Integral string topology of the symplectic group and BV rigidity for compact Lie groups

We compute the integral Batalin-Vilkovisky (BV) algebra ${\mathbb{H}}_{\ast }(L\mathrm{S}\mathrm{p}(n);\mathbb{Z})$ of the free loop space of the symplectic group. We then show that for a connected compact Lie group $G$ whose homology over a commutative ring $R$ is exterior on odd primitive generators, ${\mathbb{H}}_{\ast }(LG;R)$ is the BV algebra of the free loop space of the product of odd spheres with the classical exponents of $G$, tensored with the group ring of the torsion of ${\pi }_1(G)$. Conversely, over an integral domain the graded algebra ${\mathbb{H}}_{\ast }(LG;R)$ already determines the exponents and the group ring, hence the BV algebra. If moreover $G$ is simply connected, then ${\mathbb{H}}_{\ast }(LG;R)$, as a BV algebra, is the Hochschild cohomology of $H^{\ast }(G;R)$ with the BV operator induced by Poincaré duality.