| The Hecke algebras are ubiquitous in diverse areas, including representation theories over finite fields and Kazhdan-Lusztig theory for category and they can be constructed by a standard procedure. The theory of spin representations of they symmetric group was developed by I. Schur in 1911, and, it is parallel to the classical theory of (linear) representations of the symmetric group. The main theme of this dissertation is to develop systematically a theory of spin Hecke algebras which is a "fusion product" of the classical theory of Hecke algebras and Schur's spin (projective) representation theory. Generalizing Hecke algebras constructions and the spin symmetric group, we formulate the notion of (degenerate) spin Hecke algebras of affine and double affine type associated to the classical Weyl groups. Several closely related algebras called Hecke-Clifford algebras are also introduced. The basic properties of these algebras such as PBW basis, the intertwiner elements, and Dunkl operators are established. We further develop connections of these algebras with the usual Hecke algebras. |
