theory, but it has rich connections with combinatorics, Lie theory and algebraic geometry.
Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.- Afterword (1994).- Index of Terminology.- Index of Symbols.
Given an irreducible module for the affine Hecke algebraH
n
of type A, we consider its restriction toH
n−1. We prove that the socle of restriction is multiplicity free and moreover that the summands lie in distinct blocks. This is true regardless of the characteristic of the field or of the order of the parameterq in the definition ofH
n
. The result generalizes and implies the classical “branching rules” that describe the restriction of an irreducible representation of the symmetric groupS
n
toS
n−1.
This paper classifies and constructs explicitly all the irreducible representations of affine Hecke algebras of rank two root systems. The methods used to obtain this classification are primarily combinatorial and are, for the most part, an application of the methods used in [Ra1]. I have made special effort to describe how the classification here relates to the classifications by Langlands parameters (coming from p-adic group theory) and by indexing triples (coming from a q-analogue of the Springer correspondence). There are several reasons for doing the details of this classification: (a) The proof of the one of the main results of [Ra1] depends on this classification of representations for rank two affine Hecke algebras. Specifically, in the proof of Proposition 4.4 of [Ra1], one needs to know exactly which weights can occur in calibrated representations. The reason that this naturally depends on a rank two classification is outlined in (d) below. (b) The examples here illustrate (and clarify) results of [Ra1], [KL], [CG], [BM], [Ev], [Kr], [HO12 ]. Much of the power of the combinatorial methods which are now available is evident from the calculations in this paper, especially when one compares with the effort needed in other sources (for example [Xi], Chapt. 11). (c) The explicit information here can be very useful for obtaining results on representations of
This paper provides a combinatorial dictionary between three sets of objects: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and the irreducible modules of the affine Hecke algebra (for generic q). In particular, we compute the action of the crystal operator (a refinement of socle of Restriction) on an irreducible module both in terms of its parameterization by multisegments and by multipartitions. In other words, we give explicit crystal graph isomorphisms. A byproduct is the determination of which multisegments parameterize modules of the {\it cyclotomic} Hecke algebra . The theorems also explain why the rule for computing mirrors the rule we know for that on a tensor product of crystal graphs. We also give a construction of the irreducible module parameterized by a multipartition without relying on a choice of path in the crystal graph. The proofs given here are elementary and do not rely on any geometry.
Representations of affine Hecke algebras MR MR1320509 (96i:20058) Universität Stuttgart, Fachbereich Mathematik, Institut für Algebra und Zahlentheorie E-mail address: miemieva@mathematik
- Nan Hua
Nan Hua Xi, Representations of affine Hecke algebras, Lecture Notes in Mathematics, vol. 1587, Springer-Verlag, Berlin, 1994. MR MR1320509 (96i:20058) Universität Stuttgart, Fachbereich Mathematik, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, 70569 Stuttgart, Germany E-mail address: miemieva@mathematik.uni-stuttgart.de
Characters of finite Coxeter groups and Iwahori-Hecke alge-bras, London Mathematical Society Monographs
- Meinolf Geck
Meinolf Geck and G¨ otz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke alge-bras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press Oxford University Press, New York, 2000. MR MR1778802 (2002k:20017
Irreducible modules over the affine Hecke algebra: a strong multiplicity one result
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Vazirani, M.J.: Irreducible modules over the affine Hecke algebra: a strong multiplicity one result. Ph.D. thesis, University of California at Berkeley (1999)
Introduction to Lie algebras and representation theory Second printing, revised
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James E. Humphreys, Introduction to Lie algebras and representation theory,
Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York, 1978, Second printing, revised. MR MR499562 (81b:17007)