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We develop an equivariant Cerf theory for Morse functions onfinite-dimensional manifolds with group actions, and adapt the technique to theinfinite-dimensional setting to study the moduli space of perturbed flat$SU(n)$-connections. As a consequence, we prove the existence of perturbative$SU(n)$ Casson invariants on integer homology spheres for all $n\ge 3$, andwrite down an explicit formula when $n=4$. This generalizes the previous worksof Boden and Herald.

Equivariant Cerf theory and perturbative $SU(n)$ Casson invariants