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 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.

Spherical Functions for the Semisimple Symmetric Pair (Sp(2, ℝ), SL(2, ℂ))

2002 ◽  
Vol 54 (4) ◽  
pp. 828-865 ◽  
Author(s):  
Tomonori Moriyama
Keyword(s):  
Spherical Function ◽  
Series Representation ◽  
Principal Series ◽  
Spherical Functions ◽  
Intertwining Operators ◽  
Symmetric Pair ◽  
Radial Part ◽  
Principal Series Representation ◽  
Integral Expression ◽  
One Dimensional

AbstractLet π be an irreducible generalized principal series representation of G = Sp(2, ℝ) induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from π to the representation induced from an irreducible admissible representation of SL(2, ℂ) in G is at most one dimensional. Spherical functions in the title are the images of K-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.

scholarly journals A QUOTIENT OF THE LUBIN–TATE TOWER

2017 ◽  
Vol 5 ◽  
Author(s):  
JUDITH LUDWIG
Keyword(s):  
Borel Subgroup ◽  
Series Representation ◽  
Principal Series ◽  
Principal Series Representation ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Steinberg Representation

In this article we show that the quotient${\mathcal{M}}_{\infty }/B(\mathbb{Q}_{p})$of the Lubin–Tate space at infinite level${\mathcal{M}}_{\infty }$by the Borel subgroup of upper triangular matrices$B(\mathbb{Q}_{p})\subset \operatorname{GL}_{2}(\mathbb{Q}_{p})$exists as a perfectoid space. As an application we show that Scholze’s functor$H_{\acute{\text{e}}\text{t}}^{i}(\mathbb{P}_{\mathbb{C}_{p}}^{1},{\mathcal{F}}_{\unicode[STIX]{x1D70B}})$is concentrated in degree one whenever$\unicode[STIX]{x1D70B}$is an irreducible principal series representation or a twist of the Steinberg representation of$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

scholarly journals $0$-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Boolean Algebra ◽  
Symmetric Functions ◽  
Hecke Algebra ◽  
Type A ◽  
Quasisymmetric Function ◽  
Permutation Statistics ◽  
Nous Obtenons ◽  
Noncommutative Symmetric Functions ◽  
International Audience ◽  
Cette Action

International audience We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics. Nous définissons une action de l’algèbre de Hecke-$0$ de type A sur l’anneau Stanley-Reisner de l’algèbre de Boole. En étudiant cette action, on obtient une famille de fonctions symétriques non commutatives multivariées, qui se spécialisent pour les non commutatives fonctions de Hall-Littlewood symétriques et leur $(q,t)$-analogues introduits par Bergeron et Zabrocki. Nous obtenons également des identités de fonction quasisymmetrique multivariées, qui se spécialisent à la suite de Garsia et Gessel sur la fonction génératrice de la distribution conjointe de cinq statistiques de permutation.

scholarly journals A canonical basis for Garsia-Procesi modules

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Jonah Blasiak
Keyword(s):  
Positive Integer ◽  
Canonical Basis ◽  
Hecke Algebra ◽  
Type A ◽  
Young Tableaux ◽  
Affine Hecke Algebra ◽  
International Audience ◽  
Standard Young Tableaux

International audience We identify a subalgebra $\widehat{\mathscr{H}}^+_n$ of the extended affine Hecke algebra $\widehat{\mathscr{H}}_n$ of type $A$. The subalgebra $\widehat{\mathscr{H}}^+_n$ is a u-analogue of the monoid algebra of $\mathcal{S}_n ⋉ℤ_≥0^n$ and inherits a canonical basis from that of $\widehat{\mathscr{H}}_n$. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod $n$, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient $\mathscr{R}_1^n$ of $\widehat{\mathscr{H}}^+_n$ that is a $u$-analogue of the ring of coinvariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element $*π ∈ \widehat{\mathscr{H}}^+_n$ corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that $\mathscr{R}_1^n$ has cellular quotients $\mathscr{R}_λ$ that are $u$-analogues of the Garsia-Procesi modules $R_λ$ with left cells labeled by (a PAT version of) the $λ$ -catabolizable tableaux. On définit une sous-algèbre $\widehat{\mathscr{H}}^+_n$ de l'extension affine de l'algèbre de Hecke \$\widehat{\mathscr{H}}_n$ de type $A$. La sous-algèbre $\widehat{\mathscr{H}}^+_n$ est $u$-analogue à l'algèbre monoïde de $\mathcal{S}_n ⋉ℤ_≥0^n$ et hérite d'une base canonique de $\widehat{\mathscr{H}}_n$. On montre que ses cellules gauches sont naturellement classées par des tableaux remplis d'entiers naturels ayant chacun des restes différents modulo $n$, que l'on nomme Positive Affine Tableaux (PAT). On montre ensuite qu'un sous-quotient cellulaire $\mathscr{R}_1^n$ de $\widehat{\mathscr{H}}^+_n$ est une $u$-analogue de l'anneau des co-invariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ avec des cellules gauches classées PAT qui sont essentiellement des tableaux de Young standards avec des labels cochargés. Multiplier les éléments de la base canonique par un certain élément $π ∈ \widehat{\mathscr{H}}^+_n$ correspond à des rotations de mots, et par rapport aux cellules cela correspond à un cocyclage. Plus loin, on montre que $\mathscr{R}_1^n$ a pour quotients cellulaires $\mathscr{R}_λ$ qui sont $u$- analogues aux modules de Garsia-Procesi $R_λ$ avec des cellules gauches définies par (une version PAT) des tableaux $λ$ -catabolisable.

L-Functions for GSp(2) × GL(2): Archimedean Theory and Applications

2009 ◽  
Vol 61 (2) ◽  
pp. 395-426 ◽  
Author(s):  
Tomonori Moriyama
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Principal Series ◽  
Algebraic Number Field ◽  
Automorphic Representation ◽  
Explicit Computation ◽  
Principal Series Representation ◽  
Cuspidal Automorphic Representation ◽  
Discrete Series Representation ◽  
Totally Real

Abstract. Let Π be a generic cuspidal automorphic representation of GSp(2) defined over a totally real algebraic number field k whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product L-functions L(s,Π × σ) for an arbitrary cuspidal automorphic representation σ of GL(2). We also give an application to the spinor L-function of Π.

scholarly journals THE DOUBLE COVER OF ODD GENERAL SPIN GROUPS, SMALL REPRESENTATIONS, AND APPLICATIONS

2015 ◽  
Vol 16 (3) ◽  
pp. 609-671 ◽  
Author(s):  
Eyal Kaplan
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Automorphic Representation ◽  
Double Cover ◽  
Spin Groups ◽  
Principal Series Representation ◽  
Double Covers ◽  
Exceptional Representations ◽  
Period Integral ◽  
Exceptional Character

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of$\mathit{SO}_{2n+1}$of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

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