## Return to Answer

2 added clarification of handwavy thing; deleted 87 characters in body

Probably you already figured this out since it's been almost a month, but anyway...

If you have an arbitrary (i.e. possibly non-split) reductive group $G$ and an Iwahori $I$ and you want to write the function-theoretic Hecke algebra $\mathcal{H}(G;I)$ as one of these other combinatorial Hecke algebras, you should do the following: take $W$ to be the abstract extended affine Weyl group, take $S$ to be a base of the affine Weyl group $W_{aff} \subset W$ and form the combinatorial Hecke algebra in the usual way (e.g. chapter 7 in Humphreys's little grey book), except that when you are choosing the parameters $a_s$ and $b_s$ for the relations, you should replace $q$ with the index $[IsI:I]$.

I believe this non-split situation was done by Macdonald, but I'm not sure.

Edit - I realized that "usual way" above requires some clarification:

First, you write $W$ as the internal semidirect product $W_{aff} \rtimes \Omega$ where $\Omega$ is the subset of elements stabilizing the base alcove. The length function $\ell$ extends from $W_{aff}$ to $W$ by simply taking the length of the projections of elements in $W_{aff}$ (so now there are non-zero but zero length elements). The Bruhat order also extends but that's not necessary now. Then the "usual way" means:

• label your basis using all elements of $W$ (not just $W_{aff}$),
• interpret the relation "$T_{w_1 w_2}=T_{w_1}T_{w_2}$ whenever $\ell(w_1 w_2)=\ell(w_1)+\ell(w_2)$ for all $w_1, w_2 \in W$" verbatim, but using the extended length function, and
• interpret the relation "$(T_s)^2=\ldots$ for all $s\in S$" verbatim, not worrying that $S$ does not generate $W$.
1

Probably you already figured this out since it's been almost a month, but anyway...

If you have an arbitrary (i.e. possibly non-split) reductive group $G$ and an Iwahori $I$ and you want to write the function-theoretic Hecke algebra $\mathcal{H}(G;I)$ as one of these other combinatorial Hecke algebras, you should do the following: take $W$ to be the abstract extended affine Weyl group, take $S$ to be a base of $W$ and form the combinatorial Hecke algebra in the usual way (e.g. chapter 7 in Humphreys's little grey book), except that when you are choosing the parameters $a_s$ and $b_s$ for the relations, you should replace $q$ with the index $[IsI:I]$.

I believe this non-split situation was done by Macdonald, but I'm not sure.
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