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.
"One-dimensional double Hecke algebra and Gaussian sums." Duke Math. J. 108 (3) 511 - 538, 15 June 2001. https://doi.org/10.1215/S0012-7094-01-10834-X