This paper uses some basic concepts and results on harmonic analysis associated with Heckman-Opdam theory \(\mathbb {R}^{d}\) to study types of wavelet packets the corresponding transforms. More precisely, we two transforms related hypergeometric Fourier transform W-invariant functions, prove Plancherel, Calderón, reconstruction formulae for these

For certain negative rational numbers κ0, called singular values, and associated with the symmetric group SN on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter κ = κ0. It was shown by the author, de Jeu and Opdam (Trans. Amer. Math. Soc. 346 (1994), 237-256) that the singular values are exactly the values − n with 2 ≤ n ≤ N , m = 1, 2, . . . ...

We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for each Coxeter groups — the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.

We construct (q, t)-Catalan polynomials and q-Fuss-Catalan polynomials for any irreducible complex reflection group W . The two main ingredients in this construction are Rouquier’s formulation of shift functors for the rational Cherednik algebras of W , and Opdam’s analysis of permutations of the irreducible representations of W arising from the Knizhnik-Zamolodchikov connection.