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In this paper, we compute Tian's $\alpha_{m,k}^{K\times K}$-invariant on apolarized $G$-group compactification, where $K$ denotes a maximal compactsubgroup of a connected complex reductive group $G$. We prove that Tian'sconjecture (see Conjecture 1.1 below) is true for $\alpha_{m,k}^{K\timesK}$-invariant on such manifolds when $k=1$, but it fails in general byproducing counter-examples when $k\ge 2$.

Tian's $\alpha_{m,k}^{\hat K}$-invariants on group compactifications