Spherical Functions on SO0(p, q)/ SO(p) × SO(q)

1999 ◽  
Vol 42 (4) ◽  
pp. 486-498 ◽  
Author(s):  
P. Sawyer
Keyword(s):  
Symmetric Space ◽  
Analytic Continuation ◽  
Heat Kernel ◽  
Integral Formula ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Tubular Neighbourhood ◽  
Helpful Tool

AbstractAn integral formula is derived for the spherical functions on the symmetric space G/K = SO0(p, q)/ SO(p) × SO(q). This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra a of the abelian part in the decomposition G = KAK. The corresponding result is then obtained for the heat kernel of the symmetric space SO0(p, q)/ SO(p) × SO(q) using the Plancherel formula.In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

scholarly journals A Plancherel formula for Sp2n/Spn×Spn and its application

2009 ◽  
Vol 145 (2) ◽  
pp. 501-527 ◽  
Author(s):  
Zhengyu Mao ◽  
Stephen Rallis
Keyword(s):  
Symmetric Space ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Trace Identity ◽  
Fundamental Lemma

AbstractWe compute the spherical functions on the symmetric space Sp2n/Spn×Spn and derive a Plancherel formula for functions on the symmetric space. As an application of the Plancherel formula, we prove an identity which amounts to the fundamental lemma of a relative trace identity between Sp2n and $\widetilde {{\rm Sp}}_n$.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

2021 ◽  
pp. 1-38
Author(s):  
Yumiko Hironaka
Keyword(s):  
Laurent Polynomial ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Explicit Formulas ◽  
Hermitian Forms ◽  
Spherical Fourier Transform ◽  
Local Densities ◽  
Laurent Polynomial Ring ◽  
Injective Map ◽  
Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

Poisson’s integral formula via heat kernels

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 59-64
Author(s):  
Yoichi Miyazaki
Keyword(s):  
Fourier Transform ◽  
Heat Kernel ◽  
Half Space ◽  
Unbounded Domain ◽  
Harmonic Functions ◽  
Integral Formula ◽  
Heat Kernels ◽  
Transform Method ◽  
Fourier Transform Method ◽  
The Fourier Transform

Abstract We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

THE μ-DEFORMED SEGAL–BARGMANN TRANSFORM IS A HALL TYPE TRANSFORM

2009 ◽  
Vol 12 (02) ◽  
pp. 269-289 ◽  
Author(s):  
STEPHEN BRUCE SONTZ
Keyword(s):  
Analytic Continuation ◽  
Heat Kernel ◽  
Bargmann Transform ◽  
Kernel Analysis ◽  
Bargmann Transforms ◽  
Heat Kernel Analysis

We present an explanation of how the μ-deformed Segal–Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echavarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal–Bargmann analysis. This explanation relates the μ-deformed Segal–Bargmann transforms to the generalized Segal–Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the μ-deformed Segal–Bargmann transform can be understood as Hall type transforms. In particular, we define a μ-deformation of Hall's "Version C" generalized Segal–Bargmann transform which is then shown to be a μ-deformed convolution with a μ-deformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall.

scholarly journals A note on the heat kernel on the Heisenberg group

2002 ◽  
Vol 65 (1) ◽  
pp. 115-120
Author(s):  
Adam Sikora ◽  
Jacek Zienkiewicz
Keyword(s):  
Heisenberg Group ◽  
Analytic Continuation ◽  
Heat Kernel ◽  
Smooth Function ◽  
Convolution Kernel

We describe the analytic continuation of the heat kernel on the Heisenberg group ℍn(ℝ). As a consequence, we show that the convolution kernel corresponding to the Schrödinger operatereisLis a smooth function on ℍn(ℝ) \Ss, whereSs= {(0, 0, ±sk) ∈ ℍn(ℝ) :k=n,n+ 2,n+ 4,…}. At every point ofSsthe convolution kernel ofeisLhas a singularity of Calderón–Zygmund type.

On an Upper Bound for the Heat Kernel on SU*(2n)/ Sp(n)

1994 ◽  
Vol 37 (3) ◽  
pp. 408-418 ◽  
Author(s):  
P. Sawyer
Keyword(s):  
Symmetric Space ◽  
Heat Kernel ◽  
Upper Bound ◽  
Symmetric Spaces ◽  
Legendre Function ◽  
Noncompact Type ◽  
Image Position

AbstractJean-Philippe Anker made an interesting conjecture in [2] about the growth of the heat kernel on symmetric spaces of noncompact type. For any symmetric space of noncompact type, we can writewhere ϕ0 is the Legendre function and q, "the dimension at infinity", is chosen such that limt—>∞Vt(x) = 1 for all x. Anker's conjecture can be stated as follows: there exists a constant C > 0 such thatwhere is the set of positive indivisible roots. The behaviour of the function ϕ0 is well known (see [1]).The main goal of this paper is to establish the conjecture for the spaces SU*(2n)/ Sp(n).

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