# Questions on constructions of supercuspidal representations

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.

(1) What are the intuitions behind these constructions?

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

Much appreciated for any (partial) answer/reference.

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 $H$).

• I've always felt that Adler's thesis msp.org/pjm/1998/185-1/pjm-v185-n1-p01-s.pdf , which pre-dated and inspired Yu's paper, is one of the best entrées to Yu's approach. (If you really want to go to the beginning, then you can read the various papers of Corwin, Howe, and Corwin–Howe, but I think that Adler is a good compromise, describing a more modern perspective in a usually non-intimidating level of generality.) – LSpice Jun 5 '16 at 23:28

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.

I will try to answer the question for the theory of types. I don't know the other stuff.

What does it mean to construct the unitary dual (one of the main goals of rep-theory): Classify all reps and construct them, where constructing means often finding the as subquotient of induced representations.

1. Observation: You can construct supercuspidal reps as (compactly) induced representations from a maximal subgroup, which is compact modulo the center.

2. So it is sufficient to classify these maximal subgroups and the reps, which give something supercuspidal if induced.

Once you have accepted this, you should look at the construction in Stasinski's paper "The smooth representations of GL(2,O)" to understand the relation to definition of strata coming from Clifford theory. Bushnell-Henniart "Local Langlands for GL(2)" don't address Clifford theory explicitly, but are helpful reference except for motivating "strata".

• Maybe I have this backwards, or maybe the 'backward'ness is intentional to facilitate intuition, but isn't it the case that the only way we know (1) in full generality is by performing the construction in (2), and showing that it is exhaustive? – LSpice Oct 20 '16 at 20:07