Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces

Question

Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, respectively. Let $X_{i} = G_{i}/K_{i}$ denote their symmetric spaces. Suppose we have a representation $\phi : G_{1} \rightarrow G_{2}$ such that $\phi(K_{1}) \subset K_{2}$. Then $\phi$ naturally induces a map $\phi_{*} : X_{1} \rightarrow X_{2}$ between their symmetric spaces via $\phi_{*}(gK_{1}) = \phi(g)K_{2}$. I am interested in knowing if there are any conditions on the map $\phi$ that give us useful information about how the map $\phi_{*}$ behaves with respect to the large-scale geometry of the symmetric spaces.

More precisely, letting $\mathfrak{a}_{i}$ denote the Cartan subalgebras associated to the Lie algebras $\mathfrak{g}_{i}$ of the $G_{i}$, we can define metrics on the symmetric spaces $X_{i}$ as follows: let $\kappa_{i} : G_{i} \rightarrow \mathfrak{a}_{i}^{+}$ denote the Cartan projections of the two Lie groups, and let $\| \cdot \|_{i}$ be some choices of $K_{i}$-invariant norms on $\mathfrak{g}_{i}$. Then we obtain metrics $d_{i}$ on $X_{i}$ via $$d_{i}(gK_{i}, hK_{i}) = \|\kappa_{i}(g^{-1}h)\|_{i}.$$ My question is then the following: If $\phi : G_{1} \rightarrow G_{2}$ is a faithful, irreducible representation, is the map $\phi_{*} : (X_{1}, d_{1}) \rightarrow (X_{2}, d_{2})$ a quasi-isometric embedding? If yes, can we get away with $\phi$ only being faithful, but not necessarily irreducible? Any references or comments are greatly appreciated! Thank you!

Answer 1

Karpelevich and Mostow independently proved that there exists an equivariant totally-geodesic embedding $f: X_1\to X_2$ (irreducibility is irrelevant, all you need is compactness of the kernel). By equivariance, $f$ is an isometric embedding if we equip $X_1$ with the $G_1$-invariant pull-back metric of the metric in $X_2$. The new metric is, or course, bi-Lipschitz to the original metric on $X_1$. Now, for any other equivariant map $f': X_1\to X_2$ (such as your map $\phi_*$), by equivariance we have $$ \sup_{x\in X_1} d(f(x), f'(x)) <\infty. $$ Since $f$ is a qi embedding, so is $f'$.

F. I. Karpelevich, Surfaces of transitivity of semisimple group of motions of a symmetric space. Doklady Akad. Nauk SSSR 93:401–404, 1953.

G. Mostow, Some new decomposition theorems for semi-simple groups, Mem. Amer. Math. Soc., 14:31–54, (1955).

A. J. Di Scala, C. Olmos, Corrigendum for "A geometric proof of the Karpelevich-Mostow theorem", arXiv, 1104.0892.