## Affine Hecke algebras (Ram-Ramagge)

Last update: 25 June 2012

## Deducing the ${\stackrel{˜}{H}}_{L}$ representation theory from ${\stackrel{˜}{H}}_{P}$

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 $ℂ$ [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 ${\stackrel{~}{H}}_{L}$ from the representation theory of the affine Hecke algebra ${\stackrel{~}{H}}_{P}.$ 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 ${\stackrel{~}{H}}_{L}-$modules from the simple ${\stackrel{~}{H}}_{P}-$modules.

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

 Proof. There are two cases to consider, depending of whether the group $\Omega \cong P/Q$ is cyclic or not. $Type An-1 Bn Cn D2n-1 D2n Ω ℤ/nℤ ℤ/2ℤ ℤ/2ℤ ℤ/4ℤ ℤ/2ℤ × ℤ/2ℤ$ $Type E6 E7 E8 F4 G2 Ω ℤ/3ℤ ℤ/2ℤ 1 1 1$ In each case we construct the group $K$ and its action on ${\stackrel{˜}{H}}_{P}$ explicitly. This is necessary for the effective application of Theorem A.13 on examples. Case 1. If $\Omega$ 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 $\zeta$ be a primitive ${p}^{\mathrm{th}}$ root of unity and define an automorphism $σ: H˜P → H˜P g ↦ ζg, Ti ↦ Ti, 0≤i≤n.$ The map $\sigma$ is an algebra isomorphism since it preserves the relations in [RR, Eq. 4.6] and [RR, Eq. 4.10]. Furthermore, $\sigma$ gives rise to a $ℤ/pℤ$ action on ${\stackrel{˜}{H}}_{P}$ and $H˜L = ( H˜P ) ℤ/pℤ. (AH.RR 1)$ Case 2. If the root system ${R}^{\vee }$ is of type ${D}_{n},$ $n$ even, then $\Omega \cong ℤ/2ℤ×ℤ/2ℤ$ and the subgroups of $\Omega$ correspond to the intermediate lattices $Q\subseteq L\subseteq P.$ Suppose $Ω = { 1,g1,g2,g1g2 | g12 = g22 = 1, g1g2 = g2g1 }$ and define automorphisms of ${\stackrel{˜}{H}}_{P}$ 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 ${L}_{1}$ and ${L}_{2}$ are the two intermediate lattices strictly between $Q$ and $P.$ $\square$

## References

[RR] A. Ram and J. Ramagge, Affine Hecke Algebras, http://researchers.ms.unimelb.edu.au/~aram@unimelb/publications.html.