Non-existence of cocompact CliffordKlein forms for SL(n,R)/\psi(SL(m,R))

Prove that for any non-trivial homomorphism \psi: SL(m,\mathbb{R}) \to SL(n,\mathbb{R}) with m < n, the homogeneous space SL(n,\mathbb{R})/\psi(SL(m,\mathbb{R})) does not admit a cocompact discontinuous group.

Background

This conjecture targets a broad class of homogeneous spaces built from embeddings of smaller special linear groups into larger ones, generalizing several previously proved non-existence results (identity and certain irreducible representations).

Resolving it would unify a spectrum of obstructions to compact CliffordKlein forms in higher rank, connecting to temperedness and dynamical volume methods.