Fourier supports of K-finite Bessel integrals on classical tube domains

2018 ◽  
Vol 29 (04) ◽  
pp. 1850025
Author(s):  
Tetsuya Kobana ◽  
Kaoru Kodaira ◽  
Takuya Miyazaki
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Upper Bounds ◽  
Module Structure ◽  
Tube Domain ◽  
Principal Series Representation ◽  
Shilov Boundary ◽  
Text Types ◽  
Tube Domains ◽  
Degenerate Principal Series

Let [Formula: see text] be the symmetric tube domain associated with the Jordan algebra [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text], and [Formula: see text] be its Shilov boundary. Also, let [Formula: see text] be a degenerate principal series representation of [Formula: see text]. Then we investigate the Bessel integrals assigned to functions in general [Formula: see text]-types of [Formula: see text]. We give individual upper bounds of their supports, when [Formula: see text] is reducible. We also use the upper bounds to give a partition for the set of all [Formula: see text]-types in [Formula: see text], that turns out to explain the [Formula: see text]-module structure of [Formula: see text]. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [[Formula: see text]-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic [Formula: see text]-supports assigned to [Formula: see text]-submodules in [Formula: see text].

TWISTED MELLIN TRANSFORMS OF A REAL ANALYTIC RESIDUE OF SIEGEL–EISENSTEIN SERIES OF DEGREE 2

2009 ◽  
Vol 20 (08) ◽  
pp. 1011-1027 ◽  
Author(s):  
YASUKO HASEGAWA ◽  
TAKUYA MIYAZAKI
Keyword(s):  
Eisenstein Series ◽  
Functional Equations ◽  
Series Representation ◽  
Principal Series ◽  
Principal Series Representation ◽  
Derived Functor ◽  
Mellin Transforms ◽  
Wave Forms ◽  
Degenerate Principal Series ◽  
Real Analytic

We study a residual form of a real analytic Siegel–Eisenstein series, which generates a certain derived functor module occurring in a degenerate principal series representation. We compute its Mellin transforms twisted by various Maass wave forms to get explicit formulas as our results. We apply them to prove meromorphic continuations together with functional equations which are satisfied by those twisted Mellin transforms.

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 REGULAR FUNCTIONS ON THE SHILOV BOUNDARY

2005 ◽  
Vol 04 (06) ◽  
pp. 613-629 ◽  
Author(s):  
OLGA BERSHTEIN
Keyword(s):  
Symmetric Domain ◽  
Principal Series ◽  
Bounded Symmetric Domain ◽  
Bounded Symmetric Domains ◽  
Shilov Boundary ◽  
Generators And Relations ◽  
Regular Functions ◽  
Degenerate Principal Series

In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = Un is investigated.

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.

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