A spectral expression for a certain orbital integral

Let $F$ be a $p$-adic field, $G={\text{GL}}_{2n}(F)$ and ${\theta }_0$ be the exterior automorphism of $G$ that fixes a pinning of a Borel pair. Consider the set $\tilde{G}=G{\theta }_0$ on which $G$ acts by conjugacy and the orbital integral $J_{\tilde{G}}({\theta }_0,f)$ at ${\theta }_0$. We prove a Plancherel-Harish-Chandra type formula for this orbital integral, namely as an integral over the irreducible tempered auto-dual representations of $G$ that we call “symplectic” (meaning their Langlands parameter factors through ${\text{Sp}}_{2n}(\mathbb{C})$). This solves a problem raised by G. Chenevier and L. Clozel. Our method uses the endoscopic transfer to ${\text{SO}}_{2n+1}$. Along the way, we also prove that the Plancherel measure is constant on $L$-packets.