scholarly journals SPHERICAL FUNCTIONS ON THE SPACE OF p-ADIC UNITARY HERMITIAN MATRICES

2014 ◽  
Vol 10 (02) ◽  
pp. 513-558
Author(s):  
YUMIKO HIRONAKA ◽  
YASUSHI KOMORI
Keyword(s):  
Spherical Functions ◽  
Plancherel Formula ◽  
Symmetric Polynomials ◽  
Hermitian Matrices ◽  
Cartan Decomposition ◽  
Explicit Formulas ◽  
Spherical Fourier Transform ◽  
Rank 2 ◽  
Type C ◽  
Residual Characteristic

We investigate the space X of unitary hermitian matrices over 𝔭-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e. 2 ∉ 𝔭. From Sec. 2.2 till the end of Sec. 4, we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall–Littlewood symmetric polynomials of type Cn appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show that the Schwartz space [Formula: see text] is a free Hecke algebra [Formula: see text]-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on [Formula: see text].

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].

Some Supercuspidal Representations of Sp4(k)

1987 ◽  
Vol 39 (1) ◽  
pp. 1-7
Author(s):  
Charles Asmuth
Keyword(s):  
Orthogonal Group ◽  
Plancherel Formula ◽  
Weil Representation ◽  
Orthogonal Groups ◽  
Explicit Formulas ◽  
Weil Representations ◽  
Supercuspidal Representations ◽  
Residual Characteristic

The purpose of this paper is to produce explicit realizations of supercuspidal representations of Sp4(k) where k is a p-adic field with odd residual characteristic. These representations will be constructed using the Weil representation of Sp4(k) associated with a certain 4-dimensional compact orthogonal group OQ over k. The main problem addressed in this paper is the analysis of this representation; we need to find how the supercuspidal summands decompose into irreducible pieces.The problem of decomposing Weil representations has been studied quite a bit already. The Weil representations of SL2(k) associated to 2-dimensional orthogonal groups were used by Casselman [4] and Shalika [9] to produce all supercuspidals of SL2(k). The explicit formulas for these representations were used by Sally and Shalika ([10]) to compute the characters and finally to write down a Plancherel formula for that group.

scholarly journals Spherical Functions Approach to Sums of Random Hermitian Matrices

2017 ◽  
Vol 2019 (4) ◽  
pp. 1005-1029 ◽  
Author(s):  
Arno B J Kuijlaars ◽  
Pablo Román
Keyword(s):  
Spherical Functions ◽  
Hermitian Matrices

Olshanski spherical pairs of semigroups type

2019 ◽  
Vol 22 (03) ◽  
pp. 1950021
Author(s):  
Mohamed Bouali
Keyword(s):  
Integral Representation ◽  
Unitary Group ◽  
Inductive Limit ◽  
Complete Classification ◽  
Spherical Functions ◽  
Hermitian Matrices ◽  
Positive Type ◽  
Negative Type ◽  
Functions Of Positive Type

Let [Formula: see text] be the infinite semigroup, inductive limit of the increasing sequence of the semigroups [Formula: see text], where [Formula: see text] is the unitary group of matrices and [Formula: see text] is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on [Formula: see text], by following a general approach introduced by Olshanski and Vershik.10 Second, we prove an integral representation for functions of positive-type analog to the Bochner–Godement theorem, and a Lévy–Khinchin formula for functions of negative type defined on [Formula: see text].

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 $\widetilde {{\rm Sp}}_n$.

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.

Some Convexity Results for the Cartan Decomposition

2003 ◽  
Vol 55 (5) ◽  
pp. 1000-1018 ◽  
Author(s):  
P. Graczyk ◽  
P. Sawyer
Keyword(s):  
Symmetric Spaces ◽  
Singular Values ◽  
Product Formula ◽  
Spherical Functions ◽  
Cartan Decomposition ◽  
Simple Description ◽  
Noncompact Type

AbstractIn this paper, we consider the set = a(eXKeY) where a(g) is the abelian part in the Cartan decomposition of g. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of SL(3; F) where F= R, Cor H. In particular, we show that is convex.We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

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