I am recently troubled with a computational detail in Bushnell and Henniart's book, "The local Langlands conjecture for Gl(2)". Let $(\mathfrak{A},n,\alpha)$ be a simple stratum, and define $K\_\mathfrak{A}$ as the group
$\{ g \in G \mid g\mathfrak{A}g^{-1} = \mathfrak{A} \} $, and it can be proved that $K\_\mathfrak{A}=F[\alpha]^\times U\_\mathfrak{A}$, again we define the group $J\_\alpha=F[\alpha]^\times U\_\mathfrak{A}^{\left[\frac{n}{2}\right]+1}$


On page 173 of the book, in order to find out of the dimension of the representation $\Xi$ in the cuspidal inducing datum $(\mathfrak{A},\Xi)$, we need to compute the index $(K\_\mathfrak{A}:J\_{\alpha})$, but I cannot dope out a method. I would appreciate a lot if you could tell me an approach of how to compute it!



A detail in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)”
I do not give all details, but the main steps of the calculation. Put $E=F[\alpha ]$ and $m=[n/2]+1$. 


First $K\_{\mathfrak A}/J\_\alpha =U\_{\mathfrak A}/(U\_{\mathfrak A}\cap E^\times )U\_{\mathfrak A}^m$.


Next observe that $U\_{\mathfrak A}\cap E^\times = U\_E$, where $U\_E$ is the group of units of the ring of integers of $E$. 


We deduce that $[K\_{\mathfrak A}:J\_\alpha ] = [U\_{\mathfrak A} :U^m\_{\mathfrak A}]/ [U\_E U\_{\mathfrak A}^m : U^m\_{\mathfrak A}]$


Next $U\_E U\_{\mathfrak A}^m / U^m\_{\mathfrak A} = U\_E /(U\_E \cap U\_{\mathfrak A}^m ) = U\_E /U\_E^m$, where $U\_E^m =1+{\mathfrak p}\_E^m$, ${\mathfrak p}\_E$ the maximal ideal of ${\mathfrak o}\_E$, the ring of integers of $E$. 


So finally one gets $[K\_{\mathfrak A}:J\_\alpha ] = [U\_{\mathfrak A} :U^m\_{\mathfrak A}]/ [U\_E : U\_E^m ]$, so that your are reduced to computing the cardinality of quotients of the standard filtrations of $U\_E$ and $U\_{\mathfrak A}$ respectively. The structures of theses quotients are either described in e.g. Serre Local fields for $p$-adic fields or in Bushnell and Henniart's book for unit groups of hereditary orders. 


