Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus $\tilde{T}$. Let, $$K=\tilde{G}(O)\subset G=\tilde{G}(F)$$ be a Gelfand pair of compact and locally compact groups.


We know that for all co-character $\lambda\in Hom(\mathbb{G}\_m,\tilde{T})$ (assume $\lambda$ is in positive Weyl chamber), we have a decomposition: $$K\lambda(\pi)K=\coprod Kx\_i.$$


My question is: Is there any way to find out what $x\_i$'s are explicitly? For instance If $F=Q\_p$ and $\tilde{G}=PGL\_2$ we know that $x\_i\in\{\begin{pmatrix}p & \\ & 1\end{pmatrix},\begin{pmatrix}1 & b\\ & p\end{pmatrix}\}$. I was wondering if one can modify the Satake transform somehow so that we can recover the $x\_i$'s.


Any help or reference are highly appreciated. 



Gelfand pair and double coset decomposition
$K\pi^\lambda K$ has a transitive right action of $K$.
The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$.
Thus, $K\pi^\lambda K = \coprod\_x K\pi^\lambda x$ as $x$ runs over a set of representatives for $(K\cap \pi^{-\lambda}K\pi^\lambda)\backslash K$.


The subgroup $K\cap \pi^{-\lambda}K\pi^\lambda$ has a nice intersections with root subgroups, and I feel it should not be too hard to figure out its index in $K$. For $GL\_n$ one can write down an explicit expression in terms of $\lambda$.


