  The Sato-Tate distributions for genus 2 curves (conjecturally) describe the
statistics of numbers of rational points on the curves. In this paper, we
explicitly compute the auto-correlation functions of Sato-Tate distributions
for genus 2 curves as sums of irreducible characters of symplectic groups. Our
computations bring about families of identities involving irreducible
characters of symplectic groups $Sp(2m)$ for all $m \in \mathbb Z_{\ge 1}$,
which have interest in their own rights.
