Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

On the occurrence of admissible representations in the real Howe correspondence in stable range

Let be an irreducible real reductive dual pair of type I in stable range, with G the smaller member. In this note, we prove that all irreducible genuine representations of occur in the Howe correspondence. The proof uses structural information about the groups forming a reductive dual pair and estimates of matrix coefficients.     

Closure ordering and the Kostant-Sekiguchi correspondence

Let be a real semisimple Lie group with Lie algebra . The Kostant-Sekiguchi correspondence is a bijection between nilpotent orbits on and nilpotent orbits on . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant-Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

     

Miura opers and critical points of master functions

Critical points of a master function associated to a simple Lie algebra come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population. Supported in part by NSF grant DMS-0140460 Supported in part by NSF grant DMS-0244579     

An equivariant Novikov conjecture

We discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This will be the statement that certain G-equivariant higher signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy equivalences preserving orientation. We prove this conjecture for manifolds modeled on a complete Riemannian manifold of nonpositive curvature on which G (a compact Lie group) acts by isometries. We also use the theory of harmonic maps to construct (in some cases) G-maps into such model spaces.Dedicated to Alexander GrothendieckPartially supported by NSF Grants DMS 84-00900 and 87-00551.Partially supported by NSF Grant DMS 86-02980, a Presidential Young Investigator Award, and a Sloan Foundation Fellowship.     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

On connectedness of sets in the real spectra of polynomial rings

Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on R ⁿ can be obtained from the polynomial ring R[x ₁,..., x _n ] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points , the existence of connected sets in the real spectrum of R[x ₁,..., x _n ], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.     

Geometric parameters corresponding to representations of the classical groups with integral infinitesimal characters

In this paper we calculate the orbits of flag manifolds of the complex classical groups under the action of the sets of fixed points of Cartan involutions, and determine all the geometric parameters corresponding to representations of the classical groups with integral infinitesimal characters, which are used to discuss Arthur conjecture and the Langlands classification of the irreducible admissible representations of real classical groups(see[1])Supported by the National Natural Science Foundation of China.The second author is supported by the Post-Doctor's Foundation of China.     

The Conley conjecture for the cotangent bundle

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems, similar results have been proven by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg’s proof of the Conley conjecture for closed symplectically aspherical manifolds.     

Local geometry of orbits for an ordinary classical lie supergroup

In this paper we identify a real reductive dual pair of Roger Howe with an Ordinary Classical Lie supergroup. In these terms we describe the semisimple orbits of the dual pair in the symplectic space, a slice through a semisimple element of the symplectic space, an analog of a Cartan subalgebra, the corresponding Weyl group and the corresponding Weyl integration formula.     

The Width-Volume Inequality

We prove that a bounded open set U in has k-width less than C(n) Volume(U) ^k/n . Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in . In particular, we estimate the smallest (n – 1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor. Received: January 2006, Revision: May 2006, Accepted: June 2006     

n维可投影LOtka-Volterra竞争系统的渐近性

对于二维和三维的Lotka-Volterra竞争系统,已有文献证明:当每一个坐标轴上的平衡点均为渐近稳定时,该系统几乎所有解趋于坐标轴上平衡点所组成的点集,即,不趋于坐标轴上平衡点的解集,其测度为零.由此, van den Driessche和Zeeman于1998年提出猜测:对n(n>3)维Lotka-Volterra竞争系统,当每一个坐标轴上的平衡点均为渐近稳定时,该系统几乎所有解趋于坐标轴上平衡点所组成的点集,即,不趋于坐标轴上平衡点的解集,其在n维空间的测度为零.本文证明当n维Lotka-Volterra竞争系统可被逐维投影到一维系统时,该猜测成立,并给出了可投影条件的代数判据.本文所得结论包含了已有文献的结果.     

Boundaries and random walks on finitely generated infinite groups

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ϖΓ and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Noncommutative versions of the Singer-Wermer conjecture with linear left ϑ-derivations

The noncommutative Singer-Wermer conjecture states that every linear （possibly unbounded） derivation on a （possibly noncommutative） Banach algebra maps into its Jacobson radical. This conjecture is still an open question for more than thirty years. In this paper we approach this question via linear left θ-derivations.     

A new Fenchel dual problem in vector optimization

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.     

Groups acting on buildings, operator K-theory, and Novikov''s conjecture

The paper is devoted to the study of the KK-theory of Bruhat-Tits buildings. We develop a theory which is analogous to the corresponding theory for manifolds of nonpositive sectional curvature. We construct a C ^*-algebra and a Dirac element associated to any simplicial complex. In the case of buildings, we construct, moreover, a dual Dirac element and compute its KK-products with the Dirac element. As a consequence, we prove the Novikov conjecture for discrete subgroups of linear adelic groups. In our study, we develop a KK-theoretic Poincaré duality for non-Hausdorff manifolds.Dedicated to Alexander Grothendieck on his sixtieth birthday