Consider a local non archimedean field $k$ and its ring of integer $o$. To what extent, do we know the complex irreducible representations of $GL(2,o)$? Is there a specific list giving them all in terms of induction from certain "simple" subgroups?

I have had read, that they have been classified according to their characters, but I am not quite happy with the various presentations so far.

What I'd like to see is pretty simple to explain:

  1. Classify all irreducibles of the group (Borel subrgoup) of upper triangular matrices $B$ (easy).
  2. Given an irreducible representation $\pi$ of $B$, decompose $Ind_{B}^{GL(2,o)} \pi$ in its irreducibles (hard).

Is there such a treatment available? If not, what is the reason why not to proceed along these lines?

  • $\begingroup$ Is there any reason to expect that the completion of (1) and (2) will exhaust the spectrum? $\endgroup$ – LSpice Jul 13 '11 at 16:04
  • 1
    $\begingroup$ Yes since all irreducible complex representation of $GL(2, o)$ factor through the finite group $GL(2,o/n)$ for some ideal $n$. Then using $Ind_B^G Ind_1^B 1 = Ind_1^G 1$ gives all irreducible representations. $\endgroup$ – Marc Palm Jul 13 '11 at 16:21
  • $\begingroup$ A charming question! $\endgroup$ – paul garrett Jul 13 '11 at 23:18

All irreducible representations of ${\rm GL}(2, {\mathcal O})$ have been constructed by Alexender Statinski :

Stasinski, Alexander (2009) The smooth representations of ${\rm GL}(2, {\mathcal O})$.

Reference in Math. Arxiv : http://arxiv.org/abs/0807.4684

But he does not follow the procedure that you propose at all. He uses much more generalizable tools : Clifford theory and an adapted version of Kirillov's orbit method. These ideas are now classical in type theory for $p$-adic reductive groups (see e.g. Howe and Kutzko's works).

Some people are now working on the more general question of constructing the representations of ${\rm GL}(n,{\mathcal O})$ (A. Stasinski, A.--M. Aubert, F. Courtès, and others). This problem is known to be "wild"

  • $\begingroup$ The approach via Clifford theory and orbits was used by Hill to construct all the regular representations of $\mathrm{GL}_n(\mathcal{O}_r)$, for $n\geq 2$ and $r$ even (and also in certain cases when $r$ is odd). This makes this approach to the representations of these groups one of the most general currently available. $\endgroup$ – A Stasinski Jul 16 '11 at 7:40

Paul Broussous has already addressed the question in the title, and since it seems that you are in fact more interested in the last two questions, I will focus on these.

I don't think there is a treatment following steps 1) and 2) available in the literature, but it should be possible to work this out since the representations of $\mathrm{GL}_2(\mathcal{O})$ are known explicitly. On the other hand, it is not clear how useful this would be, because already for $\mathrm{GL}_2$ over a finite field $\mathbb{F}_q$, this is not the preferred approach to the representations (apart from the principal series, of course). This is partly because it quickly becomes unmanageable, and does not allow for an inductive approach in the same way as parabolic induction. Moreover, over a finite field even step 1) is problematic in general, because the irreps of $B(\mathbb{F}_q)$ are related (via Clifford theory) to the irreps of the upper uni-triangular subgroup $U(\mathbb{F}_q)$, and the classification of the latter is known to be a wild problem (for large enough p, one can describe the irreps of $U(\mathbb{F}_q)$ using a Kirillov orbit method formalism, but it is not clear that this would be helpful for the problem at hand).

Also, in step 2) there is going to be a lot of repetition, that is, two induced representations may have irred constituents in common, and there may be no easy way to tell when this happens and for which irred constituents.

Finally, as you mention in one of the comments, the approach via steps 1) and 2) can only work if there is a good description of the $B$-$B$ double cosets, and this is not available in general. Note however that this is not the only place where wildness enters, as the above remarks on the situation for groups over a finite field shows. One advantage of the approach via Clifford theory and orbits (following Hill), is that one can avoid wild classification problems by restricting the construction to a subcategory of representations, such as the regular representations.


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