= Mackey formula for Harish-Chandra induction
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For rational <parabolic subgroups> $P=LU$ and $Q=MV$, define $\mathcal S(M,L)=\{x:M\cap{}^xL\text{ contains a maximal algebraic torus of }G\}$. Then $\,{}^*R_M^G R_L^G$ is the sum over $M^F\backslash\mathcal S(M,L)^F/L^F$ of $R_{M\cap{}^xL}^{M}\,{}^*R_{M\cap{}^xL}^{{}^xL}\operatorname{ad}x$, 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.
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