Representations of trigonometric Cherednik algebras of rank 1 in positive characteristic

Abstract

In this paper, we classify the irreducible representations of the trigonometric Cherednik algebras of rank 1 in characteristic p > 0. There are two cases. One is the "quantum" case, where "Planck's constant" is nonzero and generic irreducible representations have dimension 2p. In this case, smaller representations exist if and only if the "coupling constant" k is in F_p; namely, if 0 <= k <= p-1, then there exist irreducible representations of dimensions p-k and p+k. The other case is the "classical" case, where "Planck's constant" is zero and generic irreducible representations have dimension 2p. In that case, one-dimensional representations exist if and only if the "coupling constant" k is zero.