TITLE: Questions on constructions of supercuspidal representations
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: In complement to the other answers I would like to add the following. 
There are of course some links between both construction. For instance for ${\rm GL}(N)$ Yu's construction amounts to Howe's construction (cf. his Pacific Journal paper) based on admissible pairs. On the other hand the Bushnell and Kutzko construction is based on the formalism of strata. It is possible to rewrite Howe's construction in terms of strata so that it appears to be a particular case of Bushnell and Kutzko's work. 
The point is that Howe's construction exhaust all supercuspidal only under a tameness condition (the residue characteristic of $F$ must not divide the $N$ of ${\rm GL}(N,F)$). If this tameness condition is not fulfilled then supercuspidal representations cannot be parametrized by the admissible pairs: they are too simple parameters.  
Indeed in the so called wild case, nobody succeeded in finding canonical parameters for the "automorphic side". The only known canonical parameter is the Galois parameter which in the tame case only can be expressed in terms of admissible pairs. 
In the formalism of strata one must make choices to construct the type of a supercuspidal representation. 
The formalism of Bushnell and Kutzko is in some sense more powerfull since it exhaust the supercuspidal representations. On the other hand it uses techniques
that force to restrict to the case of classical groups. Indeed the theory needs to consider anisotropic tori in the group that split under wild extensions and the Bruhat-Tits building behaves very badly under wild extensions.