# Cartography of the duals of GL, PGL, SL, etc

A short version of this question could be

What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?

I should obviously add some precisions.

• there are different notions of duals: admissible, unitary, tempered and generic; all are of interest;
• having a parametrization similar to the well-known one for $SL(2, \mathbf{R})$ would be perfect, where principal series, complementary series, discrete series, special representations and parameters involved in their construction appear, as well as the admissible, generic, unitary and tempered parts. Do other drawings of this type exist? even for $SL(2, \mathbf{Q}_p)$?
• I stress on $PGL$ because it suits my research purposes, yet having some ideas of the interrelations between $SL$, $PGL$ and $GL$ is probably of interest also. Representation theorists probably know how to switch from one to another, and it could be the reason I always have seen this drawing for $SL(2)$. I would like a reference to be able to do other ones;
• the question is about completions of $\mathbf{Q}$, yet more general answers on how to think/visualize representations on every local fields is obviously welcome. I believe it is enough to keep things simple, since the question in mainly about having toy models to think about.

The aim of all that is to develop some kind of intuition and to have toy models for representation theoretics properties, questions, conjectures, that are as handable and easy to think about as the $SL(2, \mathbf{R})$ case.

Any comment, idea or critics by more enlightened people on this topic to help me formulate the question are obviously welcome.

I'm assuming you're interested in complex representations of these groups. In each case the unitary, tempered and generic duals are contained in the smooth dual, so I'll start by describing that. Let $G$ be any of the groups you're interested in. Actually, most of what I'll say goes through pretty nicely for $G$ the group of points of a connected reductive group over a local field, with some modifications to take the more complicated structure into account.

The first key step in classifying representations into the broad classes you describe is parabolic induction. In $G$ there are certain parabolic subgroups $P$, which have a Levi factorization into a semidirect product $P=MN$, where $M$ is another nice group of the type you're interested in (a Levi subgroup) and $N$ is a normal subgroup. When $G=GL_2$, every parabolic is conjugate to either $GL_2$ itself (which obviously isn't going to help), or to the group $B$ of upper-triangular matrices (the Borel subgroup). The Borel subgroup has a Levi decomposition $B=TU$, where $T$ is the diagonal torus and $U$ is the group of upper-triangular matrices with 1 along the diagonal. In $SL_2$ and $PGL_2$, the same picture is true, but you restrict and project these subgroups, respectively.

Now take $\chi$ an irreducible (so one-dimensional) representation of $T$. You can inflate $\chi$ to $B$ by $B\rightarrow B/U\stackrel{\sim}{\rightarrow}T\stackrel{\chi}{\rightarrow}\Bbb{C}^\times$, and then form the induced representation $Ind_B^G\ \chi$. This is a finite-length representation. Subquotients of this induced representation break into families:

• Finite-dimensional representations. If you're working over a $p$-adic field then these are one-dimensional and factor through $\det$; if not then for each $n\geq 1$ there is a unique family of $n$-dimensional representations, where any two in each family are twists of one another by a character through the determinant.
• Principal series representations, which occur when $Ind_B^G \chi$ is irreducible.
• If $Ind_B^G\chi$ is reducible, it may or may not be semisimple. If it's semisimple then it splits as a direct sum of two irreducibles, which are called the limit of discrete series. This doesn't happen over a $p$-adic fieldd. Otherwise, it has a unique irreducible subrepresentation, which is one of the finite-dimensional representations. The quotient by this representation is formed of discrete series representations.

Over the reals or complexes, going through this process gives you all irreducibles. Over a $p$-adic field, there is a family which you don't see in this way: the cuspidal representations. These are much, much harder to construct.

The general picture is similar, but combinatorially it becomes much more complicated as the rank of the group grows. In general there are multiple conjugacy classes of parabolic subgroups, and one has to consider parabolic induction from each of these. Principal series always refers to representations parabolically induced from a Borel subgroup, and discrete series has a more general meaning as well.

Once you know the dual of $GL_2$, you basically know that of the other groups. Given $\pi$ an irreducible representation of $GL_2$, $\pi$ identifies with an irreducible representation of $PGL_2$ if and only if it has trivial central character, and all representations arise in this way. On the other hand, $SL_2$ is a normal subgroup of $GL_2$, and by Clifford theory the restriction of $\pi$ to $SL_2$ splits as a direct sum of pairwise $GL_2$-conjugate irreducible representations (moreover, each appears with multiplicity one). This restriction to $SL_2$ preserves containment in each of the above families of representations.

• As someone whose training is in the representation theory of $p$-adic groups, and knows about real groups only by (more or less flawed) analogy, I think of the discrete-series representations of a real group as being the best possible analogue of cuspidal representations of a $p$-adic group. (Of course, this is not perfect; the perfect analogy is to take the discrete series on the $p$-adic side as well.) In this setting, I think that it may be appropriate to remark that it just shakes out that the discrete-series representations of $\mathrm{SL}_2$ are finite dimensional. (Does … – LSpice Feb 28 '18 at 20:58
• … that always happen?) It may also be appropriate to mention that, although, as you say, much of what you write generalises, the existence of a discrete series of real $\mathrm{SL}_2$ is a very special thing (tied to the existence of a compact Cartan subgroup), whereas the existence of cuspidal representation of a $p$-adic group is not special at all (tied, in some sense, to the fact that a semisimple $p$-adic group has compact Cartan subgroups in abundance). – LSpice Feb 28 '18 at 21:00
• Also, I should have started by saying that this is a very nice survey answer to an extremely broad question. – LSpice Feb 28 '18 at 21:00
• Many thanks for this great answer! It encapsulates so much that efficiently, on a topic where references are usually far too broad to be digested by someone not used to it, or not able to spend months learning the theory! – Desiderius Severus Apr 9 '18 at 7:58
• If I can afford a further question, are the unitary/generic/tempered properties straightforwardly classified among the admissible dual or is it very contextual/another theory? – Desiderius Severus Apr 9 '18 at 7:59