smooth cohomology of Lie groups

Question

Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)=\{ f: G^n \to A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of abstract groups. I think this cohomology must be well studied. Can somebody provide me some references for this cohomology. A similar cohomology for topological groups has been studied by Hu and Heller using continuous cochains.

Also, can somebody tell me what kind of Lie group extensions $H^2(G,A)$ correspoind to.

Answer 1

This is indeed very well studied. The standard references are Borel-Wallach, Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Math. Studies 94, Princeton University Press (1980) and the more gentle book by A. Guichardet, Cohomologie des groupes topologiques et des algèbres de Lie Textes Mathématiques 2 Fernand Nathan, Paris (1980). But there has been a lot of progress since. For instance, there is some recent work by Crainic extending the theory to Lie groupoids and algebroids.

The continuous cohomology is usually attributed to Hochschild-Mostow and coincides with the smooth cohomology under reasonably weak hypotheses.

The interesting fact is that there is an intimate interplay between the continuous cohomology of, say a semi-simple Lie group and the cohomology of its Lie algebra and the de Rham cohomology of the associated symmetric space. One of the most important results is the van Est-isomorphism:

Let $G$ be a semi-simple Lie group with finite center and no compact factors. Let $M = G/K$ be the associated symmetric space. Then the de Rham complex \[ \mathbb{R} \to \Omega^{0}(M) \to \Omega^{1}(M) \to \cdots \] is an injective resolution of $\mathbb{R}$. In particular, since a $G$-invariant differential form on $M$ is automatically closed, there is a natural isomorphism \[ H_{c}^{\ast}(G,\mathbb{R}) \cong \Omega^{*}(M)^{G} \] where the right hand side are the $G$-invariant differential forms on $M$. If $A$ is a sufficiently nice smooth $G$-module then the $A$-valued differential forms need not be closed, but one still has \[ H_{c}^{\ast}(G,A) \cong H^{\ast}(\Omega^{\ast}(M,A)^{G}) \] Moreover, one may identify this with relative Lie algebra cohomology $H^{\ast}(\mathfrak{g},\mathfrak{k};A)$.

This immediately gives us an interpretation of $H_{c}^{2}(G,\mathbb{R}) = \Omega^{2}(M)^{G}$. Namely, if $G$ is simple, then $\Omega^{2}(M)^{G}$ is one-dimensional if and only if $G$ is Hermitian (when it is generated by the Kähler form on $M$), and otherwise it is zero. So the dimension of $H_{c}^{2}(G,\mathbb{R})$ corresponds to the number of of Hermitian simple factors of $G$.

Of course, one may also bring discrete subgroups into play, together with all their rich and beautiful connections to geometry and number theory. There are far too many things to mention here, so I'd better stop now.

Before I forget: I don't know of a direct interpretation of $H_{c}^{2}(G,A)$ as equivalence classes of suitable extensions of $G$. One problem is that in the topological category, it is not clear at all that there should be a smooth (or continuous) section of non-trivial extension.