A Nekrasov-Okounkov Type formula for $\widetilde{C}$

In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function attributed to Nekrasov and Okounkov (which was actually first proved the same year by Westbury) by using a famous identity of Macdonald in the framework of type $\widetilde{A}$ affine root systems. In this paper, we obtain new combinatorial expansions of powers of $\eta$, in terms of partition hook lengths, by using the Macdonald identity in type $\widetilde{C}$ and a new bijection between vectors with integral coordinates and a subset of $t$-cores for integer partitions. As applications, we derive a symplectic hook formula and an unexpected relation between the Macdonald identities in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$. We also generalize these expansions through the Littlewood decomposition and deduce in particular many new weighted generating functions for subsets of integer partitions and refinements of hook formulas.