Kähler-Einstein metrics on symmetric Fano T-varieties

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kähler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being Kähler-Einstein but admitting deformations without Kähler-Einstein metric.