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Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which are geralizations of quasimorphisms. In the present paper, we prove a Bavard-type duality theorem between conjugation-invariant (pseudo-)norms and subset-controlled quasimorphisms on stable groups. We also pose a generalization of our main theorem and prove that "stably non-displaceable subsets of symplectic manifolds are heavy" in a rough sense if that generalization holds.

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Biran, Polterovich and Salamon defined a relative symplectic capacity which indicates the existence of 1-periodic non-contractible closed trajectories of Hamiltonian isotopies. Many of researches have used the Hamiltonian Floer theory on non-contractible trajectories for giving upper bounds of Biran-Polterovich-Salamon's capacities. However, in the present paper, we use the Oh-Schwarz spectral invariants which are defined in terms of the Hamiltonian Floer theory on contractible trajectories for a similar purpose.