One - dimensional double Hecke algebras and Gaussians

These notes are about applications of the one-dimensional double affine Hecke algebra to q-Gauss integrals and Gaussian sums. The double affine Hecke algebras were designed for a somewhat different purpose: to clarify the quantum Knizhnik-Zamolodchikov equation. Eventually (through the Macdonald polynomials) they led to a unification of the Harish-Chandra transform (the zonal case) and the p-adic spherical transform. It is not just a unification. The new transform is self-dual in contrast to its celebrated predecessors. Actually it is very close to the Hankel transform (Bessel functions) and its recent generalizations from [O3,D,J] (see also [H]). The q-Gauss integrals are in the focus of the q-theory. The transfer to the roots of unity and Gaussian sums is quite natural as well. Quantum groups are the motivation. We generalize and, at the same time, simplify the Verlinde algebras, the reduced categories of representations of quantum groups at roots of unity. Another interpretation of the Verlinde algebras is via the Kac-Moody algebras [KL] (due to Finkelberg for roots of unity). A valuable feature is the projective action of PSL(2,Z) (cf. [K, Theorem 13.8]). It is the foundation of the double Hecke algebra technique and an extension of the framework of the theory of metaplectic representations. The classification of irreducible spherical unitary representations of onedimensional double Hecke algebra matches well and generalizes the classical formulas for the Gaussian sums. The complete version of this paper (arbitrary root systems) is [C5]. See also [C1]. There are of course other applications of double affine Hecke algebras. We will not discuss them here (see [C6]). This paper is based on my Harvard mini-course (1999) and lectures at University Roma I (La Sapienza). I am thankful to D. Kazhdan, C. De Concini, and the organizers of the CIME school for the kind invitation and publication of the paper.