Difference Macdonald - Mehta conjecture
نویسنده
چکیده
with respect to the Gaussian measure. Macdonald extended it from An−1 to other root systems and verified his conjecture for classical ones by means of the Selberg integrals [M1]. It was established by Opdam in [O1] in full generality using the shift operators. The integral is an important normalization constant for a k-deformation of the Hankel transform introduced by Dunkl [D]. The generalized Bessel functions [O3] multiplied by the Gaussian are eigenfunctions of this transform. The eigenvalues are given in terms of this constant. See [D,J] for the detail. The Hankel transform is a rational degeneration of the Fourier transform in the Harish-Chandra theory of spherical functions when the symmetric space G/K is replaced by its tangent space Te(G/K) with the adjoint action of G (see [H]). The harmonic analysis for G/K is much more complicated than in the rational case. The reproducing kernel of the Fourier transform is not symmetric, the Gaussian is not Fourier-invariant, and so on. The zonal spherical functions for dominant weights, are (trigonometric) polynomials and have no counterparts in the rational theory. They play a great role in mathematics and physics. For k = 1 they are the characters of finite dimensional representations of G. Unfortunately the Fourier transforms of spherical polynomials are no good, but for the characters.
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