One-dimensional Double Hecke Algebra and Gaussian Sums

We introduce and describe one-dimensional cyclotomic Gauss-Selberg sums generalizing the classical Gaussian sums. They correspond to irreducible self-dual unitary spherical representations of the one-dimensional double affine Hecke algebra. Introduction This paper is 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 KnizhnikZamolodchikov 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 transform from [C2] is self-dual, in contrast to its predecessors. Actually, it is close to the Hankel transform (Bessel functions) and its recent generalizations from [O2], [D], and [J] (see also [H]). The Gaussian is an eigenfunction of the q-transform. The eigenvalue generalizes the classical Gauss-Selberg integrals. The case when q is a root of unity is of obvious importance. The main result of this paper is a list of one-dimensional cyclotomic Gauss-Selberg sums corresponding to irreducible spherical self-dual representations of the double Hecke algebra. It is worth mentioning that our cyclotomic sums are different from the modular GaussSelberg sums (see, e.g., [E, (1.2b)]). The list includes major classical Gaussian sums and new ones that do not seem to have classical counterparts. We focus on explicit formulas in this paper. We describe the corresponding representations but do not discuss in what sense the list is complete. Paper [C5] is devoted to this question (arbitrary root systems). See also [C2] and [C3] for other aspects of the theory of double affine Hecke algebras. Note the relation to the Verlinde algebras formed by reduced representations of quantum groups at roots of unity and, equivalently, by integrable representations of DUKE MATHEMATICAL JOURNAL Vol. 108, No. 3, c © 2001 Received 8 May 2000. Revision received 5 September 2000. 2000 Mathematics Subject Classification. Primary 22E; Secondary 43. Author’s work partially supported by National Science Foundation grant number DMS 98-77048.