I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.

Let me first describe the book a little.

The book uses hereditary orders $\mathfrak{A}$ arising from lattice chains and then defines simple strata using the special integer $k_0(\beta,\mathfrak{A})$ (cf. page $43$). Then in page $95$ the authors construct the subgroups $H^m(\beta,\mathfrak{A})$ and $J^m(\beta,\mathfrak{A})$ and uses them (specifically $J^0$ and $J^1$ ) in the later chapters to extend representations of $J^1$ to $J^0$ and then they define the simple types $(J,\lambda)$ (cf. page $184$). In page $162$ they define the maximal and minimal orders $\mathfrak{B}_M $ and $\mathfrak{B}_m$ and uses them throughout the theory. Chapter $7$ proceeds with the Iwahori decomposition where they define (cf. page $213$) the parabolic subgroups $P$ with Levi decomposition $P=MU$, where $U$ is the unipotent radical of the group $G$ in consideration. The book finishes (chapter $8$) with the classification of smooth representations: the supercuspidals containing a simple type and the 'atypical' representations containing the split types.

Being a beginner in this field it is necessary for me to have a concrete example to keep in mind while reading the above book because it is too general (construction using lattices and so on). Therefore I would like to have a concrete example of the following:

$(a)$ the hereditary order $\mathfrak{A}$,

$(b)$ the groups $U^n(\mathfrak{A})$ in page $21$,

$(c)$ the integer $k_0(\beta,\mathfrak{A})$,

$(d)$ the groups $H^m(\beta,\mathfrak{A})$ and $J^m(\beta,\mathfrak{A})$, specifically when $m=0,1$,

$(e)$ the utility of the maximal and the minimal hereditary orders $\mathfrak{B}_M$ and $\mathfrak{B}_m$ and there concrete examples,

$(f)$ concrete examples of the groups $P,M,U,U^-$ defined in $(7.1.13)$.

$(g)$ the importance of computing the interwining of representations (cf ($5.1.8$), ($6.1.3)$, e.t.c).

Regarding the Interwining of characters, it is not clear to me why the Interwining is impotant in representation theory. What role does it play? I agree that in chapter $8$ there is a theorem which says that the supercuspidal representations twisted by unramified characters have the same simple types. But I do not understand the impotantance of computing the Interwining of representations.

I know that answering these questions above will consume long typing in latex. I thank you in advance for your efforts.