Representations of Cherednik algebras in positive characteristic

Abstract

In this thesis, we first classify the irreducible representations of the rational 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 pr, where r is the order of the cyclic group contained in the algebra. The other is the "classical" case, where "Planck's constant" is zero and generic irreducible representations have dimension r. Secondly, we classify the irreducible representations of the trigonometric Cherednik algebras of rank 1 in characteristic p > 0. There are two cases. In one case, the "Planck's constant" is zero, and generic irreducible representations have dimension 2; one-dimensional irreducible representations exist when the "coupling constant" is also zero. In the other case, the "Planck's constant" is nonzero, and generic irreducible representations have dimension 2p; if the "coupling constant" is an even integer 0 =/< k =/< p - 1, then there exist smaller irreducible representations of dimensions p + k and p - k.