Difference Fourier transforms for nonreduced root systems

Abstract

Abstract. In the first part of the paper kernels are constructed which meromorphically extend the Macdonald-Koornwinder polynomials in their degrees. In the second part of the paper the kernels associated with rank one root systems are used to define nonsymmetric variants of the spherical Fourier transform on the quantum SU(1, 1) group. Related Plancherel and inversion formulas are derived using double affine Hecke algebra techniques.

Generalized Weyl modules and nonsymmetric q-Whittaker functions

“…Then G and Ξ can be regarded as meromorphic functions on (X ⊗ Z C) 2 , where Z µ X ν is the function sending (z, x) ∈ (X ⊗ Z C) 2 to q (µ,z)+ (ν,x) . In this setting, all of the infinite sums above are uniformly absolutely convergent on compact sets-see [St1,Lemma 9.2] for the relevant estimates.…”

Fourier transforms related to a root system of rank 1

“…However, it turns out here that this sum does not converge absolutely. In [3], [23] the absolute convergence of the series comes from the Gaussian, which, in the rank 1 case, contains the factor q n 2 for 0 < q < 1, where n is the summation index. Although we cannot define the kernel in this way, it does give us an idea what properties the kernel E is expected to have; if the above expansion would converge absolutely, E(x, λ) would be an entire function in x and λ, and E would satisfy the duality property E(x, λ) = E σ (λ, x), provided that k σ = k.…”

“…Cherednik's double affine Hecke algebras and their degenerate versions are very useful for studying Macdonald orthogonal polynomials and generalized Fourier transforms, see, e.g., Cherednik [2], [3], Macdonald [15], Opdam [19], Sahi [20], Stokman [23]. In rank 1 this approach leads to new interpretations and new results for many well-known orthogonal polynomials of (basic) hypergeometric type and corresponding integral transforms.…”

“…Sahi's algebra in rank 1 is very useful for studying Askey-Wilson polynomials [1] and Askey-Wilson functions [11], see Noumi and Stokman [18], Stokman [23], and Sahi [21]. In fact, Sahi's double affine Hecke algebra H q is used in [20], [21], [22] to study the Koornwinder-Macdonald polynomials, the multivariable versions of the Askey-Wilson polynomials introduced by Koornwinder [13].…”

Macdonald difference operators and Harish-Chandra series

“…Some initial steps towards the harmonic analysis on quantum noncompact Riemannian symmetric spaces can be found in e.g. [18], [19], [30] and [22].…”