Aane Hecke Algebras, Cyclotomic Hecke Algebras and Cliiord Theory

We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, can be obtained, in all cases, from the aane Hecke algebra of type A. The Young tableaux theory was extended to aane Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general aane Hecke algebras can be constructed from the representations of simply connected aane Hecke algebras by using an extended form of Cliiord theory. This extension of Cliiord theory is given in the Appendix. 0. Introduction Recent work of A. Ram Ra2,5] gives a straightforward combinatorial construction of the simple calibrated modules of aane Hecke algebras (of general Lie type as well as type A). The rst aim of this paper is to show that Young's seminormal construction and all of its previously known generalizations are special cases of the construction in Ra5]. In particular, the representation theory of (a) Weyl groups of types A, B, and D, can be derived entirely from the representation theory of aane Hecke algebras of type A. Furthermore , the relationship between the aane Hecke algebra and the objects in (a)-(d) always produces a natural set of Jucys-Murphy type elements and can be used to prove the standard Jucys-Murphy type theorems. In particular, we are able to use Bernstein's results about the center of the aane Hecke algebra to show that, in the semisimple case, the center of the cyclotomic Hecke algebra H r;1;n is the set of symmetric polynomials in the Jucy-Murphy elements. A. Young's seminormal construction of the irreducible representations of the symmetric group dates from 1931 Yg1]. Young himself generalized his tableaux to treat the representation theory of Weyl groups of types B and D Yg2]. In 1974 P.N. Hoefsmit Hf] generalized the seminormal construction to Iwahori-Hecke algebras of types A, B, and D. Hoefsmit's work has never been published and, in 1985, H. Wenzl Wz] independently found the seminormal construction for irreducible representations for Iwahori-Hecke algebras of type A. In 1994 Ariki and Koike AK] introduced (some of) the cyclotomic Hecke algebras and generalized Hoefsmit's construction to these algebras. The construction was generalized to a larger class of cyclotomic Hecke algebras in Ar2]. For a summary of this work see Ra1] and HR]. General aane Hecke algebras are naturally associated to a reductive algebraic group …