Estimating the eigenvalues on Quaternionic K\"ahler Manifolds

We study geometric first order differential operators on quaternionic K"ahler manifolds. Their principal symbols are related to the enveloping algebra and Casimir elements for $\text{Sp}(1)\text{Sp}(n)$. This observation leads to anti-symmetry of the principal symbols and Bochner-Weitzenb"ock formulas for operators. As an application, we estimate the first eigenvalues of them.