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

Auto-correlation functions of Sato-Tate distributions and identities of symplectic characters