Twisted Kähler–Einstein metrics on flag varieties

In this paper, we present a description of invariant twisted Kähler–Einstein (tKE) metrics on flag varieties. Additionally, we delve into the applications of the concepts utilized in proving our main result, particularly concerning the existence of the invariant twisted constant scalar curvature Kähler metrics. Moreover, we provide a precise description of the greatest Ricci lower bound for arbitrary Kähler classes on flag varieties. From this description, we establish a sequence of inequalities linked to optimal upper bounds for the volume of Kähler metrics, relying solely on tools derived from the Lie theory. Further, we illustrate our main results through various examples, encompassing full flag varieties, the projectivization of the tangent bundle of $P^{n+1}$, and families of flag varieties with a Picard number 2.