A remark on the genericity of multiplicity results for forced oscillations on manifolds

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.?In particular, the genericity of the following property is investigated: if the differentiable manifold M is compact, then the equation _π=h(x,)+f(t,x,) on M has |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f. Received: January 24, 2000; in final form: January 16, 2001?Published online: March 19, 2002     

A remark on the failure of multiplicity one for GSp(4)

We revisit a classical result of Howe and Piatetski-Shapiro on the failure of strong multiplicity one for GSp(4).     

U(3,Z_p),U(4,Z_p)及U(5,Z_p)的共轭类

通过计算相关矩阵的秩,求出了U(3,Z_p),U(4,Z_p)及U(5,Z_p)的共轭类.     

On strong multiplicity one for automorphic representations

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let π be a unitary, cuspidal, automorphic representation of . Let S be a set of finite places of K, such that the sum is convergent. Then π is uniquely determined by the collection of the local components finite} of π. Combining this theorem with base change, it is possible to consider sets S of positive density, having appropriate splitting behavior with respect to a solvable extension L of K, and where π is determined up to twisting by a character of the Galois group of L over K.     

A density theorem for (multiplicity,rank) pairs

For measure-preserving dynamical systems, we investigate the possible values of the pair (m, r), wherer is the maximal number of Rokhlin towers approximating the system andm is the maximal number of cyclic spaces approximating the associated unitary operator. We prove that in this way we can get every pair (d, n) for everyn≥3, and every divisord of some arithmetic function ψ(n) related to the Euler function. We show that for fixedn, the set of possibled is of density one.     

The multiplicity one conjecture for local theta correspondences

Over a non-archimedean local field of characteristic zero, we prove multiplicity preservation of local theta correspondences for orthogonal-symplectic dual pairs. The same proof works for dual pairs of unitary groups.     

A uniqueness theorem for representations of GSO(6) and the strong multiplicity one theorem for generic representations of GSp(4)

Consider the θ-correspondence from GSp(4) to GSO(6). We prove that locally over a nonarchimedean fieldF, this correspondence is injective on generic representations (i.e. with Whittaker model) of GSp(4,F). We use this to show the strong multiplicity one property for irreducible, automorphic, cuspidal representations of GSp(4,A), which are generic.     

Global Jacquet–Langlands correspondence,multiplicity one and classification of automorphic representations

In this paper we generalize the local Jacquet-Langlands correspondence to all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one Theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Mœglin–Waldspurger and Jacquet–Shalika for GL(n).     

The index of an Eisenstein ideal and multiplicity one

Mathematische Zeitschrift - Mazur’s fundamental work on the Eisenstein ideal of prime level has a variety of arithmetic applications. In this article, we generalize some of his work to...     

Strong multiplicity one theorems for affine Hecke algebras of type A

Given an irreducible module for the affine Hecke algebraH _n of type A, we consider its restriction toH _n–1. We prove that the socle of restriction is multiplicity free and moreover that the summands lie in distinct blocks. This is true regardless of the characteristic of the field or of the order of the parameterq in the definition ofH _n . The result generalizes and implies the classical branching rules that describe the restriction of an irreducible representation of the symmetric groupS _n toS _n–1.Both authors would like to thank Gus Lehrer and the University of Sydney for their hospitality while this paper was edited into its final form.     

Discrete Series Characters for GL(n, q)

The discrete series characters of the finite general linear group GL(n, q) are expressed as uniquely defined integral linear combinations of characters induced from linear characters on certain subgroups Hd, n of GL(n, q). The coefficients in these linear combinations are determined (for all n, q) by a family of polynomials r(T) Z[T] indexed by the set of all partitions .     

SU(3)-manifolds of cohomogeneity one

In this article, we classify 7- and 8-dimensional manifolds M admitting an SU(3) action of cohomogeneity one such that (i) M is simply connected and the orbit space M/G is isomorphic to [0, 1], and (ii) \({M/G\cong S^{1}}\) and the principal orbits are simply connected. We discuss applications to the study of the group manifold SU(3) and to 8-dimensional quaternion-Kähler spaces, and links between dimension 7 and 8 given by circle actions.