Affine Hecke algebras (Ram-Ramagge)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 25 June 2012

Deducing the H˜L representation theory from H˜P

Although we have not taken this point of view in our presentation, the affine Hecke algebras defined above are naturally associated to a reductive algebraic group G over [KL] or a p-adic Chevalley group [IM]. In this formulation, the lattice L is determined by the group of characters of the maximal torus of G. It is often convenient to work only with the adjoint version or only with the simply connected representation theory of the affine Hecke algebras H~L from the representation theory of the affine Hecke algebra H~P. The following theorem shows that this can be done in a simple way by using the extension of Clifford theory in the Appendix. In particular, Theorem A.13, can be used to construct all of the simple H~L-modules from the simple H~P-modules.

Let L be a lattice such that QLP, where Q is the root lattice and P is the weight lattice of the root system R. Let H˜ = H˜L be the affine Hecke algebra corresponding to L. Then there is an action of a finite group K of H˜P, acting by automorphisms, such that H˜L = (H˜P)K, is the subalgebra of fixed points under the action of the group K.

There are two cases to consider, depending of whether the group ΩP/Q is cyclic or not. Type An-1 Bn Cn D2n-1 D2n Ω /n /2 /2 /4 /2 × /2 Type E6 E7 E8 F4 G2 Ω /3 /2 1 1 1 In each case we construct the group K and its action on H˜P explicitly. This is necessary for the effective application of Theorem A.13 on examples.
  1. Case 1. If Ω is a cyclic group then the subgroup L/Q is a cyclic subgroup. Suppose Ω = { 1,g,...,gr-1 } and L/Q = { 1,gd,...,gd(p-1) }, where pd=r. Let ζ be a primitive pth root of unity and define an automorphism σ: H˜P H˜P g ζg, Ti Ti, 0in. The map σ is an algebra isomorphism since it preserves the relations in [RR, Eq. 4.6] and [RR, Eq. 4.10]. Furthermore, σ gives rise to a /p action on H˜P and H˜L = ( H˜P ) /p. (AH.RR 1)
  2. Case 2. If the root system R is of type Dn, n even, then Ω /2 × /2 and the subgroups of Ω correspond to the intermediate lattices QLP. Suppose Ω = { 1,g1,g2,g1g2 | g12 = g22 = 1, g1g2 = g2g1 } and define automorphisms of H˜P by σ1: H˜P H˜P g1 -g1, g2 g2, Ti Ti, and σ2: H˜P H˜P g1 g1, g2 -g2, Ti Ti. Then H˜L1 = ( H˜P )σ1, H˜L2 = ( H˜P )σ2, and H˜Q = ( H˜P ) σ1,σ2, (AH.RR 2) where L1 and L2 are the two intermediate lattices strictly between Q and P.


[RR] A. Ram and J. Ramagge, Affine Hecke Algebras,

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