Affine Hecke algebras (RamRamagge)
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 $\u2102$ [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
$\tilde{H}={\tilde{H}}_{L}$
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
$${\tilde{H}}_{L}={\left({\tilde{H}}_{P}\right)}^{K},$$
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.
$$\begin{array}{cccccc}Type& {A}_{n1}& {B}_{n}& {C}_{n}& {D}_{2n1}& {D}_{2n}\\ \Omega & \mathbb{Z}/n\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}& \mathbb{Z}/4\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\end{array}$$
$$\begin{array}{cccccc}Type& {E}_{6}& {E}_{7}& {E}_{8}& {F}_{4}& {G}_{2}\\ \Omega & \mathbb{Z}/3\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}& 1& 1& 1\end{array}$$
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 $\Omega $ is a cyclic group then the subgroup $L/Q$ is a cyclic subgroup. Suppose
$$\Omega =\{1,g,...,{g}^{r1}\}\phantom{\rule{2em}{0ex}}and\phantom{\rule{2em}{0ex}}L/Q=\{1,{g}^{d},...,{g}^{d(p1)}\},$$
where $pd=r.$ Let $\zeta $ be a primitive ${p}^{\mathrm{th}}$ root of unity and define an automorphism
$$\begin{array}{rrcl}\sigma :& {\tilde{H}}_{P}& \to & {\tilde{H}}_{P}\\ & g& \mapsto & \zeta g,\\ & {T}_{i}& \mapsto & {T}_{i},\phantom{\rule{4em}{0ex}}0\le i\le n.\end{array}$$
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 $\mathbb{Z}/p\mathbb{Z}$ action on ${\tilde{H}}_{P}$ and
$$\begin{array}{ll}{\tilde{H}}_{L}={\left({\tilde{H}}_{P}\right)}^{\mathbb{Z}/p\mathbb{Z}}.& \phantom{\rule{3em}{0ex}}(AH.RR\; 1)\end{array}$$
 Case 2. If the root system ${R}^{\vee}$ is of type ${D}_{n},$ $n$ even, then
$\Omega \cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$
and the subgroups of $\Omega $ correspond to the intermediate lattices
$Q\subseteq L\subseteq P.$
Suppose
$$\Omega =\{1,{g}_{1},{g}_{2},{g}_{1}{g}_{2}\phantom{\rule{.5em}{0ex}}\phantom{\rule{.5em}{0ex}}{g}_{1}^{2}={g}_{2}^{2}=1,\phantom{\rule{.5em}{0ex}}{g}_{1}{g}_{2}={g}_{2}{g}_{1}\}$$
and define automorphisms of ${\tilde{H}}_{P}$ by
$$\begin{array}{rrcl}{\sigma}_{1}:& {\tilde{H}}_{P}& \to & {\tilde{H}}_{P}\\ & {g}_{1}& \mapsto & {g}_{1},\\ & {g}_{2}& \mapsto & {g}_{2},\\ & {T}_{i}& \mapsto & {T}_{i},\end{array}\phantom{\rule{4em}{0ex}}and\phantom{\rule{4em}{0ex}}\begin{array}{rrcl}{\sigma}_{2}:& {\tilde{H}}_{P}& \to & {\tilde{H}}_{P}\\ & {g}_{1}& \mapsto & {g}_{1},\\ & {g}_{2}& \mapsto & {g}_{2},\\ & {T}_{i}& \mapsto & {T}_{i}.\end{array}$$
Then
$$\begin{array}{ll}{\tilde{H}}_{{L}_{1}}={\left({\tilde{H}}_{P}\right)}^{{\sigma}_{1}},\phantom{\rule{2em}{0ex}}{\tilde{H}}_{{L}_{2}}={\left({\tilde{H}}_{P}\right)}^{{\sigma}_{2}},\phantom{\rule{2em}{0ex}}and\phantom{\rule{2em}{0ex}}{\tilde{H}}_{Q}={\left({\tilde{H}}_{P}\right)}^{\u27e8{\sigma}_{1},{\sigma}_{2}\u27e9},& \phantom{\rule{3em}{0ex}}(AH.RR\; 2)\end{array}$$
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.
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