The cartan decomposition

1970 ◽  
pp. 24-27
Author(s):  
Ian Stewart
Keyword(s):  
Cartan Decomposition

scholarly journals Type III factors with unique Cartan decomposition

2013 ◽  
Vol 100 (4) ◽  
pp. 564-590 ◽  
Author(s):  
Cyril Houdayer ◽  
Stefaan Vaes
Keyword(s):  
Cartan Decomposition ◽  
Type Iii

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].

scholarly journals CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 470-485 ◽  
Author(s):  
BOUDJEMÂA ANCHOUCHE ◽  
SANJIV KUMAR GUPTA
Keyword(s):  
Positive Integer ◽  
Symmetric Space ◽  
Lie Algebra ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Double Coset ◽  
Cartan Decomposition ◽  
Absolutely Continuous ◽  
Orbital Measure ◽  
Noncompact Symmetric Space

AbstractLet G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

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