To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, Kutzko, Stevens etc. The other is the construction given by Yu,Jiu-Kang (for more general groups).


Both constructions are very difficult, especially the construction via theory of types, which relies heavily on some huge algebra machinery, and is sort of artificial to me. 


So here I'm asking some questions about these constructions.


(1) What are the intuitions behind these constructions? 


(2) Are there some links between these two constructions? 


Much appreciated for any (partial) answer/reference. 



Questions on constructions of supercuspidal representations
There are certainly links between the two; a good place to start would be looking at the theory for $GL\_2$ and $SL\_2$. Henniart's appendix to Breuil-Mezard's *Multiplicités modulaires et représentations de $GL\_2(Z\_p)$ et de $Gal(\bar{Q\_p}/Q\_p)$ en $l=p$* makes explicit the construction for $GL\_2$ via strata, and it's simple enough to see that you can do more or less the same thing for $SL\_2$ (now requiring your stratum to be simple $[\mathfrak{A},n,0,\beta]$ with $\beta\in\mathfrak{sl}\_2$), while Nevins' *Branching rules for supercuspidal representations of $SL\_2$* gives an introduction to Yu's construction specified somewhat to the case of $SL\_2$.


In this case, it's pretty clear that both constructions are doing more or less the same thing. Yu starts with a "tamely ramified cuspidal $G$-datum" $(T,y,r,\phi)$, i.e. a quadruple where:


* $T$ is an anisotropic torus of $SL\_2$, i.e. the group of points in $E^\times\cap SL\_2$ for $E/F$ a quadratic extension embedded into $\mathrm{Mat}\_2(F)$ in a suitable way;
* $y$ is a point in the building of $SL\_2$, where it turns out that we only ever need three points, $0$, $1/2$ and $1$ in order to determine the necessary parahorics;
* $r$ is (essentially) the level of the supercuspidal that you're constructing;
* $\phi$ is a "generic" character of $T$.


Yu then defines a suitable character $\psi$ on a group $H$ arising from the Moy-Prasad filtration, such that $\psi$ agree with $\phi$ on $T\cap H$, and extends this to a larger group from which the supercuspidal is induced.


On the other hand, here's the approach via strata. You start with a "simple stratum" $[\mathfrak{A},n,0,\beta]$, where:


* $\mathfrak{A}$ is a hereditary $\mathfrak{o}\_F$-order, which is essentially defining which parahoric we use, just as the $y$ in Yu's construction did;
* $n$ is (essentially) the level of the supercuspidal;
* $\beta$ defines a quadratic extension $F[\beta]/F$, as well as a character $\psi\_\beta$ of a group $H$ in the filtration of the parahoric $\mathfrak{A}^\times$.


Then, as before, you extend, this time going from your filtration subgroup $H$ to the product $TH$, where $T=F[\beta]^\times$ is an anisotropic torus, then extend in to the same larger group from which Yu induced the supercuspidal (Heisenberg and $\beta$-extensions, although for $SL\_2$ you can be more naive and just choose any suitable extension in exactly the same way as Henniart does for $GL\_2$).


Both of these constructions look very similar: they specify more or less the same information about your supercuspidal in slightly different forms, and they end up inducing the supercuspidal from the same group $TH$ for $T$ an anisotropic torus and $H$ a filtration subgroup. The main difference is that Yu begins by defining the representation on $T$, while Bushnell-Kutzko begin by defining the representation on $H$ (or at least on some slightly smaller group contained in $H$).


