Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory

Abstract

We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r, p, n) and the corresponding cyclotomic Hecke algebras H r,p,n , can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.

Discrete series characters for affine Hecke algebras and their formal degrees

“…It follows that twisting by ψ acts freely on ∆ H (B n , V, F b 1 , (k, 0)) as well. Using [RR,Theorems A.6 and A.13] we see that all characters in ∆ H (B n , V, F b 1 , (k, 0)) remain irreducible when restricted to H(D n , V, F 1 , k)=H(B n , V, F b 1 , (k, 0)) Ψ , that all δ∈∆ H (D n , V, F 1 , k) arise in this way, that there always exist precisely two irreducible characters δ + , δ − ∈∆ H (B n , V, F b 1 , (k, 0)) restricting to δ, and that these two characters are ψ -twists of each other. This proves the required result.…”

The two-boundary Temperley–Lieb algebra

“…We also define the Murphy elements of type B: [20,32,33] is given by adding to the Hecke algebra of type B an additional boundary generator g N with relations:…”

Analytic R-groups of affine Hecke algebras

“…Since H e P is merely a product of such type A-factors and a type C factor, this shows in view of the above that ''twisting by e'' acts on the set D e P of equivalence classes of discrete series representations of H e P without fixed points. In turn this implies (by elementary Cli¤ord theory, see [27], and Lemma 6.3) that the restriction functor sends irreducible discrete series of H e P to irreducible discrete series of H P , and all discrete series of H P are obtained in this way. Hence the action groupoid W e P; D e P of the group K e P z W P; P acting on the set of equivalence classes D e P of irreducible discrete series representations of H e P via twisting automorphisms on H e P , is Morita equivalent to W P; D P .…”