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1. CMB Online first
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 |
2. CMB 2011 (vol 54 pp. 663)
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 |
3. CMB 2007 (vol 50 pp. 291)
Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type |
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 |
4. CMB 2004 (vol 47 pp. 389)
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 |
5. CMB 1999 (vol 42 pp. 169)
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 |