scholarly journals On the global Gan-Gross-Prasad conjecture for general spin groups

2018 ◽  
Author(s):  
◽  
Melissa Emory
Keyword(s):  
Classical Groups ◽  
Spin Groups ◽  
Automorphic Representations ◽  
Spin Group ◽  
Central Value ◽  
Period Integral

In the 1990s, Benedict Gross and Dipendra Prasad formulated an intriguing conjecture connected with restriction laws for automorphic representations of a particular group. More recently, Gan, Gross, and Prasad extended this conjecture, now known as the Gan-Gross-Prasad Conjecture, to the remaining classical groups. Roughly speaking, they conjectured the non-vanishing of a certain period integral is equivalent to the non-vanishing of the central value of a certain L- function. Ichino and Ikeda refined the conjecture to give an explicit relationship between this central value of a L-function and the period integral. We propose a similar conjecture for a nonclassical group, the general spin group, and prove one case.

scholarly journals A reciprocal branching problem for automorphic representations and global Vogan packets

2020 ◽  
Vol 2020 (765) ◽  
pp. 249-277 ◽  
Author(s):  
Dihua Jiang ◽  
Baiying Liu ◽  
Bin Xu
Keyword(s):  
Irreducible Representation ◽  
Symmetry Breaking ◽  
Descent Method ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Automorphic Representations ◽  
Branching Rule ◽  
Branching Problem

AbstractLet G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.

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 On fields of rationality for automorphic representations

2014 ◽  
Vol 150 (12) ◽  
pp. 2003-2053 ◽  
Author(s):  
Sug Woo Shin ◽  
Nicolas Templier
Keyword(s):  
Isomorphism Class ◽  
Discrete Series ◽  
Automorphic Representation ◽  
Linear Groups ◽  
Classical Groups ◽  
Automorphic Representations ◽  
General Linear Groups ◽  
Mild Conditions ◽  
Finite Set ◽  
Level Aspect

AbstractThis paper proves two results on the field of rationality$\mathbb{Q}({\it\pi})$for an automorphic representation${\it\pi}$, which is the subfield of$\mathbb{C}$fixed under the subgroup of$\text{Aut}(\mathbb{C})$stabilizing the isomorphism class of the finite part of${\it\pi}$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations${\it\pi}$such that${\it\pi}$is unramified away from a fixed finite set of places,${\it\pi}_{\infty }$has a fixed infinitesimal character, and$[\mathbb{Q}({\it\pi}):\mathbb{Q}]$is bounded. The second main result is that for classical groups,$[\mathbb{Q}({\it\pi}):\mathbb{Q}]$grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed$L$-packet under mild conditions.

Ternary composition algebras II. Automorphism groups and subgroups

1990 ◽  
Vol 431 (1881) ◽  
pp. 21-36 ◽  
Keyword(s):  
Lie Algebra ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Complex Numbers ◽  
Composition Algebra ◽  
Spin Groups ◽  
Spin Group ◽  
Composition Algebras ◽  
Complex Construction

We study an eight-dimensional ternary composition algebra ε = ( E , ⟨ ⟩, { }) of signature (8, 0) or (4, 4). Such an algebra is associated with an involutary outer automorphism M of the Lie algebra so ( E )of ⟨, ⟩. The automorphisms and (in the (4, 4) case) counter-automorphisms of ε are obtained in terms of the d = 7 spin group determined by M . Using M we derive a ‘principle of duplicity’ for the ternary multiplication { } (related to the well-known principle of triality for the associated octonionic binary multiplication). However, rather than construct ε out of the octonions as done (in effect) by previous authors, we give a four-dimensional ‘complex’ construction of ε . This non-octonionic view of ε highlights certain (15-dimensional) d = 6 spin groups, rather than the customary (14-dimensional) octonionic automorphism groups. In the (4, 4) case we can choose to interpret ‘complex’ in terms of the split complex numbers, and are thereby led to consider the subgroup chain SO + (4, 4) ⊃ Spin + (3, 4) ⊃ SL (4; ℝ) (⋍ Spin + (3, 3)) instead of the chain SO + (4, 4) ⊃ Spin + (3, 4) ⊃ SU (2, 2) (⋍ Spin + (2, 4)).

A Representation of the Euclidean Group by Spin Groups, and Spatial Kinematics Mappings

1990 ◽  
Vol 112 (1) ◽  
pp. 42-49 ◽  
Author(s):  
J. M. Rico ◽  
J. Duffy
Keyword(s):  
Three Dimensional ◽  
Spin Representation ◽  
Euclidean Group ◽  
Spin Groups ◽  
Spin Group ◽  
Dimensional Vector Space ◽  
Orthogonal Space ◽  
Algebra Representation ◽  
Euclidean Motion ◽  
Spatial Kinematics

A new derivation of the spin and biquaternion representation of the Euclidean group is presented. The derivation is based upon the even Clifford algebra representation of the orientation preserving orthogonal automorphisms of nondegenerate orthogonal spaces, also called spin representation. Embedding the degenerate orthogonal space IR1,0,3 into the nondegenerate orthogonal space IR1,4, and imposing certain conditions on the orthogonal automorphisms of IR1,4, one obtains a subgroup of the spin group. The action of this subgroup, on a subspace of IR1,4, is isomorphic to IR1,0,3, is precisely a Euclidean motion. The conditions imposed on the orthogonal automorphisms of IR1,4 lead to the biquaternion representation. Furthermore, the invariants of the representations are easily obtained. The derivation also allows the spin representation to be related to the action of the representation over an element of a three-dimensional vector space proposed by Porteous, and used by Selig. As a byproduct, the derivation provides an insightful interpretation of the dual unit used in both the spin representation and the biquaternion representation.

Classical Groups, Derangements and Primes

2015 ◽  
Author(s):  
Timothy C. Burness ◽  
Michael Giudici
Keyword(s):  
Classical Groups
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