Equivariant Morse index of min-max $G$-invariant minimal hypersurfaces

For a closed Riemannian manifold $M^{n+1}$ with a compact Lie group $G$ acting as isometries, the equivariant min-max theory gives the existence and the potential abundance of minimal $G$-invariant hypersurfaces provided $3\leq {\rm codim}(G\cdot p) \leq 7$ for all $p\in M$. In this paper, we show a compactness theorem for these min-max minimal $G$-hypersurfaces and construct a $G$-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a $C^\infty_G$-generic finiteness result for min-max $G$-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min-max minimal hypersurfaces to the equivariant setting. Namely, the closed $G$-invariant minimal hypersurface $\Sigma\subset M$ constructed by the equivariant min-max on a $k$-dimensional homotopy class can be chosen to satisfy ${\rm Index}_G(\Sigma)\leq k$.