On representations of quantum groups and Cherednik algebras

Abstract

In the first part of the thesis, we study quantum groups associated to a semisimple Lie algebra g. The classical Chevalley theorem states that for [ a Cartan subalgebra and W the Weyl group of g, the restriction of g-invariant polynomials on g to [ is an isomorphism onto the W-invariant polynomials on , Res: C[g]1 -+ C[]w. A recent generalization of [36] to the case when the target space C of the polynomial maps is replaced by a finite-dimensional representation V of g shows that the restriction map Res: (C[g] 0 V)9 -+ C[] 0 V is injective, and that the image can be described by three simple conditions. We further generalize this to the case when a semisimple Lie algebra g is replaced by a quantum group. We provide the setting for the generalization, prove that the restriction map Res: (Oq(G) 0 V)Uq(9) -+ O(H) 0 V is injective and describe the image. In the second part we study rational Cherednik algebras Hi,c(W, j) over the field of complex numbers, associated to a finite reflection group W and its reflection representation . We calculate the characters of all irreducible representations in category 0 of the rational Cherednik algebra for W the exceptional Coxeter group H3 and for W the complex reflection group G12 . In particular, we determine which of the irreducible representations are finite-dimensional, and compute their characters. In the third part, we study rational Cherednik algebras Ht,c(W, [) over the field of finite characteristic p. We first prove several general results about category 0, and then focus on rational Cherednik algebras associated to the general and special linear group over a finite field of the same characteristic as the underlying algebraically closed field. We calculate the characters of irreducible representations with trivial lowest weight of the rational Cherednik algebra associated to GL,(Fp,) and SL,(Fpr), and characters of all irreducible representations of the rational Cherednik algebra associated to GL2(F,).