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

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!

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.