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For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connectedsubgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version ofthe $\delta$-threshold. We prove that for a Fano variety $(X, -K_X)$ and aconnected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes$G$-equivariant uniform $K$-stability. We also use this invariant toinvestigate $G$-equivariant $K$-stability of some Fano varieties with largegroups of symmetries, including spherical Fano varieties. We also consider thecase of $G$ being a finite group.

Delta-invariants for Fano varieties with large automorphism groups