Affine Hecke algebras (Ram-Ramagge)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 25 June 2012

Deducing the ${\tilde{H}}_{L}$ representation theory from ${\tilde{H}}_{P}$

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 ${\tilde{H}}_{L}$ from the representation theory of the affine Hecke algebra ${\tilde{H}}_{P} .$ 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 ${\tilde{H}}_{L} -$ modules from the simple ${\tilde{H}}_{P} -$ 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 $\tilde{H} = {\tilde{H}}_{L}$ be the affine Hecke algebra corresponding to $L .$ Then there is an action of a finite group $K$ of ${\tilde{H}}_{P},$ acting by automorphisms, such that ${\tilde{H}}_{L} = {({\tilde{H}}_{P})}^{K},$ 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

Ω ≅ P / Q

is cyclic or not.

\begin{array}{c} Type & A_{n - 1} & B_{n} & C_{n} & D_{2 n - 1} & D_{2 n} \\ Ω & ℤ / n ℤ & ℤ / 2 ℤ & ℤ / 2 ℤ & ℤ / 4 ℤ & ℤ / 2 ℤ \times ℤ / 2 ℤ \end{array}

\begin{array}{c} Type & E_{6} & E_{7} & E_{8} & F_{4} & G_{2} \\ Ω & ℤ / 3 ℤ & ℤ / 2 ℤ & 1 & 1 & 1 \end{array}

In each case we construct the group

K

and its action on

{\tilde{H}}_{P}

explicitly. This is necessary for the effective application of Theorem A.13 on examples.

Case 1. If $Ω$ is a cyclic group then the subgroup $L / Q$ is a cyclic subgroup. Suppose $Ω = {1, g, ..., g^{r - 1}} and L / Q = {1, g^{d}, ..., g^{d (p - 1)}},$ where $p d = r .$ Let $ζ$ be a primitive $p^{th}$ root of unity and define an automorphism $\begin{array}{rrcl} σ : & {\tilde{H}}_{P} & \to & {\tilde{H}}_{P} \\ g & \mapsto & ζ g, \\ T_{i} & \mapsto & T_{i}, 0 \leq i \leq n . \end{array}$ 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 $ℤ / p ℤ$ action on ${\tilde{H}}_{P}$ and $\begin{array}{ll} {\tilde{H}}_{L} = {({\tilde{H}}_{P})}^{ℤ / p ℤ} . & (AH.RR 1) \end{array}$
Case 2. If the root system $R^{\lor}$ is of type $D_{n},$ $n$ even, then $Ω ≅ ℤ / 2 ℤ \times ℤ / 2 ℤ$ and the subgroups of $Ω$ correspond to the intermediate lattices $Q \subseteq L \subseteq P .$ Suppose $Ω = {1, g_{1}, g_{2}, g_{1} g_{2} | g_{1}^{2} = g_{2}^{2} = 1, g_{1} g_{2} = g_{2} g_{1}}$ and define automorphisms of ${\tilde{H}}_{P}$ by $\begin{array}{rrcl} σ_{1} : & {\tilde{H}}_{P} & \to & {\tilde{H}}_{P} \\ g_{1} & \mapsto & - g_{1}, \\ g_{2} & \mapsto & g_{2}, \\ T_{i} & \mapsto & T_{i}, \end{array} and \begin{array}{rrcl} σ_{2} : & {\tilde{H}}_{P} & \to & {\tilde{H}}_{P} \\ g_{1} & \mapsto & g_{1}, \\ g_{2} & \mapsto & - g_{2}, \\ T_{i} & \mapsto & T_{i} . \end{array}$ Then $\begin{array}{ll} {\tilde{H}}_{L_{1}} = {({\tilde{H}}_{P})}^{σ_{1}}, {\tilde{H}}_{L_{2}} = {({\tilde{H}}_{P})}^{σ_{2}}, and {\tilde{H}}_{Q} = {({\tilde{H}}_{P})}^{⟨ σ_{1}, σ_{2} ⟩}, & (AH.RR 2) \end{array}$ where $L_{1}$ and $L_{2}$ are the two intermediate lattices strictly between $Q$ and $P .$

$□$

References

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

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