Focusing on the Lie group SU(n) and associated symmetric spaces,
I investigate two related topics. The main result is that the bi-invariant
Einstein metric on SU(2n+1) is isolated in the moduli space of Einstein
metrics, even though it admits infinitesimal deformations. This gives a
non-Kaehler, non-product example of this phenomenon adding to the famous
example of Koiso from the eighties. I also explore the relationship between
the question of rigidity and instability (under the Ricci flow) of Einstein
metrics, and present results in this direction for complex Grassmannians.