Tian’s $$\alpha _{m,k}^{{\hat{K}}}$$-invariants on group compactifications

In this paper, we compute Tian’s $$\alpha _{m,k}^{K\times K}$$ -invariant on a polarized G-group compactification, where K denotes a maximal compact subgroup of a connected complex reductive group G. We prove that Tian’s conjecture (see Conjecture 1.1 below) is true for $$\alpha _{m,k}^{K\times K}$$ -invariant on such manifolds when $$k=1$$ , but it fails in general by producing counter-examples when $$k\ge 2$$ .