Here is an issue that thoroughly confuses me. I hope I can express it in a way that is clear cut enough for this site.
Let $G$ be a real reductive Lie group and $\mathfrak{g}$ be the complexification of its Lie algebra. (We work with complexifications since we are interested in complex representations.) Let $P$ be a minimal parabolic subgroup of $G$. The complexification of its Lie algebra inside $\mathfrak{g}$ has Langlands decomposition $\mathfrak{m} \oplus \mathfrak{a} \oplus \mathfrak{n}$. Let $\mathfrak{t}$ be maximal abelian subspace of $\mathfrak{m}$. Then if I'm not mistaken (and, if I am mistaken, please edit this statement accordingly) $\mathfrak{b} := \mathfrak{t} \oplus \mathfrak{a} \oplus \mathfrak{n}$ is a Borel subalgebra of $\frak{g}$.
Let $\lambda: \mathfrak{a} \to \mathbb{C}$ be character, seen as a one dimensional representation of $\mathfrak{a}$ on a representation space that we will denote $\mathbb{C}_\lambda$. We extend $(\lambda, \mathbb{C}_\lambda)$ to a representation of $\mathfrak{b}$ and $\mathfrak{m} \oplus \mathfrak{a} \oplus \mathfrak{n}$ by letting $\mathfrak{m}$ and $\mathfrak{n}$ act trivially.
Now there seem to be two common ways to create a representation of $\mathfrak{g}$ out of $\lambda$.
1) [Parabolic induction] Integrate $\lambda$ to a representation of the group $P$. Induce this to a representation of G as follows: consider all smooth functions $f$ on $G$ with values in $\mathbb{C}_\lambda$ with the additional property that $f(pg) = \lambda(p)g$ for every $p \in P, g \in G$, where the group G acts in the right regular representation. Now since we are looking at smooth functions, the action of the Lie algebra $\mathfrak{g}$ in this representation is well defined and since, for the sake of this question, we are [pretending to be] only interested in this $\mathfrak{g}$-action we restrict our attention to the (dense) subspace of $\mathfrak{g}$-finite vectors.
2) [Verma-type modules] We consider the enveloping algebra $U(\mathfrak{g})$ as simultaneous being a left $U(\mathfrak{b})$- and a right $U(\mathfrak{g})$-module in the obvious way and create a right $U(\mathfrak{g})$-module (hence a $\mathfrak{g}$-rep) out of the $\mathfrak{b}$-rep $\lambda$ by taking the tensor product $\mathbb{C}_\lambda \otimes_{U(\mathfrak{b})} U(\mathfrak{g})$.
Now in spite of these two $\mathfrak{g}$-representations 'feeling' similar and both constructions being called induction on occasion, they are not the same. (This only became clear to me recently.) Hence my question:
what is the precise relationship between the two constructions?
Concretely:
- Can the $\mathfrak{g}$-representations constructed under 1) be expressed in terms of the representations under 2) and vice versa?
- In particular: is there a way of constructing the modules under 1) that avoid passing via the group? (Not that I dislike the group, the globalization of $\lambda$ is as painless as it could be given that $\mathbb{C}_\lambda$ is finite dimensional and the analytic subgroup $A$ of $G$ associated to $\mathfrak{a} \cap Lie(G)$ is simply connected, but just out of curiosity.)
Here are some links that I found useful. Vogan [1] explains the difference between the two notions of induced representation found in the literature by comparing them to their analogues for finite groups. (Where, ironically, they are the same, but not trivially so.) However he does not discuss construction 2) above explicitly. Stroppel [2] on the other hand gives a clear and detailed account of many constructions related to 2) and in particular constructs 'principal series modules' as $\mathfrak{g}$-finite homomorphisms between a Verma module and the dual of a Verma module. In the introduction of that text there is a remark in which the wishful thinker might read that the modules constructed under 1) above (and which are also known as (spherical) principal series) are indeed special cases of this construction. However, that particular subject seems not to be mentioned again in the main text.
I hope someone can clarify the relation between these two constructions for me. In fact a better understanding of what is going on only in the special case of $G = SL(2, \mathbb{R})$ would already make me more than happy.
Best Answer
I don't know whether this helps, but there is a nice description of the relation between the two modules in geometric terms. The principal series representation can be viewed as the space of smooth sections of the homogeneous line bundle $E$ over $G/P$ induced by the representation $\lambda$ of $P$. Then you can form the infinite jet prolongation $J^\infty E$ of this line bundle, so the fiber of this in a point is the formal Taylor series of a section in that point. This is a homgeneous vector bundle over $G/P$ since it is obtained from $E$ by a functorial construction, so in particular the fiber over $o=eP$ naturally inherits a representation of $P$. While this fiber is not preserved by the action of $G$, it is stable under the infintiesimal action of $\mathfrak g$, thus becoming a $(\mathfrak g,P)$-Module. This module is the dual to the Verma module induced by $\lambda$.
This dualtity extends to generalized verma modules and in that form it is the basis for the relation between conformally invariant differential operators on a sphere and homomorphisms of generalized Verma modules, which motivated many developments in conformal geometry and, more generally, parabolic geometries.
I don't know whether this is written up somewhere in detail, but the preprint version an artilce of J. Slovak, V. Soucek and myself which is available at http://arxiv.org/abs/math/0001164 contains an exposition.