Spectra for Gelfand pairs associated with the Heisenberg group

Chal Benson; Joe Jenkins; Gail Ratcliff; Tefera Worku

Displaying similar documents to “Spectra for Gelfand pairs associated with the Heisenberg group”

Explicit fundamental solutions of some second order differential operators on Heisenberg groups

Isolda Cardoso, Linda Saal (2012)

Colloquium Mathematicae

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Let p,q,n be natural numbers such that p+q = n. Let be either ℂ, the complex numbers field, or ℍ, the quaternionic division algebra. We consider the Heisenberg group N(p,q,) defined ⁿ × ℑ , with group law given by (v,ζ)(v’,ζ’) = (v + v’, ζ + ζ’- 1/2 ℑ B(v,v’)), where $B (v, w) = \sum_{j = 1}^{p} v_{j} \bar{w_{j}} - \sum_{j = p + 1}^{n} v_{j} \bar{w_{j}}$ . Let U(p,q,) be the group of n × n matrices with coefficients in that leave the form B invariant. We compute explicit fundamental solutions of some second order differential operators on N(p,q,) which are canonically associated...

On a certain generalization of spherical twists

Yukinobu Toda (2007)

Bulletin de la Société Mathématique de France

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This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of $(0, - 2)$ -curves on threefolds, or deforming $ℙ$ -objects introduced by D.Huybrechts and R.Thomas.

Twisted spherical means in annular regions in $ℂ^{n}$ and support theorems

Rama Rawat, R.K. Srivastava (2009)

Annales de l’institut Fourier

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Let $Z (Ann (r, R))$ be the class of all continuous functions $f$ on the annulus $Ann (r, R)$ in $ℂ^{n}$ with twisted spherical mean $f \times μ_{s} (z) = 0,$ whenever $z \in ℂ^{n}$ and $s > 0$ satisfy the condition that the sphere $S_{s} (z) \subseteq Ann (r, R)$ and ball $B_{r} (0) \subseteq B_{s} (z) .$ In this paper, we give a characterization for functions in $Z (Ann (r, R))$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in $ℂ^{n}$ which improve some of the earlier results.

Reduced spherical polygons

Marek Lassak (2015)

Colloquium Mathematicae

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For every hemisphere K supporting a spherically convex body C of the d-dimensional sphere $S^{d}$ we consider the width of C determined by K. By the thickness Δ(C) of C we mean the minimum of the widths of C over all supporting hemispheres K of C. A spherically convex body $R \subset S^{d}$ is said to be reduced provided Δ(Z) < Δ(R) for every spherically convex body Z ⊂ R different from R. We characterize reduced spherical polygons on S². We show that every reduced spherical polygon is of thickness at...

Asymptotic spherical analysis on the Heisenberg group

Jacques Faraut (2010)

Colloquium Mathematicae

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The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair....

A new perspective on some approximations used in neutron transport modeling

Hanuš, Milan

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In this contribution, we will use the (introduced in [1,2]) to derive a system of partial differential equations governing transport of neutrons within an interacting medium. This system forms an alternative to the well known $P_{N}$ approximation, which is based on the expansion of the directional dependence into tesseral spherical harmonics ([3,p.197]). In comparison with this latter set of equations, the Maxwell-Cartesian system posesses a much more regular structure, which may be used...

Moebius-invariant algebras in balls

Walter Rudin (1983)

Annales de l'institut Fourier

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It is proved that the Fréchet algebra $C (B)$ has exactly three closed subalgebras $Y$ which contain nonconstant functions and which are invariant, in the sense that $f \circ Ψ \in Y$ whenever $f \in Y$ and $Ψ$ is a biholomorphic map of the open unit ball $B$ of $C^{n}$ onto $B$ . One of these consists of the holomorphic functions in $B$ , the second consists of those whose complex conjugates are holomorphic, and the third is $C (B)$ .

Spherical means on the Heisenberg group and a restriction theorem for the symplectic Fourier transform.

Sundaram Thangavelu (1991)

Revista Matemática Iberoamericana

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The aim of this paper is to study mean value operators on the reduced Heisenberg group H/Γ, where H is the Heisenberg group and Γ is the subgroup {(0,2πk): k ∈ Z} of H.

A central limit theorem on the space of positive definite symmetric matrices

Piotr Graczyk (1992)

Annales de l'institut Fourier

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A central limit theorem is proved on the space $𝒫_{n}$ of positive definite symmetric matrices. To do this, some natural analogs of the mean and dispersion on $𝒫_{n}$ are defined and investigated. One uses a Taylor expansion of the spherical functions on $𝒫_{n}$ .

Reduction of invariant differential operators and expansions of conical distributions for $S O_{0} (n, 1)$

Aleksander Strasburger (1989)

Annales Polonici Mathematici

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An inequality for spherical Cauchy dual tuples

Sameer Chavan (2013)

Colloquium Mathematicae

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Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\binom{\sum_{| β | = k} {(β!)}^{- 1} {(T^{})}^{β} (T^{}) *^{β} \leq (k + m - 1}{k) \sum_{| β | = k} {(β!)}^{- 1} (T^{}) *^{β} {(T^{})}^{β}}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

Jacek Zienkiewicz (1997)

Studia Mathematica

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Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f \in L^{2} (ℍ^{n})$ such that ${(I - L)}^{s / 2} f \in L^{2} (ℍ^{n})$ , s > 3/4, for a $ϕ \in C_{c} (ℍ^{n})$ we have $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t L} f (x) |}^{2} d x \leq C_{ϕ} ∥ f ∥_{W^{s}}^{2}$ . On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$ , then for every s < 1 there exists a sequence $f_{n} \in L^{2} (ℍ^{n})$ and $C_{n} > 0$ such that ${(I - L)}^{s / 2} f_{n} \in L^{2} (ℍ^{n})$ and for a $ϕ \in C_{c} (ℍ^{n})$ we have $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t Δ} f_{n} (x) |}^{2} d x \geq C_{n} ∥ f_{n} ∥_{W^{s}}^{2}, l i m_{n \to \infty} C_{n} = + \infty$ .