Espaces symétriques de Drinfeld et correspondance de Langlands locale

We study the Galois action on the equivariant cohomology complex of Drinfeld's p-adic symmetric spaces and show how it encodes Langlands' correspondence for the so-called “principal elliptic” representations of GLd. This is the first stage of an expected generalization of Carayol's non-Abelian Lubin-Tate theory from supercuspidal to elliptic representations. In the process we obtain a new proof of Deligne's weight-monodromy conjecture for those varieties which admit p-adic uniformization by these spaces, we compute Ext groups and cup-products for elliptic representations, and we give a new computation of the compactly supported cohomology of p-adic symmetric spaces.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

Filtrations de Radon et fonctions sphériques pour Sn et son q-analogue GL(n,q)

We define a sequence of generalized Radon transforms, which are intertwining operators for natural representations associated to Gel'fand spaces for the symmetric group S_n. This sequence enables us to decompose in a recursive way these natural representations and to compute explicitly the associate spherical functions. We prove analogous results for a sequence of generalized Radon transforms between natural representations for the general linear group GL(n,q), which are a q-analogue of the preceding ones.     

Monodromy of Fuchsian systems on complex linear spaces

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.     

Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G_r ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G_R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G_R, we study the convex cone of entire functions on G whose restrictions to G_R are positive-definite, and determine the minimal order of growth at infinity of such functions.     

Integral representation of the W-weighted Drazin inverse for Hilbert space operators

This paper studies the integral representation of the W-weighted Drazin inverse for bounded linear operators between Hilbert spaces. By using operator matrix blocks, some integral representations of the W-weighted Drazin inverse for Hilbert space operators are established.     

Analysis of a class of nonlinear and non-separable multiscale representations

In this paper, we introduce a particular class of nonlinear and non-separable multiscale representations which embeds most of these representations. After motivating the introduction of such a class on one-dimensional examples, we investigate the multi-dimensional and non-separable case where the scaling factor is given by a non-diagonal dilation matrix M. We also propose new convergence and stability results in L ^p and Besov spaces for that class of nonlinear and non-separable multiscale representations. We end the paper with an application of the proposed study to the convergence and the stability of some nonlinear multiscale representations.     

Arc spaces and DAHA representations

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of the symmetric group S _n . We compare certain multiplicity spaces in its decomposition into irreducible representations of S _n with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity x ^m = y ⁿ .     

Representations of certain semidirect product groups

We study a class of semidirect product groups G = N · U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincaré group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces.     

Hardy-Sobolev Spaces Decomposition in?Signal Analysis

Some fundamental formulas and relations in signal analysis are based on the amplitude-phase representations s(t)=A(t)e ^{i ??(t)} and $\hat{s}(\omega)=B(\omega)e^{i\psi(\omega)}$ , where the amplitude functions A(t) and B(??) and the phase functions ??(t) and ??(??) are assumed to be differentiable. They include the amplitude-phase representations of the first and second order means of the Fourier frequency and time, and the equivalence between two forms of the covariance. A proof of the uncertainty principle is also based on the amplitude-phase representations. In general, however, signals of finite energy do not necessarily have differentiable amplitude-phase representations. The study presented in this paper extends the classical formulas and relations to general signals of finite energy. Under the formulation of the phase and amplitude derivatives based on the Hardy-Sobolev spaces decomposition the extended formulas reveal new features, and contribute to the foundations of time-frequency analysis. The established theory is based on the equivalent classes of the L ² space but not on particular representations of the classes. We also give a proof of the uncertainty principle by using the amplitude-phase representations defined through the Hardy-Sobolev spaces decomposition.     

Unitary representations of unimodular Lie groups in Bergman spaces

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. These representation spaces are infinite-dimensional and have compact kernels. In particular, the Bergman spaces of these natural manifolds are infinite-dimensional.     

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.     

An analogue of the BGG resolution for locally analytic principal series

Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.     

A Theorem à la Fatou for the Square Root of Poisson Kernel in $$\delta $$-Hyperbolic Spaces and Decay of Matrix Coefficients of Boundary Representations

In this note, we prove a theorem à la Fatou for the square root of Poisson Kernel in the context of quasi-convex cocompact discrete groups of isometries of \(\delta \)-hyperbolic spaces. As a corollary we show that some matrix coefficients of boundary representations cannot satisfy the weak inequality of Harish-Chandra. Nevertheless, such matrix coefficients satisfy an inequality which can be viewed as a particular case of the inequality coming from property RD for boundary representations. The inequality established in this paper is based on a uniform bound which appears in the proof of the irreducibility of boundary representations. Moreover this uniform bound can be used to prove that the Harish-Chandra’s Schwartz space associated with some discrete groups of isometries of \(\delta \)-hyperbolic spaces carries a natural structure of a convolution algebra. Then in the context of CAT(?1) spaces we show how our elementary techniques enable us to apply an equidistribution theorem of Roblin to obtain information about the decay of matrix coefficient of boundary representations associated with continuous functions.     

Fuzzy topological vector spaces I

Some of the properties of fuzzy topological vector spaces are investigated. Also, there are given necessary and sufficient conditions for a family of fuzzy sets, in a vector space E, to be the family of all neighborhoods of zero for a fuzzy linear topology.     

Cyclic gradings of Lie algebras and lax pairs for <Emphasis Type="Italic">σ</Emphasis>-models

We study a class of σ-models with complex homogeneous target spaces and zero-curvature representations. We find a relation between these models and σ-models with certain m-symmetric target spaces. We also describe a model with the hypercomplex target space S ¹ × S ³ in detail.     

Representations for Moore-Penrose inverses in Hilbert spaces

Let H₁, H₂ be two Hilbert spaces, and let T : H₁ → H₂ be a bounded linear operator with closed range. We present some representations of the perturbation for the Moore-Penrose inverse in Hilbert spaces for the case that the perturbation does not change the range or the null space of the operator.     

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p1+2r and of exponent p (Theorems 1.5 and 1.6).     

Almost, Asymptotic and Fredholm Representations of Discrete Groups

An almost representation of a group is a map from this group into the unitary group of a Hilbert space, such that the group relations hold only approximately. We give a survey of the recent results on almost representations and on their relations to asymptotic representations. Applications to K-theory of classifying spaces are also discussed.     

Off-Shell Supersymmetry and Filtered Clifford Supermodules

An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1|N}\), correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1|p,q}\), correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.