Geometric structure in smooth dual and local Langlands conjecture

This expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.     

The essentially tame local Langlands correspondence, I

Let be a non-Archimedean local field (of characteristic or ) with finite residue field of characteristic . An irreducible smooth representation of the Weil group of is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension is denoted . The Langlands correspondence induces a bijection of with a certain set of irreducible supercuspidal representations of . We consider the set of isomorphism classes of certain pairs , called ``admissible', consisting of a tamely ramified field extension of degree and a quasicharacter of . There is an obvious bijection of with . Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of with , generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of with . We show that one obtains the Langlands correspondence by composing the map with a permutation of of the form , where is a tamely ramified character of depending on . This answers a question of Moy (1986). We calculate the character in the case where is totally ramified of odd degree.

     

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.     

Between local and global logarithmic averages

We obtain an approximation for the logarithmic averages of I{k^1/2a(k) S(k) k^1/2b(k)}, where a(k) → 0, b(k) → 0 (k → ∞) and S(k) is partial sum of independent, identically distributed random variables.     

Factorization and Poisson correspondences

The Darboux transformation as an example of an integrable infinite-dimensional Poisson correspondence is discussed in the context of the general factorization problem. Generalizations related to energy-dependent Schrödinger operators and to Kac-Moody algebras are considered. We also present the finite-dimensional reductions of the Darboux transformation to stationary flows.Dedicated to the memory of Ira DorfmanTranslated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 225–245, November, 1995.     

Clone and hypervariety correspondences

The general correspondence between clones and hypervarieties was set out by Walter Taylor in [5], By mapping each varietyV to its cloneClone V Taylor obtained inverse isomorphisms between the lattice of all clone varieties and the lattice of all hypervarieties. Taylor also described a more specific correspondence in [6], using the 1-clone Cl₁ V of a varietyV. Since 1-clones are just monoids, he found a one-to-one correspondence from the uncountably infinite lattice of monoid varieties into the lattice of all hypervarieties. In this paper we show how Taylor's construction may be carried out forn-clones, for anyn 1. Thus we usen-clones to restrict the general correspondence to a family (_n) of correspondences, including the monoid correspondence as the special casen=1. These correspondences pick out certain families of hypervarieties, then-closed hypervarieties (forn 1), and we show that there are at least countably many such hypervarieties for eachn. This represents some progress towards the goal of understanding the structure of the lattice of all hypervarieties.Presented by W. Taylor.Research supported by NSERC of Canada.     

Analyzing local and global properties of multigraphs

The local structure of undirected multigraphs under two random multigraph models is analyzed and compared. The first model generates multigraphs by randomly coupling pairs of stubs according to a fixed degree sequence so that edge assignments to vertex pair sites are dependent. The second model is a simplification that ignores the dependency between the edge assignments. It is investigated when this ignorance is justified so that the simplified model can be used as an approximation, thus facilitating the structural analysis of network data with multiple relations and loops. The comparison is based on the local properties of multigraphs given by marginal distribution of edge multiplicities and some local properties that are aggregations of global properties.     

Tableaux and matrix correspondences

The Robinson-Schensted correspondence, a bijection between nonnegative matrices and pair of tableaux, is investigated. The representation, in the tableau, of the dihedral symmetries of the matrix are derived in a simple manner using a shape-preserving anti-isomorphism of the platic monoid. The Robinson-Schensted correspondence is shown to be equivalent to the Hillman-Grassl bijection between reverse plane partitions and tabloids. A generalized insertion method for the Robinson-Schensted correspondence is obtained.     

Balancing global and local search in parallel efficient global optimization algorithms

Most parallel efficient global optimization (EGO) algorithms focus only on the parallel architectures for producing multiple updating points, but give few attention to the balance between the global search (i.e., sampling in different areas of the search space) and local search (i.e., sampling more intensely in one promising area of the search space) of the updating points. In this study, a novel approach is proposed to apply this idea to further accelerate the search of parallel EGO algorithms. In each cycle of the proposed algorithm, all local maxima of expected improvement (EI) function are identified by a multi-modal optimization algorithm. Then the local EI maxima with value greater than a threshold are selected and candidates are sampled around these selected EI maxima. The results of numerical experiments show that, although the proposed parallel EGO algorithm needs more evaluations to find the optimum compared to the standard EGO algorithm, it is able to reduce the optimization cycles. Moreover, the proposed parallel EGO algorithm gains better results in terms of both number of cycles and evaluations compared to a state-of-the-art parallel EGO algorithm over six test problems.     

Multiplicity one,local and global conjugacy

The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO_(2N) where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.     

Compositions of theta correspondences

Theta correspondence θ over is established by Howe (J. Amer. Math. Soc. 2 (1989) 535). In He (J. Funct. Anal. 199 (2003) 92), we prove that θ preserves unitarity under certain restrictions, generalizing the result of Li (Invent. Math. 97 (1989) 237). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method (J. Algebra 190 (1997) 518; Representation Theory of Lie Groups, Park City, 1998, pp. 179-238; The Orbit Correspondence for real and complex reductive dual pairs, preprint, 2001), one can compose theta correspondences to obtain unitary representations. We call this process quantum induction.     

Random correspondences and nonlinear equations

The question of existence of random solutions of nonlinear equations involving random correspondences is studied by direct use of the results from the deterministic case without resorting to iterative techniques. Applications to stochastic control and nonlinear random operator equations are given.