 Mackey formula for Harish-Chandra induction

For rational parabolic subgroups $P=LU$ and $Q=MV$, define $\mathcal{S}(M,L)=\{x:M\cap {}^{x}L\text{ contains a maximal algebraic torus of }G\}$. Then ${}^{\ast }{R}_M^G{R}_L^G$ is the sum over $M^F\backslash \mathcal{S}(M,L{)}^F/L^F$ of $R_{M\cap {}^{x}L}^M{}^{\ast }{R}_{M\cap {}^{x}L}^{{}^{x}L}\mathrm{ad}x$, with the intersection parabolics understood. It follows from ordinary finite-group Mackey restriction formula, taking $V^F$-invariants, and the compatible-Levi intersection lemma for parabolics.