Log-canonical thresholds under torus quotients

The log-canonical threshold of a Fano variety $X$ is an invariant with applications in birational geometry as well as in Kahler geometry. It is defined with respect to a finite subgroup $G$ of $Aut(X)$. After chosing a maximal torus $T$ in the automorphismen group of our Fano variety $X$ we would like to reduce the computation of the log-canonical threshold on X to that of a log-canonical threshold on some torus quotient $X=X/T$. As it turns out, this works well if the following conditions are fulfilled:
(i) $G$ is contained in the normalizer of the maximal torus,
(ii) the $G$-action on the characters given by conjugation has the trivial character as its unique fixed point.
In this situation the $T$-variety is called symmetric. As an application we provide a criterion for the existence of Kahler–Einstein metrics on symmetric Fano $T$-varieties of complexity one.