A cyclic sieving phenomenon for symplectic tableaux

We give a cyclic sieving phenomenon for symplectic $\lambda$-tableaux, $SP(\lambda, 2m)$, where $\lambda$ is a partition of an odd positive integer $n$ and $gcd(m,p)=1$ for any odd prime $p\leq n$. We use the crystal structure on Kashiwara-Nakashima symplectic tableaux to get a cyclic sieving action as the product of simple reflections in the Weyl group. The cyclic sieving polynomial is the $q$-anologue of the hook-content formula for symplectic tableaux.