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2021 Alibaba Global Math Competition, 16
Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$, one has that \[\text{dim} V^{H_1}=\text{dim} V^{H_2},\] where $V^{H_i}$ is the subspace of $H_i$-invariant vectors in $V$ ($i=1,2$). Prove that \[Z(G) \cap H_1=Z(G) \cap H_2.\] Here $Z(G)$ denotes the center of $G$.
group theory abstract algebra center invariant representation theory algebra combinatorics invariants