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As explained by Torsten Ekedahl in this reply, the problem of classifying the irreducible (infinite dimensional) representations of $sl_2(\mathbb{C})$ is a wild one. The notion of wild classification problem was discussed in this question. It means that the problem in a sense contains the problem of reducing pairs of matrices over a field to a canonical form by simultaneous conjugation.

Let $F$ be a non-Archimedean local field. Is the classification of supercuspidal representations of $\mathrm{GL}_n(F)$ a wild classification problem?

The work of Bushnell and Kutzko [The admissible dual of $\mathrm{GL}(N)$ via compact open subgroups] gives a classification of the supercuspidal representations of $\mathrm{GL}_n(F)$, and the local Langlands correspondence for $\mathrm{GL}_n$ also gives a parametrisation of the supercuspidal representations. However, it is not obvious to me whether any of these classifications answers the question.

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I cannot directly answer your question because I am not a specialist of wild classification problems. However I want to make the following remark that could help.

I was told that the problem of classifying the irreducible (smooth) complex representations of ${\rm GL}_n ({\mathfrak o}_F )$, where ${\mathfrak o}_F$ is the ring of integers of a local field $F$, is wild (at least for $n$ large).

Moreover to construct and classify supercuspidal representations, Bushnell and kutzko obtain them as compactly induced representations from irreducible smooth representations of compact mod center subgroups. In particular we obtain a large class of supercuspidals by inducing certain irreducible complex representations of $F^{\times}{\rm GL}_n ({\mathfrak o}_F)$.

However in Bushnell and Kutzko's theory not all representations of ${\rm GL}_n ({\mathfrak o}_F )$ are needed, but only very particular ones and one knows how to construct them effectively. Indeed the whole Bushnell and Kutzko's construction is based on the theory of 'simple characters' which is in principal entirely effective.

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  • $\begingroup$ In general there exist many supercuspidals which are not induced from $F^\times \mathrm{GL}_n(\mathfrak{o}_F)$, so the classification of all supercuspidals may well be wild even if the classification of the supercuspidals induced from $F^\times \mathrm{GL}_n(\mathfrak{o}_F)$ is tame. In what sense is the theory of simple characters in principle entirely effective? $\endgroup$
    – A Stasinski
    Commented Apr 11, 2010 at 12:47
  • $\begingroup$ For the other supercuspidal representations one has to induce from the stabilizer of a uniform ${\mathfrak o}_F$-lattice chain in $F^n$, that is from a maximal compact mod centre subgroup. The number of conjugacy classes of such subgroups equals the number of divisors of $n$. For such a subgroup $K$ one only needs to induce certain very particular irreducible representations. As in the case of $F^{\times}{\rm GL}_n ({\mathfrak o}_F)$, there are constructed by using simple characters. $\endgroup$
    – Paul Broussous
    Commented Apr 11, 2010 at 16:00
  • $\begingroup$ When I write that the theory of simple characters is in principle effective I mean that one should be able to explicitely write down all simple characters needed to construct the supercuspidal representations. But maybe some difficulties are hidden. $\endgroup$
    – Paul Broussous
    Commented Apr 11, 2010 at 16:05

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