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 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 Ellipticity and discrete series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bernhard Krötz ◽  
Job J. Kuit ◽  
Eric M. Opdam ◽  
Henrik Schlichtkrull
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Reductive Group ◽  
Cartan Subgroup ◽  
Discrete Series Representation ◽  
Spherical Spaces

Abstract We explain by elementary means why the existence of a discrete series representation of a real reductive group G implies the existence of a compact Cartan subgroup of G. The presented approach has the potential to generalize to real spherical spaces.

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 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 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 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.

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ivan Matić
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Discrete Series ◽  
Series Representation ◽  
General Linear ◽  
Composition Series ◽  
Induced Representations ◽  
Discrete Series Representation

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals Analytic Continuation of Equivariant Distributions

2018 ◽  
Vol 2019 (23) ◽  
pp. 7160-7192 ◽  
Author(s):  
Dmitry Gourevitch ◽  
Siddhartha Sahi ◽  
Eitan Sayag
Keyword(s):  
Analytic Continuation ◽  
Reductive Group ◽  
Principal Series ◽  
Real Algebraic Varieties ◽  
Series Representations ◽  
Degenerate Principal Series ◽  
Special Case ◽  
Adic Case ◽  
Principal Series Representations ◽  
Whittaker Models

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

On the Super-Unitarity of Discrete Series Representations of Orthosymplectic Lie Superalgebras

1998 ◽  
Vol 10 (04) ◽  
pp. 467-497
Author(s):  
Amine M. El Gradechi
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Lie Superalgebras ◽  
Point Of View ◽  
Low Rank ◽  
Holomorphic Discrete Series ◽  
Computational Point ◽  
First Order ◽  
Series Representations ◽  
Discrete Series Representations

We investigate the notion of super-unitarity from a functional analytic point of view. For this purpose we consider examples of explicit realizations of a certain type of irreducible representations of low rank orthosymplectic Lie superalgebras which are super-unitary by construction. These are the so-called superholomorphic discrete series representations of osp (1/2,ℝ) and osp (2/2,ℝ) which we recently constructed using a ℤ2–graded extension of the orbit method. It turns out here that super-unitarity of these representations is a consequence of the self-adjointness of two pairs of anticommuting operators which act in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su (1,1) such that the difference of the respective lowest weights is [Formula: see text]. At an intermediate stage, we show that the generators of the considered orthosymplectic Lie superalgebras can be realized either as matrix-valued first order differential operators or as first order differential superoperators. Even though the former realization is less convenient than the latter from the computational point of view, it has the advantage of avoiding the use of anticommuting Grassmann variables, and is moreover important for our analysis of super-unitarity. The latter emphasizes the fundamental role played by the atypical (or degenerate) superholomorphic discrete series representations of osp (2/2,ℝ) for the super-unitarity of the other representations considered in this work, and shows that the anticommuting (unbounded) self-adjoint operators mentioned above anticommute in a proper sense, thus connecting our work with the analysis of supersymmetric quantum mechanics.

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