scholarly journals Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces

1996 ◽  
Vol 123 (1) ◽  
pp. 16-77 ◽  
Author(s):  
Masatoshi Noumi
Keyword(s):  
Homogeneous Spaces ◽  
Spherical Functions ◽  
Symmetric Polynomials ◽  
Quantum Homogeneous Spaces

scholarly journals Subgroups of quantum groups

2018 ◽  
Vol 70 (2) ◽  
pp. 509-533
Author(s):  
Georgia Christodoulou
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Homogeneous Spaces ◽  
General Definition ◽  
Monoidal Categories ◽  
Categorical Approach ◽  
Quantum Homogeneous Spaces

Abstract We investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups, where some faithful flatness conditions apply. Furthermore, we proceed with a categorical approach to the problem of finding quantum subgroups. We translate all existing results into the language of module and monoidal categories and give another characterization of the notion of a quantum subgroup.

ON THE QUANTUM RUIJSENAARS MODEL, SOME QUANTUM HOMOGENEOUS SPACES AND HECKE ALGEBRAS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 224-239
Author(s):  
N. IORGOV
Keyword(s):  
Homogeneous Space ◽  
Explicit Form ◽  
Homogeneous Spaces ◽  
Hecke Algebras ◽  
The Other ◽  
Difference Operators ◽  
Quantum Homogeneous Spaces ◽  
Quantum Analogue ◽  
Affine Hecke Algebras

The aim of the article is to derive in the explicit form the radial components of Casimir elements of Uq( gl n) corresponding to a quantum analogue of the homogeneous space GL (n)/ SO (n). They coincide with the Macdonald–Ruijsenaars difference operators (MRDOs), if one starts from a special set of Casimir elements from the center of Uq( gl n). The derivation is essentially based on Cherednik's approach to MRDOs by means of affine Hecke algebras. From the other side, MRDOs coincide with commuting Hamiltonians of quantum trigonometric n-particle Ruijsenaars model.

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].

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