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Search: MSC category 43A85 ( Analysis on homogeneous spaces )

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1. CMB 2016 (vol 60 pp. 111)

Ghaani Farashahi, Arash
Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$.

Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula
Categories:20G05, 43A85, 43A32, 43A40

2. CMB 2015 (vol 58 pp. 632)

Silberman, Lior
Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift
Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

Keywords:quantum unique ergodicity, microlocal lift, spherical dual
Categories:22E50, 43A85

3. CMB 2011 (vol 54 pp. 663)

Haas, Ruth; G. Helminck, Aloysius
Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.

Categories:20G15, 20G20, 22E15, 22E46, 43A85

4. CMB 2007 (vol 50 pp. 291)

Sarkar, Rudra P.; Sengupta, Jyoti
Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type
We prove Beurling's theorem for rank $1$ Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Keywords:Beurling's Theorem, Riemannian symmetric spaces, uncertainty principle
Categories:22E30, 43A85

5. CMB 2004 (vol 47 pp. 389)

He, Jianxun
An Inversion Formula of the Radon Transform Transform on the Heisenberg Group
In this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

Keywords:wavelet transform, Radon transform, Heisenberg group
Categories:43A85, 44A15

6. CMB 1999 (vol 42 pp. 169)

Ding, Hongming
Heat Kernels of Lorentz Cones
We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, Laplace-Beltrami operator, eigenvalues
Categories:35K05, 43A85, 35K15, 80A20

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