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 &#34;quantum&#34; case, where &#34;Planck&#39;s constant&#34; is nonzero and generic irreducible representations have dimension 2p. In this case, smaller representations exist if and only if the &#34;coupling constant&#34; 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 &#34;classical&#34; case, where &#34;Planck&#39;s constant&#34; is zero and generic irreducible representations have dimension 2p. In that case, one-dimensional representations exist if and only if the &#34;coupling constant&#34; k is zero.