Twisted Kähler-Einstein metrics on flag varieties

In this paper, we present a description of invariant twisted K & auml;hler-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 K & auml;hler metrics. Moreover, we provide a precise description of the greatest Ricci lower bound for arbitrary K & auml;hler classes on flag varieties. From this description, we establish a sequence of inequalities linked to optimal upper bounds for the volume of K & auml;hler 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 Pn+1${\mathbb {P}}<^>{n+1}$, and families of flag varieties with a Picard number 2.