scholarly journals Theta series liftings from orthogonal groups to semi-simple groups

2000 ◽  
Vol 69 (1) ◽  
pp. 127-142
Author(s):  
Min Ho Lee
Keyword(s):  
Half Space ◽  
Lie Group ◽  
Orthogonal Group ◽  
Automorphic Form ◽  
Automorphic Forms ◽  
Theta Series ◽  
Symmetric Domain ◽  
Simple Groups ◽  
Orthogonal Groups ◽  
Spin Groups

AbstractWe study a correspondence between automorphic forms on an orthogonal group and automorpbic forms on a semi-simple Lie group associated to an equivariant holomorphic map of a symmetric domain into a Siegel upper half space. We construct an automorphic form on the symmetric domain thatg corresponds to an automorphic form on an orthogonal group using theta series, and prove that such a correspondence is compatible with the appropriate Hecks operator actions on the corresponding automorphic forms. As an example, we describe the case of spin groups.

scholarly journals On standard L-functions attached to automorphic forms on definite orthogonal groups

1998 ◽  
Vol 152 ◽  
pp. 57-96 ◽  
Author(s):  
Atsushi Murase ◽  
Takashi Sugano
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Orthogonal Group ◽  
Automorphic Form ◽  
Automorphic Forms ◽  
Algebraic Number Field ◽  
Constant Function ◽  
Orthogonal Groups ◽  
Totally Real

Abstract.We show an explicit functional equation of the standard L-function associated with an automorphic form on a definite orthogonal group over a totally real algebraic number field. This is a completion and a generalization of our previous paper, in which we constructed standard L-functions by using Rankin-Selberg convolution and the theory of Shintani functions under certain technical conditions. In this article we remove these conditions. Furthermore we show that the L-function of f has a pole at s = m/2 if and only if f is a constant function.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals RANKIN–COHEN TYPE DIFFERENTIAL OPERATORS FOR SIEGEL MODULAR FORMS

1998 ◽  
Vol 09 (04) ◽  
pp. 443-463 ◽  
Author(s):  
WOLFGANG EHOLZER ◽  
TOMOYOSHI IBUKIYAMA
Keyword(s):  
Half Space ◽  
Modular Forms ◽  
Automorphic Form ◽  
Differential Operators ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Elliptic Case ◽  
Vector Valued

Let ℍn be the Siegel upper half space and let F and G be automorphic forms on ℍn of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on ℍn × ℍn such that the restriction of [Formula: see text] to Z = Z1 = Z2 is again an automorphic form of weight k + l + v on ℍn. Since the elliptic case, i.e. n = 1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin–Cohen type operators. We also discuss a generalisation of Rankin–Cohen type operators to vector valued differential operators.

scholarly journals WEIGHT CHANGING OPERATORS FOR AUTOMORPHIC FORMS ON GRASSMANNIANS AND DIFFERENTIAL PROPERTIES OF CERTAIN THETA LIFTS

2016 ◽  
Vol 228 ◽  
pp. 186-221 ◽  
Author(s):  
SHAUL ZEMEL
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Orthogonal Groups ◽  
Basic Properties ◽  
Differential Properties

We define weight changing operators for automorphic forms on Grassmannians, that is, on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.

scholarly journals Boundary behaviour of special cohomology classes arising from the Weil representation

2012 ◽  
Vol 12 (3) ◽  
pp. 571-634 ◽  
Author(s):  
Jens Funke ◽  
John Millson
Keyword(s):  
Modular Forms ◽  
Symmetric Spaces ◽  
Theta Functions ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Boundary Behaviour ◽  
Orthogonal Groups ◽  
Local Coefficients ◽  
Generating Series ◽  
Vector Valued

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

On Limit Multiplicities for Spaces of Automorphic Forms

1999 ◽  
Vol 51 (5) ◽  
pp. 952-976 ◽  
Author(s):  
Anton Deitmar ◽  
Werner Hoffmann
Keyword(s):  
Lie Group ◽  
Automorphic Forms ◽  
Plancherel Measure ◽  
Semisimple Lie Group ◽  
Selberg Trace Formula ◽  
Arithmetic Subgroup ◽  
Trivial Group ◽  
Rank One ◽  
Discrete Part ◽  
Cocompact Lattices

AbstractLet Γ be a rank-one arithmetic subgroup of a semisimple Lie group G. For fixed K-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of G, whose discrete part encodes the dimensions of the spaces of square-integrable Γ-automorphic forms. It is shown that this distribution converges to the Plancherel measure of G when Γ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices Γ follows from results of DeGeorge-Wallach and Delorme.

The Construction of Automorphic Forms From the Derivatives of a Given Form II

1985 ◽  
Vol 28 (3) ◽  
pp. 306-316
Author(s):  
R. A. Rankin
Keyword(s):  
Automorphic Form ◽  
Symmetric Functions ◽  
Automorphic Forms ◽  
Explicit Constructions ◽  
Derivatives Of

AbstractExplicit constructions of polynomials of preassigned degree and weight in the derivatives of a given automorphic form are described and studied, supplementing the results of an earlier paper. It turns out that the problem is essentially one concerning symmetric functions rather than automorphic forms.

Representations of Lie Groups by Contact Transformations, II: Non-Compact Simple Groups

1993 ◽  
Vol 45 (4) ◽  
pp. 778-802
Author(s):  
Carl Herz
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Compact Manifold ◽  
Transitive Group ◽  
Simple Groups ◽  
Representations Of Lie Groups ◽  
Contact Transformations

AbstractIf a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

scholarly journals Harmonic analysis on the quotient spaces of Heisenberg groups

1991 ◽  
Vol 123 ◽  
pp. 103-117 ◽  
Author(s):  
Jae-Hyun Yang
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Unitary Representation ◽  
Lie Group ◽  
Theta Series ◽  
Foundations Of Quantum Mechanics ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Quotient Spaces

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

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