TITLE: Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: 1) On the groups $U({\mathfrak A}))$ and $U^n ({\mathfrak A})$. Their structure theory are very well explained in:
Bushnell, C. J.; Fröhlich, A. Nonabelian congruence Gauss sums and $p$-adic simple algebras. Proc. London Math. Soc. (3) 50 (1985), no. 2, 207–264. 
But as PL wrote, you should start with the case $GL_p$, $p$ prime. The structure of these groups in that case is explained (maybe with a different notation) in Carayol's paper. 
2) Before tackling Bushnell and Kutzko's theory, except looking at the $GL_p$ case, you should first read introductory papers :
Henniart, Guy Représentations des groupes réductifs $p$-adiques. (French) [Representations of $p$-adic reductive groups] Séminaire Bourbaki, Vol. 1990/91. Astérisque No. 201-203 (1991), Exp. No. 736, 193–219 (1992). 
Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm GL}_N$ via restriction to compact open subgroups. Harmonic analysis on reductive groups (Brunswick, ME, 1989), 89–99, Progr. Math., 101,
Kutzko, Philip C. Smooth representations of reductive $p$-adic groups: an introduction to the theory of types. Geometry and representation theory of real and $p$-adic groups (Córdoba, 1995), 175–196, Progr. Math., 158, Birkhäuser Boston, Boston, MA, 1998.