Conductors and newforms for U(1,1)

Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms forU (1, 1)(F), building on previous work onSL ₂(F). This theory is analogous to the results of Casselman forGL ₂(F) and Jacquet, Piatetski-Shapiro, and Shalika forGL _n(F). To a representation π ofU(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on theL-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.     

On newforms for Kohnen plus spaces

In this article, we investigate the plus space of level N, where 4^?1 N is a square-free (not necessarily odd) integer. This is a generalization of Kohnen’s work. We define a Hecke isomorphism ${\wp_k}In this article, we investigate the plus space of level N, where 4⁻¹ N is a square-free (not necessarily odd) integer. This is a generalization of Kohnen’s work. We define a Hecke isomorphism ?_k{\wp_k} of M _k+1/2(4M) onto M_k+1/2⁺(8M){M_{k+1/2}^+(8M)} for any odd positive integer M. The methods of the proof of the newform theory are this isomorphism, Waldspurger’s theorem, and the dimension identity.     

The local theta correspondence for unramified unitary groups

We study the local theta correspondences for dual reductive pairs consisting of quasi-split unitary groups defined over a non-archimedean local field. We construct Howe’s correspondence between the set of spherical representations of the one group and that of the other group by using the Whittaker model.     

Basic functions and unramified local L-factors for split groups

According to a program of Braverman, Kazhdan and Ng, for a large class of split unramified reductive groups G and representations ρ of the dual group G, the unramified local L-factor L(s, π, ρ) can be expressed as the trace of π(f_(ρ,s)) for a function f_(ρ,s) with non-compact support whenever Re(s)0. Such a function should have useful interpretations in terms of geometry or combinatorics, and it can be plugged into the trace formula to study certain sums of automorphic L-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jacquet theory for(G, ρ). In this paper, we derive some basic properties for the basic functions f_(ρ,s) and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.     

On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D

Let D be a quaternion division algebra with an involution of the second kind. We show that the quotient group SU(1,D)/[U(1,D),U(1,D)] is nontrivial in general. For global fields, we completely determine this group. Received: 11 May 2007, Revised: 15 December 2007     

Finite Subgroups of Arithmetic Lattices in U (2, 1)

The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U (2, 1).     

不动点集为RP(4)∪ P(4,2n-1)的对合

设(M,T)是一个光滑闭流形上的对合,不动点集为F=RP(4)UP(4,2n-1),则它的每一个对合(M,T)必协边(RP(4)×RP(4),twist)和(P(4,2n),T')之一.     

不动点集为RP(8)(U) P(8,2n-1)的对合

设(M,T)是一个带有光滑对合T的光滑闭流形,T的不动点集为RP(8)() P(8,2n-1).本文证明了(M,T)必协边于(RP(8)×RP(8),twist)和(P(8,RP(2n)),T')之一.     

Topological automorphic forms on U(1, 1)

The homotopy type and homotopy groups of some spectra TAF _GU of topological automorphic forms associated to a unitary similitude group GU of type (1, 1) are explicitly described in quasi-split cases. The spectrum TAF _GU is shown to be closely related to the spectrum TMF in these cases, and homotopy groups of some of these spectra are explicitly computed.     

On 2-chromatic (v, 5, 1)-designs

In this note, we completely settle the existence of 2-chromatic (v, 5, 1)-designs. This settles a problem posed by Rosa and Colbourn.     

Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: the unramified case

In this note we partially extend results on the finite-ness properties of local cohomology modules from the case of a regular local ring containing a field to the unramified case of a regular local ring of mixed characteristic.     

U(n + 1) × U(p + 1)-Hermitian metrics on the manifold S 2n+1 × S 2p+1

A two-parameter family of invariant almost-complex structures J _α,c is given on the homogeneous space M × M’ = U(n + 1)/U(n) × U(p + 1)/U(p); all these structures are integrable. We consider all invariant Riemannian metrics on the homogeneous space M × M’. They depend on five parameters and are Hermitian with respect to some complex structure J _α,c . In this paper, we calculate the Ricci tensor, scalar curvature, and obtain estimates of the sectional curvature for any metric on M × M’. All the invariant metrics of nonnegative curvature are described. We obtain the extremal values of the scalar curvature functional on the four-parameter family of metrics g _{α,c,λ,λ’;1} .     

关于空间(k1,k2)-拟正则映射

本文考虑空间 (k1,k2 ) -拟正则映射的 Lp(p >n)可积性 ,以及当 k1→ 1 ,k2 → 0时 p的渐近行为 .