SOME VECTOR-VALUED SINGULAR AUTOMORPHIC FORMS ON U(2, 2) AND THEIR RESTRICTION TO Sp(1, 1)

2012 ◽  
Vol 23 (10) ◽  
pp. 1250104
Author(s):  
ATSUO YAMAUCHI ◽  
HIRO-AKI NARITA
Keyword(s):  
Hyperbolic Space ◽  
Symplectic Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Discrete Series ◽  
Theta Series ◽  
Explicit Construction ◽  
Discrete Series Representations ◽  
Vector Valued ◽  
Real Symplectic Group

In this paper we provide a construction of theta series on the real symplectic group of signature (1,1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2,2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

scholarly journals MORITA'S THEORY FOR THE SYMPLECTIC GROUPS

2011 ◽  
Vol 07 (08) ◽  
pp. 2115-2137 ◽  
Author(s):  
ZHI QI ◽  
CHANG YANG
Keyword(s):  
Symplectic Group ◽  
Discrete Series ◽  
Principal Series ◽  
Symplectic Groups ◽  
Holomorphic Discrete Series ◽  
Algebraic Interpretation ◽  
Series Representations ◽  
Discrete Series Representations ◽  
Principal Series Representations

We construct and study the holomorphic discrete series representations and the principal series representations of the symplectic group Sp (2n, F) over a p-adic field F as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for SL (2, F) defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.

scholarly journals Relative Discrete Series Representations for Two Quotients ofp-adic GLn

2018 ◽  
Vol 70 (6) ◽  
pp. 1339-1372 ◽  
Author(s):  
Jerrod Manford Smith
Keyword(s):  
Symmetric Space ◽  
Local Field ◽  
Galois Extension ◽  
Symmetric Spaces ◽  
Discrete Series ◽  
Explicit Construction ◽  
Series Representations ◽  
Discrete Series Representations ◽  
Square Integrability ◽  
Residual Characteristic

AbstractWe provide an explicit construction of representations in the discrete spectrum of twop-adic symmetric spaces. We consider GLn(F) × GLn(F)\GL2n(F) and GLn(F)\GLn(E), whereEis a quadratic Galois extension of a nonarchimedean local fieldFof characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman’s Criterion for square integrability due to Kato and Takano.

scholarly journals NOTE ON A PAPER OF J. HOFFSTEIN

2007 ◽  
Vol 49 (2) ◽  
pp. 243-255 ◽  
Author(s):  
S. J. PATTERSON
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Field ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Arbitrary Order ◽  
Theta Functions ◽  
Theta Series ◽  
Arithmetic Groups ◽  
Recent Developments

AbstractThe concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

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