A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group

1990 ◽  
Vol 42 (1) ◽  
pp. 28-49 ◽  
Author(s):  
Robert A. Proctor
Keyword(s):  
Representation Theory ◽  
Orthogonal Group ◽  
The Body ◽  
Orthogonal Groups ◽  
Tensor Representations ◽  
Combinatorial Construction ◽  
Theoretic Motivation

This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

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.

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 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 Singular Isometries in Orthogonal Groups

1977 ◽  
Vol 20 (2) ◽  
pp. 189-198 ◽  
Author(s):  
Georg Gunther
Keyword(s):  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Orthogonal Groups ◽  
Singular Subspace

In this paper, we study the behaviour of singular isometries in orthogonal groups. These are isometries whose path is a singular subspace. We shall prove that the path of such a singular isometry is always even-dimensional. We shall use this result to show that the subgroup of the orthogonal group On(K, Q) which is generated by the singular isometries is the commutator subgroup Ωn(K, Q).

scholarly journals Arithmetic of Orthogonal Groups (II)

1955 ◽  
Vol 9 ◽  
pp. 129-146 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Vector Space ◽  
Orthogonal Group ◽  
The Other ◽  
Vector Spaces ◽  
Structure Theory ◽  
Orthogonal Groups

In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

scholarly journals Involutions and commutators in orthogonal groups

1998 ◽  
Vol 65 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Frieder Knüppel ◽  
Gerd Thomsen
Keyword(s):  
Vector Space ◽  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Dimensional Vector ◽  
Orthogonal Groups ◽  
Commutative Field ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractSuppose we are given a regular symmetric bilinear from on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2. We want to write given elements of the commutator subgroup ω(V) (of the orthogonal group O(V)) and also of the kernel of the spinorial norm ker(Θ) as (short) products of involutions and as products of commutators

Some Supercuspidal Representations of Sp4(k)

1987 ◽  
Vol 39 (1) ◽  
pp. 1-7
Author(s):  
Charles Asmuth
Keyword(s):  
Orthogonal Group ◽  
Plancherel Formula ◽  
Weil Representation ◽  
Orthogonal Groups ◽  
Explicit Formulas ◽  
Weil Representations ◽  
Supercuspidal Representations ◽  
Residual Characteristic

The purpose of this paper is to produce explicit realizations of supercuspidal representations of Sp4(k) where k is a p-adic field with odd residual characteristic. These representations will be constructed using the Weil representation of Sp4(k) associated with a certain 4-dimensional compact orthogonal group OQ over k. The main problem addressed in this paper is the analysis of this representation; we need to find how the supercuspidal summands decompose into irreducible pieces.The problem of decomposing Weil representations has been studied quite a bit already. The Weil representations of SL2(k) associated to 2-dimensional orthogonal groups were used by Casselman [4] and Shalika [9] to produce all supercuspidals of SL2(k). The explicit formulas for these representations were used by Sally and Shalika ([10]) to compute the characters and finally to write down a Plancherel formula for that group.

scholarly journals Commutators in pseudo-orthogonal groups

1995 ◽  
Vol 59 (3) ◽  
pp. 353-365 ◽  
Author(s):  
F. A. Arlinghaus ◽  
L. N. Vaserstein ◽  
Hong You
Keyword(s):  
Commutative Ring ◽  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Stable Condition ◽  
Stable Rank ◽  
Orthogonal Groups

AbstractWe study commutators in pseudo-orthogonal groups O2nR (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR) ≤ 4 if R satisfies the ∧-stable condition and either n ≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR) ≤ 3 when the involution is trivial and ∧ = R∈. We also show that c(O2nR) ≤ 3 and c(GO2nR) ≤ 2 for the ordinary orthogonal group O2nR over a commutative ring R of absolute stable rank 1 where either n ≥ 3 or n = 2 and 1 is the sum of two units in R.

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