Motivation and reducibility of principal series

Question

I am trying to understand the representation theory of $GL_2(\mathbb{R})$, and I have a few questions. I know that it is generally inadvisable to put a few questions in the same topic, but since they are quite inter-related I will give it a shot like this.

I can understand the idea of classifying irreducible unitary $GL_2(\mathbb{R})$-representations by first classifying $(gl_2(\mathbb{R}), O(2))$-modules, then picking out the unitary ones from there. However,

1. To classify $(gl_2(\mathbb{R}), O(2))$-modules, one is led to study the parabolic induction. But why induce from parabolics? Is it because flag manifolds are well-studied and we can compute a lot of things about the cohomology of its vector bundles?

2. A more confusing point, is the study of irreducibility of parabolic induction. How does one study its (ir)reducibility? My main concern is that Schur's lemma fails in this case. Several places I looked at are very unsatisfying,

   • Bump first calculated the necessary conditions on irreducible admissible $(gl_2(\mathbb{R}, O(2))$-modules, then show that they all arise from parabolic induction, by actually calculating the $K$-types and the infinitesimal character. Such approach does not look very generalizable to arbitrary reductive groups, mainly because the necessary conditions on irreducible $({\frak{g}},K)$-modules in general cannot be written as easily I imagine.
   • Jacquet-Langlands directly quoted Harish-Chandra's paper, oh well..
   • Moeglin's exposition "Representations of $GL(n,\mathbb{R})$" (on Speh-Vogan's method) seems to rely on Langlands classification, and I really want to know if there are direct ways to study the reducibility.
   • Some other places focus on the unitary representations and thus use Schur's lemma. I can accept this, but what if I just want to look at $({\frak{g}}, K)$-modules in general?

I know that there are a lot of questions here, and I would be immensely grateful if someone can at least tell me where to look. Thanks!

Answer 1

Regarding your question 1., there is an important underlying fact at work here, which you may not know:

It is a theorem of Casselman (the subrepresentation theorem) that for a real reductive Lie group $G$, any irreducible admissible $(\mathfrak g, K)$-module can be embedded as a subrepresentation of a parabolically induced representation. (This generalizes an earlier result of Harish Chandra, which was the slightly weaker statement in which subrepresentation is replaced by subquotient. For more discussion, you could read Casselman's 1978 ICM talk.)

Given this general fact, it certainly makes sense to study parabolic inductions and their reducibility in order to classify irreducible admissible $(\mathfrak gl_2, O_2)$-modules (and hence also unitary representations of $\mathrm{GL}_2(\mathbb R)$).

As to how Harish Chandra and Casselman discovered their theorems, or (more-or-less equivalently) how they realized that it was useful/important to focus on parabolic inductions, I don't know if there is simple answer. Ceratinly it is related to the geometry of flag manifolds, as well as analogies with finite dimensional representations and Borel--Weil--Bott theory.

From a structural/algebraic point of view, the Iwasawa decomposition $G = KAN$ leads to a tensor decomposition of the enveloping algebra $U\mathfrak g = U\mathfrak k \otimes U\mathfrak a \otimes U\mathfrak n.$ Now on an admissible representation $U\mathfrak k$ acts through finite dimensional quotients, while $U\mathfrak a$ is closely related to the centre of $U\mathfrak g$ via the Harish Chandra isomorphism, and so also acts through a finite quotient on an irreducible representation. The upshot is that an irreducible (or more generally a f.g.) $(\mathfrak g, K)$-module will actually be f.g. over $U\mathfrak n$ (this is the so-called Lemma of Osborne). This suggests a possible relationship to parabolically induced representations.

From a geometric perspective, one knows that f.g. $(\mathfrak g, K)$-modules can be described as $K$-equivariant coherent $\mathcal D$-modules on the flag variety (this is Beilinson--Bernstein theory), while $\mathfrak g$-modules in so-called category $\mathcal O$ (so Verma-type modules, which are closely related to parabolically induced representations) can be described as $N$-equivariant $\mathcal D$-modules on the flag variety. Geometrically, one can relate the two geometric situations by using a certain flow on the flag variety to move the $K$-orbits to the $N$-orbits, and so give a geometric proof of Casselman's theorem. This is done in a paper of Emerton--Nadler--Vilonen (see e.g. here).

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Regarding 2, for general groups, the reducibility of principal series can be subtle, and even for tempered reps. is the subject of a general theory, due to Knapp--Zuckerman (see e.g. Langlands notes).

But in the case of $\mathfrak gl_2$ --- or, more-or-less equivalently --- of $\mathfrak sl_2$, the analysis is pretty easy, and very similar to the finite-dimensional situation. The induction of a character from the Borel is a direct sum of weight spaces for $SO(2)$, each of dimension one, and each composite $X^+ X^-$ and $X^- X^+$ (lowering and then raising, or raising and then lowering) acts on a given weight space by a scalar depending just on the weight and on the Casimir eigenvalue. With this information computed, it is then easy to determine any reducibility's, just by looking for weights where $X^+ X^- =0$ or $X^- X^+ =0$).

Answer 2

Let me just give a very basic answer to 1 (which may be too simple for you, I don't know). In general, the basic way to study representations of a group $G$ is to try to reduce the problem to the case of simpler subgroups $H$. For $G=GL(n)$, one of the simplest kinds of subgroups you can consider are the (proper) Levi subgroups $M$, which are direct products of $GL(m)$'s for smaller $m$'s. So you can try to construct representations of $G$ by inducing from $M$, but if you directly induce your representation will be too big. So you first extend representations of $M$ to an appropriate parabolic $P \supset M$, and induce from $P$, which gives you parabolic induction.

For $GL(2,\mathbb R)$, there is only one interesting Levi (up to isomorphism), $M = \mathbb R^\times \times \mathbb R^\times$. Thus parabolic induction lets you study representations of the very non-abelian group $GL(2,\mathbb R)$ by reducing to the abelian case of pairs of characters of $\mathbb R^\times$. Here we luck out because you can describe all irreducible unitary representations of $GL(2, \mathbb R)$ from parabolic induction. In the case of $GL(2, \mathbb Q_p)$ parabolic induction does not suffice, in some sense because there are a bunch of other interesting subgroups, and one also need to consider inducing from non-split tori or open compact subgroups like $GL(2, \mathbb Z_p)$ (which gives the so-called supercuspidal representations).