Symmetric pairs and associated commuting varieties
  We obtain a series of new results on the problem of irreducibility of commuting varieties associated with symmetric pairs or, in other words, $Z_2$-graded simple Lie algebras. In particular, we present many examples of reducible commuting varieties and show that the number of irreducible components can be arbitrarily large. 
Dmitri Panyushev, Oksana Yakimova