Related papers: Gelfand triples for the Kohn-Nirenberg quantizatio…
This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and $P\subset G$ is a parabolic subgroup. We use methods from differential geometry and very…
Many stochastic processes are defined on special geometrical objects like spheres and cones. We describe how tools from harmonic analysis, i.e. Fourier analysis on groups, can be used to investigate probability density functions (pdfs) on…
A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of…
Let G be a locally compact group and let K be a compact subgroup of Aut(G), the group of automorphisms of G. The pair $(G, K )$ is a Gelfand pair if the algebra $L^{1}_{K}(G)$ of K-invariant integrable functions on G is commutative under…
In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second…
In this paper we use the quantization of fields based on Geometric Langlands Correspondence \cite{diep1} to realize the automorphic representations of some concrete series of groups: for the affine Heisenberg (loop) groups it is reduced to…
We extend the Gelfand and Graev construction of generalized Fourier transforms on basic affine space from split groups to quasi-split groups over a local non-archimedean field $F$.
In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn-Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on…
We discuss a fine tuning of the co- and contra-variant transforms through construction of specific fiducial and reconstructing vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg…
Let $G$ be a compact connected simple Lie group and let $M=G^{\bb{C}}/P=G/K$ be a generalized flag manifold. In this article we focus on an important invariant of $G/K$, the so called $\fr{t}$-root system $R_{\fr{t}}$, and we introduce the…
We study, in a global uniform manner, the quotient of the ring of polynomials in l sets of n variables, by the ideal generated by diagonal quasi-invariant polynomials for general permutation groups W=G(r,n). We show that, for each such…
One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic…
Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. q-Deformations of Weyl or Clifford algebrae that were covariant under the action of a simple Lie algebra g are…
We study invariant distributions on the tangent space to a symmetric space. We prove that an invariant distribution with the property that both its support and the support of its Fourier transform are contained in the set of…
A review of the multiparametric linear quantum group GL_qr(N), its real forms, its dual algebra U(gl_qr(N)) and its bicovariant differential calculus is given in the first part of the paper. We then construct the (multiparametric) linear…
We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-covariant differential calculi. For each we find a frame, a metric and…
We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand-Kirillov dimension the Fourier coefficients are completely determined by certain…
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a…
In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gr\"ochenig of the mixed, weighted Lebesgue spaces $L_{v}^{p,q}$ with respect to the quasi-regular representation…
The main result of this paper is an explicit formula for the Fourier transform of the canonical measure on a nilpotent coadjoint orbit for GL(n,R). This paper also includes some results on limit formulas for reductive Lie groups including…