scholarly journals Hecke group algebras as degenerate affine Hecke algebras

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Florent Hivert ◽  
Anne Schilling ◽  
Nicolas M. Thiéry
Keyword(s):  
Weyl Group ◽  
Group Algebra ◽  
Series Representation ◽  
Principal Series ◽  
Hecke Algebra ◽  
Affine Hecke Algebra ◽  
Hecke Group ◽  
Finite Coxeter Group ◽  
Nous Montrons ◽  
Combinatorial Result

International audience The Hecke group algebra $\operatorname{H} \mathring{W}$ of a finite Coxeter group $\mathring{W}$, as introduced by the first and last author, is obtained from $\mathring{W}$ by gluing appropriately its $0$-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $\mathring{W}$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $\operatorname{H} \mathring{W}$ is the natural quotient of the affine Hecke algebra $\operatorname{H}(W)(q)$ through its level $0$ representation. The proof relies on the following core combinatorial result: at level $0$ the $0$-Hecke algebra acts transitively on $\mathring{W}$. 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 classical $0$-Hecke algebra and that of the affine Hecke algebra at this specialization. L'algèbre de Hecke groupe $\operatorname{H} \mathring{W}$ d'un groupe de Coxeter fini $\mathring{W}$, introduite par le premier et le dernier auteur, est obtenue en recollant de manière appropriée son algèbre de Hecke dégénérée et son algèbre de groupe. Dans cet article, nous donnons une construction alternative dans le cas où $\mathring{W}$ est un groupe de Weyl associé à un groupe de Weyl affine $W$. Plus précisément, nous montrons que quand $q$ n'est ni nul ni une racine de l'unité, $\operatorname{H} \mathring{W}$ est le quotient naturel de l'algèbre de Hecke affine $\operatorname{H}(W)(q)$ dans sa représentation de niveau $0$. Nous montrons de plus que la représentation de niveau $0$ est une représentation de série principale calibrée $M(t)$ pour un certain caractère $t$, de sorte que le quotient se factorise par la spécialisation centrale principale. Ce fait explique en particulier les similarités entre les théories des représentations de l'algèbre de Hecke dégénérée et de l'algèbre de Hecke affine sous cette spécialisation.

scholarly journals The biHecke monoid of a finite Coxeter group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Florent Hivert ◽  
Anne Schilling ◽  
Nicolas M. Thiéry
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Coxeter Group ◽  
Projective Modules ◽  
Permutation Matrices ◽  
Hecke Group ◽  
Finite Coxeter Group ◽  
Nous Montrons ◽  
Combinatorial Model ◽  
International Audience

arXiv : http://arxiv.org/abs/0912.2212 International audience For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0-Hecke$ algebra $H_0(W)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each $w∈W$ a combinatorial module $T_w$ whose support is the interval $[1,w]_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset. Pour tout groupe de Coxeter fini $W$, nous définissons deux nouveaux objets : son ordre de coupures et son monoïde de Hecke double. L'ordre de coupures, construit au moyen d'une généralisation de la notion de bloc dans les matrices de permutations, est presque un treillis sur $W$. La construction du monoïde de Hecke double s'appuie sur le modèle combinatoire usuel de la $0-algèbre$ de Hecke $H_0(W)$, pour le groupe symétrique, l'algèbre (ou le monoïde) engendré par les opérateurs de tri par bulles élémentaires. Les auteurs ont introduit précédemment l'algèbre de Hecke-groupe, construite comme l'algèbre engendrée conjointement par les opérateurs de tri et d'anti-tri, et décrit sa théorie des représentations. Dans cet article, nous considérons le monoïde engendré par ces opérateurs. Nous montrons qu'il admet $|W|$ modules simples et projectifs. Afin de construire ses modules simples, nous introduisons pour tout $w∈W$ un module combinatoire $T_w$ dont le support est l'intervalle [$1,w]_R$ pour l'ordre faible droit. Ce module détermine une algèbre dont la théorie des représentations généralise celle de l'algèbre de Hecke groupe, en remplaçant la combinatoire des descentes par celle des blocs et de l'ordre de coupures.

scholarly journals ON THE FORMAL AFFINE HECKE ALGEBRA

2014 ◽  
Vol 14 (4) ◽  
pp. 837-855 ◽  
Author(s):  
Changlong Zhong
Keyword(s):  
Weyl Group ◽  
Group Algebra ◽  
Hecke Algebra ◽  
Formal Group ◽  
Affine Hecke Algebra ◽  
Formal Group Law ◽  
Group Law

We study the action of the formal affine Hecke algebra on the formal group algebra, and show that the the formal affine Hecke algebra has a basis indexed by the Weyl group as a module over the formal group algebra. We also define a concept called the normal formal group law, which we use to simplify the relations of the generators of the formal affine Demazure algebra and the formal affine Hecke algebra.

scholarly journals Algebraic and analytic Dirac induction for graded affine Hecke algebras

2013 ◽  
Vol 13 (3) ◽  
pp. 447-486 ◽  
Author(s):  
Dan Ciubotaru ◽  
Eric M. Opdam ◽  
Peter E. Trapa
Keyword(s):  
Weyl Group ◽  
Discrete Series ◽  
Hecke Algebras ◽  
Reductive Group ◽  
Dirac Operators ◽  
Hecke Algebra ◽  
Semisimple Lie Groups ◽  
Affine Hecke Algebra ◽  
Affine Hecke Algebras ◽  
Dirac Induction

AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

scholarly journals Rodier type theorem for generalized principal series

2021 ◽  
Author(s):  
Caihua Luo
Keyword(s):  
Weyl Group ◽  
Coxeter Group ◽  
Structure Theorem ◽  
Borel Subgroup ◽  
Discrete Series ◽  
Type Theorem ◽  
Series Representation ◽  
Reductive Group ◽  
Principal Series ◽  
Levi Subgroup

AbstractGiven a regular supercuspidal representation $$\rho $$ ρ of the Levi subgroup M of a standard parabolic subgroup $$P=MN$$ P = M N in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $$JH(Ind^G_P(\rho ))$$ J H ( I n d P G ( ρ ) ) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation $$Ind^G_P(\rho )$$ I n d P G ( ρ ) , vastly generalizing Rodier structure theorem for $$P=B=TU$$ P = B = T U Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group $$W_M=N_G(M)/M$$ W M = N G ( M ) / M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group $$W_T=N_G(T)/T$$ W T = N G ( T ) / T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split $$(G)Sp_{2n}$$ ( G ) S p 2 n and $$SO_{2n+1}$$ S O 2 n + 1 , and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

scholarly journals Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maki Nakasuji ◽  
Hiroshi Naruse
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Hecke Algebra ◽  
Type A ◽  
Principal Series Representation ◽  
International Audience ◽  
Definition Of ◽  
Math Phys

International audience We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.

scholarly journals Kazhdan-Lusztig Basis and a Geometric Filtration of an Affine Hecke Algebra

2006 ◽  
Vol 182 ◽  
pp. 285-311 ◽  
Author(s):  
Toshiyuki Tanisaki ◽  
Nanhua Xi
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Hecke Algebra ◽  
Reductive Algebraic Group ◽  
Affine Hecke Algebra ◽  
Nilpotent Variety ◽  
Affine Weyl Group

AbstractAccording to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant K-group of Steinberg’s triple variety. The K-group is equipped with a filtration indexed by closed G-stable subvarieties of the nilpotent variety, where G is the corresponding reductive algebraic group over ℂ. In this paper we will show in the case of type A that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

scholarly journals R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

2021 ◽  
pp. 1-61
Author(s):  
Fan Gao
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Symplectic Groups ◽  
Principal Series Representation ◽  
Connected Group ◽  
Covering Group ◽  
Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

scholarly journals Generalized Descent Algebras

2007 ◽  
Vol 50 (4) ◽  
pp. 535-546
Author(s):  
Christophe Hohlweg
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Descent Algebra ◽  
Finite Coxeter Group ◽  
Solomon Descent Algebra ◽  
Descent Algebras

AbstractIf A is a subset of the set of reflections of a finite Coxeter group W, we define a sub-ℤ-module of the group algebra ℤW. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if W is of type B, the Mantaci–Reutenauer algebra.

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