J un 2 00 8 Zhedanov ’ s algebra AW ( 3 ) and the double affine Hecke algebra in the rank one case . II . The spherical subalgebra
نویسنده
چکیده
This paper builds on the previous paper by the author, where a relationship between Zhedanov’s algebra AW (3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW (3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra.
منابع مشابه
Zhedanov’s Algebra AW (3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
This paper builds on the previous paper by the author, where a relationship between Zhedanov’s algebra AW (3) and the double affine Hecke algebra (DAHA) corresponding to the Askey–Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW (3) with an additional relation that the Casimir operator equals an explicit constant. A similar resu...
متن کاملIwahori-hecke Algebras of Sl2 over 2-dimensional Local Fields
Hecke algebras were first studied because of their role in the representation theory of p-adic groups, or algebraic groups over 1-dimensional local fields. There are two important classes of Hecke algebras. One is spherical Hecke algebras attached to maximal compact open subgroups, and the other is Iwahori-Hecke algebras attached to Iwahori subgroups. A spherical Hecke algebra is isomorphic to ...
متن کاملThe Relationship between Zhedanov’s Algebra AW (3) and the Double Affine Hecke Algebra in the Rank One Case
Zhedanov’s algebra AW (3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey–Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW (3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double ...
متن کاملM ar 2 00 7 The relationship between Zhedanov ’ s algebra AW ( 3 ) and the double affine Hecke algebra in the rank one case
Zhedanov’s algebra AW (3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW (3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double ...
متن کامل0 O ct 2 00 3 DOUBLE AFFINE HECKE ALGEBRAS OF RANK 1 AND AFFINE CUBIC SURFACES
We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is ...
متن کامل