Representations of Affine Hecke Algebras and Related Algebras

We introduce the wreath Hecke algebra Hn (G) associated to an arbitrary finite group G as a generalization of the degenerate affine Hecke algebra of type A and develop its representation theory over the field F of characteristic p ≥ 0. We classify the simple representations of Hn (G) and of its associated cyclotomic algebras. We establish the modular branching rule for these algebras and its interpretation via crystal graphs of quantum affine algebras.;We also generalize some classical results in the representation theory of the symmetric group algebra FSn to the spin symmetric group algebra FS-n . We classify and construct the irreducible completely splittable representations of affine Hecke-Clifford algebras over F with p ≠ 2. This further leads to a family of FS-n -modules which afford a form similar to the Young's orthogonal form on Specht modules.;Generalizing the classical invariant theory of the symmetric group Sn, we formulate an invariant theory for the spin symmetric group algebra CS-n . We solve the corresponding graded multiplicity problem in terms of specializations of Schur Q-functions and a shifted q-hook formula.