Question

Let $\ \mathrm{Hom}(H,G)\ $ be the space of Lie group homomorphisms between compact connected Lie groups $H$, $G$. What is known about homology (or homotopy) groups of $\mathrm{Hom}(H,G)$?

UPDATE: $G$ acts on $\mathrm{Hom}(H,G)$ by conjugation, and the orbits are
a) connected, as $G$ is connected,
b) closed, as easily follows from compactness of $G$, and
c) open for it is an classical result that any nearby representations of $H$ into $G$ are conjugate; this uses compactness of $H$, and a proof can be found in the book by Conner-Floyd, "Differentiable periodic maps", Chapter VIII, Lemma 38.1.

Thus $\mathrm{Hom}(H,G)$ is the disjoint union of the $G$-orbits, and the $G$-orbit that contains a representation $r$ is homeomorphic to $G/Z_G(r(H))$, where $Z_G(r(H))$ is the centralizer of $r(H)$ in $G$.

What I do not yet understand is how to see whether $\mathrm{Hom}(H,G)$ has infinitely many connected components.

UPDATE: the topology on $\mathrm{Hom}(H,G)$ is that of uniform convergence.

Answer 1

In general there are infinitely many connected components. Look at the case when both groups are the one dimensional torus, then every $k\in\mathbb Z$ gives a homomorphism by $z\mapsto z^k$. So the set is isomorphic to $\mathbb Z$ in this case. But this phenomenon has to do this central tori. In general, the groups are products of simple Lie-groups and central tori. If both groups are simple, they have only nontrivial homomorphisms, if they are isomorphic, so it boils down to determining isomorphisms, these induce isomorphisms on the Lie-algebra which preserve the Killing form. The latter is definite, hence its isometry group is compact, which should suffice to make the set of conjugacy classes finite.