Representations of trigonometric Cherednik algebras of rank one in positive characteristic

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 k is an even integer such that 0 <= k <= p - 1, then there exist irreducible representations of dimensions p-k and p+k, and if k is an odd integer such that 1 <= k <= p - 2, then there exist irreducible representations of dimensions k and 2p - k. The other case is the "classical" case, where "Planck's constant" is zero and generic irreducible representations have dimension 2. In that case, one-dimensional representations exist if and only if the "coupling constant" k is zero. (C) 2010 Elsevier B.V. All rights reserved.