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 TILING BRANCHING MULTIPLICITY SPACES WITH GL PATTERN BLOCKS

2013 ◽  
Vol 94 (3) ◽  
pp. 362-374 ◽  
Author(s):  
SANGJIB KIM
Keyword(s):  
Classical Groups ◽  
Orthogonal Groups ◽  
Branching Rules

AbstractWe study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

Poles ofL-functions and theta liftings for orthogonal groups

2009 ◽  
Vol 8 (4) ◽  
pp. 693-741 ◽  
Author(s):  
David Ginzburg ◽  
Dihua Jiang ◽  
David Soudry
Keyword(s):  
Eisenstein Series ◽  
Orthogonal Group ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Automorphic Representations ◽  
Domain Of Holomorphy ◽  
First Occurrence ◽  
Cuspidal Automorphic Representations

AbstractIn this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standardL-functions.

scholarly journals Symmetry breaking for representations of rank one orthogonal groups

2015 ◽  
Vol 238 (1126) ◽  
pp. 0-0 ◽  
Author(s):  
Toshiyuki Kobayashi ◽  
Birgit Speh
Keyword(s):  
Symmetry Breaking ◽  
Orthogonal Groups ◽  
Rank One

Restriction of irreducible representations of groups to a subgroup

1971 ◽  
Vol 322 (1549) ◽  
pp. 181-189 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Irreducible Representation ◽  
Symmetry Breaking ◽  
Irreducible Representations ◽  
Representations Of Groups ◽  
Group Play ◽  
Special Case

The number and character of the irreducible representations of a subgroup, contained in an irreducible representation of the whole group (if this representation is restricted to the sub­ group) play an important role in quantum mechanics. They give the number and type of the states, generated by a symmetry breaking perturbation, from a state which has the symmetry of the whole group. Three equations are derived here for the number and character of the representations of the subgroup, resulting from the restriction of the irreducible represen­tations of the whole group. These equations contain an earlier rule as a special case.

scholarly journals A theorem on power series with applications to classical groups over finite fields

2001 ◽  
Vol 64 (1) ◽  
pp. 121-129
Author(s):  
Andrew J. Spencer
Keyword(s):  
Power Series ◽  
Finite Fields ◽  
Generating Functions ◽  
General Linear ◽  
Classical Groups ◽  
Orthogonal Groups

For some of the classical groups over finite fields it is possible to express the proportion of eigenvalue-free matrices in terms of generating functions. We prove a theorem on the monotonicity of the coefficients of powers of power series and apply this to the generating functions of the general linear, symplectic and orthogonal groups. This proves a conjecture on the monotonicity of the proportions of eigenvalue-free elements in these groups.

scholarly journals Factorial Characters and Tokuyama's Identity for Classical Groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Recurrence Relations ◽  
Function Theory ◽  
Lattice Paths ◽  
Schur Function ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Schubert Polynomial ◽  
International Audience ◽  
Symplectic Case ◽  
Polynomial Theory

International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.

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.

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