Any Unitary Principal Series Representation of GL n over a p-Adic Field is Irreducible

1976 ◽  
Vol 54 (1) ◽  
pp. 376
Author(s):  
Roger Howe ◽  
Allan Silberger
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Principal Series Representation ◽  
Unitary Principal Series

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$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

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.

scholarly journals Highest Weights for Categorical Representations

2018 ◽  
Vol 2020 (24) ◽  
pp. 9988-10004 ◽  
Author(s):  
David Ben-Zvi ◽  
Sam Gunningham ◽  
Hendrik Orem
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Reductive Groups ◽  
Monoidal Categories ◽  
Principal Series Representation ◽  
Categorical Representation ◽  
Morita Equivalent ◽  
Highest Weight ◽  
D Modules ◽  
De Rham

Abstract We present a criterion for establishing Morita equivalence of monoidal categories and apply it to the categorical representation theory of reductive groups $G$. We show that the “de Rham group algebra” $\mathcal D(G)$ (the monoidal category of $\mathcal D$-modules on $G$) is Morita equivalent to the universal Hecke category $\mathcal D({N}\backslash{G}/{N})$ and to its monodromic variant $\widetilde{\mathcal D}({B}\backslash{G}/{B})$. In other words, de Rham $G$-categories, that is, module categories for $\mathcal D(G)$, satisfy a “highest weight theorem”—they all appear in the decomposition of the universal principal series representation $\mathcal D(G/N)$ or in twisted $\mathcal D$-modules on the flag variety $\widetilde{\mathcal D}(G/B)$.

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