Remarks on Certain Metaplectic Groups

1998 ◽  
Vol 41 (4) ◽  
pp. 488-496 ◽  
Author(s):  
Heng Sun
Keyword(s):  
Automorphic Forms ◽  
Reductive Group ◽  
Open Subgroup ◽  
Global Section ◽  
List Type ◽  
Metaplectic Groups ◽  
Unipotent Subgroup ◽  
Simple Chevalley Group ◽  
Simply Connected ◽  
Archimedean Place

AbstractWe study metaplectic coverings of the adelized group of a split connected reductive group G over a number field F. Assume its derived group G′ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we1.construct metaplectic coverings of G(A) from those of G′(A);2.for any non-archimedean place v, show the section for a covering of G(Fv) constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of G(Fv);3.define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

On some arithmetic properties of automorphic forms ofGLmover a division algebra

2014 ◽  
Vol 10 (04) ◽  
pp. 963-1013 ◽  
Author(s):  
Harald Grobner ◽  
A. Raghuram
Keyword(s):  
Division Algebra ◽  
Automorphic Forms ◽  
Local Version ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Central Division ◽  
Arithmetic Properties ◽  
Central Division Algebra ◽  
Archimedean Place ◽  
Split Form

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GLm/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e. D = F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke–Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G′ to statements on automorphic forms of its split form using the global Jacquet–Langlands correspondence developed by Badulescu and Badulescu–Renard. Beside that we prove that the local version of the Jacquet–Langlands transfer at an archimedean place preserves the property of being cohomological.

scholarly journals TOWARDS THE STANDARD MODEL SPECTRUM FROM ELLIPTIC CALABI–YAU MANIFOLDS

2004 ◽  
Vol 19 (12) ◽  
pp. 1987-2014 ◽  
Author(s):  
BJÖRN ANDREAS ◽  
GOTTFRIED CURIO ◽  
ALBRECHT KLEMM
Keyword(s):  
Standard Model ◽  
Global Section ◽  
Elliptic Surfaces ◽  
Spectral Cover ◽  
Elliptic Fiber ◽  
The Standard Model ◽  
Free Involution ◽  
Simply Connected ◽  
Rational Elliptic Surfaces ◽  
Standard Model Gauge Group

We show that it is possible to construct supersymmetric three-generation models with the Standard Model gauge group in the framework of non-simply-connected elliptically fibered Calabi–Yau threefolds, without section but with a bi-section. The fibrations on a cover Calabi–Yau threefold, where the model has six generations of SU(5) and the bundle is given via the spectral cover description, use a different description of the elliptic fiber which leads to more than one global section. We present two examples of a possible cover Calabi–Yau threefold with a free involution: one is a fiber product of rational elliptic surfaces dP9; another example is an elliptic fibration over a Hirzebruch surface. We compute the necessary amount of chiral matter by "turning on" a further parameter which is related to singularities of the fibration and the branching of the spectral cover.

scholarly journals A CRITERION FOR HOMOGENEOUS PRINCIPAL BUNDLES

2010 ◽  
Vol 21 (12) ◽  
pp. 1633-1638
Author(s):  
INDRANIL BISWAS ◽  
GÜNTHER TRAUTMANN
Keyword(s):  
Vector Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Reductive Group ◽  
Linear Algebraic Group ◽  
Principal Bundles ◽  
Simply Connected

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle EH → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad (EH) is homogeneous. Fix a faithful H-module V. We also show that EH is homogeneous if the vector bundle EH ×H V associated to it for the H-module V is homogeneous.

scholarly journals ON A CERTAIN LOCAL IDENTITY FOR LAPID–MAO'S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE

2021 ◽  
pp. 1-55
Author(s):  
Kazuki Morimoto
Keyword(s):  
Explicit Formula ◽  
Unitary Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Unitary Groups ◽  
Reductive Groups ◽  
Local Identity ◽  
Metaplectic Groups ◽  
Group Case ◽  
Formal Degree

Abstract Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

scholarly journals Sur une conjecture de Breuil–Herzig

2019 ◽  
Vol 2019 (751) ◽  
pp. 91-119 ◽  
Author(s):  
Julien Hauseux
Keyword(s):  
Parabolic Subgroup ◽  
Reductive Group ◽  
Principal Series ◽  
Nous Obtenons ◽  
Derived Subgroup ◽  
Nous Montrons ◽  
Simply Connected ◽  
Simplement Connexe ◽  
Do So

AbstractSoit G un groupe réductif p-adique de centre connexe et de groupe dérivé simplement connexe. Nous montrons que certaines “chaînes ” de séries principales de G n’existent pas et nous établissons plusieurs propriétés de la construction \Pi(\rho)^{\mathrm{ord}} de Breuil–Herzig. En particulier, nous obtenons une caractérisation naturelle de cette dernière et nous démontrons une conjecture de Breuil–Herzig. Pour cela, nous calculons le δ-foncteur \mathrm{H^{\bullet}Ord}_{P} des parties ordinaires dérivées d’Emerton relatif à un sous-groupe parabolique P de G sur une série principale. Nous énonçons une nouvelle conjecture sur les extensions entre représentations lisses modulo p de G obtenues par induction parabolique à partir de représentations supersingulières de sous-groupes de Levi de G et nous la démontrons pour les extensions par une série principale. Let G be a split p-adic reductive group with connected centre and simply connected derived subgroup. We show that certain “chains” of principal series of G do not exist and we establish several properties of the Breuil–Herzig construction \Pi(\rho)^{\mathrm{ord}}. In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil–Herzig. In order to do so, we partially compute Emerton’s δ-functor \operatorname{H^{\bullet}Ord}_{P} of derived ordinary parts with respect to a parabolic subgroup on a principal series. We formulate a new conjecture on the extensions between smooth mod p representations of G parabolically induced from supersingular representations of Levi subgroups of G and we prove it in the case of extensions by a principal series.

scholarly journals p-ADIC INTERPOLATION OF METAPLECTIC FORMS OF COHOMOLOGICAL TYPE

2012 ◽  
Vol 08 (07) ◽  
pp. 1613-1660
Author(s):  
RICHARD HILL ◽  
DAVID LOEFFLER
Keyword(s):  
Number Field ◽  
Automorphic Forms ◽  
Reductive Group

Let G be a reductive group over a number field k. It is shown how Emerton's methods may be applied to the problem of p-adically interpolating the metaplectic forms on G, i.e. the automorphic forms on metaplectic covers of G, as long as the metaplectic covers involved split at the infinite places of k.

scholarly journals GLUING OF ABELIAN CATEGORIES AND DIFFERENTIAL OPERATORS ON THE BASIC AFFINE SPACE

2002 ◽  
Vol 1 (4) ◽  
pp. 543-557 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Alexander Braverman ◽  
Leonid Positselskii
Keyword(s):  
Fourier Transform ◽  
Affine Space ◽  
Differential Operators ◽  
General Notion ◽  
Unipotent Subgroup ◽  
D Modules ◽  
Abelian Categories ◽  
The Fourier Transform ◽  
Simply Connected ◽  
Basic Affine

The notion of gluing of abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied further by Polishchuk. We observe that this notion is a particular case of a general categorical construction.We then apply this general notion to the study of the ring of global differential operators $\mathcal{D}$ on the basic affine space $G/U$ (here $G$ is a semi-simple simply connected algebraic group over $\mathbb{C}$ and $U\subset G$ is a maximal unipotent subgroup).We show that the category of $\mathcal{D}$-modules is glued from $|W|$ copies of the category of $D$-modules on $G/U$ where $W$ is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that the algebra $\mathcal{D}$ is Noetherian, and get some information on its homological properties.AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20

PERIODS OF AUTOMORPHIC FORMS: THE TRILINEAR CASE

2015 ◽  
Vol 17 (1) ◽  
pp. 59-74 ◽  
Author(s):  
Shunsuke Yamana
Keyword(s):  
Eisenstein Series ◽  
Automorphic Forms ◽  
Metaplectic Groups ◽  
Periods Of Automorphic Forms

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

Convex polygons as carriers

2016 ◽  
Vol 100 (547) ◽  
pp. 93-102
Author(s):  
G. C. Shephard
Keyword(s):  
List Type ◽  
Similar Material ◽  
Convex Polygons ◽  
Regular Octahedron ◽  
Simply Connected

We shall use the word ‘polyhedron’ to mean a connected, simply-connected 3-polytope of positive (non-zero) volume. The idea of the net of a polyhedron P is well known. For example, the regular octahedron has eleven distinct nets, three of which are shown in Figure 1. A net consists of three parts:(a) A plane connected and simply-connected polygon Q (denoted by heavy lines in the diagrams), known as the carrier of the net;(b) A set of lines known as, fold-lines in the interior of Q;(c) A labelling of the edges of Q.If one cuts Q out of paper or similar material, folds it along the fold-lines, and then pastes together edges with matching labels, one obtains a model of the polyhedron P. We say that a net is convex if, and only if, its carrier Q is convex.

scholarly journals Cuspidal cohomology of stacks of shtukas

2020 ◽  
Vol 156 (6) ◽  
pp. 1079-1151
Author(s):  
Cong Xue
Keyword(s):  
Finite Field ◽  
Compact Support ◽  
Function Field ◽  
Automorphic Forms ◽  
Finite Dimension ◽  
Reductive Group ◽  
Constant Term ◽  
Cuspidal Cohomology

Let $G$ be a connected split reductive group over a finite field $\mathbb{F}_{q}$ and $X$ a smooth projective geometrically connected curve over $\mathbb{F}_{q}$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

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