A detail in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)”

Question

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!

Answer 1

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.