Harish-Chandra induction is left adjoint to restriction

Frobenius adjunction for $P\le G$. By [L3] the Harish–Chandra induction is the composite functor $R_L^G(V)={\mathrm{Ind}}_P^G\text{\hspace{0.17em}⁣}\left({\mathrm{Inf}}_L^{P}V\right)$, so with $W:={\mathrm{Inf}}_L^{P}V$ the adjunction of [L1], applied to the finite group $G$, its subgroup $P$, the $P$-module $W$ and the $G$-module $X$, gives a natural bijection $\mathrm{\Theta }:{\mathrm{Hom}}_G(R_L^{G}V,X)\to {\mathrm{Hom}}_P(W,X)$, where $X$ is viewed as a $P$-module by restricting its $G$-action.

Images of $P$-linear maps lie in $X^U$. Let $\phi :W\to X$ be $P$-linear, $u\in U$ and $w\in W$. Since $U\le P$ we have $\phi (u\cdot w)=u\cdot \phi (w)$, and $u\cdot w=w$ because every element of $U$ acts as the identity on the inflation $W={\mathrm{Inf}}_L^{P}V$ by [L2]; hence $u\cdot \phi (w)=\phi (w)$ for all $u\in U$, that is $\phi (W)\subseteq X^U$.

The identification with $L$-linear maps into $X^U$. Define $\mathrm{\Lambda }:{\mathrm{Hom}}_P(W,X)\to {\mathrm{Hom}}_L(V,X^U)$ by $\mathrm{\Lambda }(\phi ):=\phi {\mathrm{\mid }}_V$. This is well defined: a $P$-linear map is $L$-linear because $L\le P$, and it takes values in $X^U$ by step 1.2. It is injective: $W$ and $V$ have the same underlying set, so a $P$-linear map is determined by its values on $V$. It is surjective: for an $L$-linear $\psi :V\to X^U$ define ${\mathrm{\Phi }}_{\psi }(w):=\psi (w)$ for $w\in W$; writing an arbitrary $p\in P$ as $p=lu$ with $l\in L$ and $u\in U$, the unique factorisation available in $P=L⋉U$ with $P=LU$ and $L\cap U=\{1\}$ by [L2], one computes ${\mathrm{\Phi }}_{\psi }(p\cdot w)={\mathrm{\Phi }}_{\psi }(l\cdot w)=\psi (l\cdot w)=l\cdot \psi (w)=lu\cdot \psi (w)=p\cdot {\mathrm{\Phi }}_{\psi }(w),$ where the middle steps use that $u$ acts trivially on $W$ and that $\psi (w)$ lies in $X^U$, and the last is the $L$-action on $X^U$ of [L3]. Thus ${\mathrm{\Phi }}_{\psi }$ is $P$-linear with $\mathrm{\Lambda }({\mathrm{\Phi }}_{\psi })=\psi$, so $\mathrm{\Lambda }$ is a bijection.

The composite is a natural bijection. Composing the bijections of steps 1.1 and 2.1 gives a bijection $\mathrm{\Lambda }\circ \mathrm{\Theta }:{\mathrm{Hom}}_G(R_L^{G}V,X)⟶{\mathrm{Hom}}_L(V,{}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G}X),$ the target being ${\mathrm{Hom}}_L(V,X^U)$ by [L3]. It is natural in $V$: for an $L$-linear $\alpha :V\to V^{\mathrm{\prime }}$ with induced $P$-linear $\mathrm{Inf}\alpha :W\to W^{\mathrm{\prime }}$, the functoriality of [L4] gives $R_L^G(\alpha )={\mathrm{Ind}}_P^G(\mathrm{Inf}\alpha )$, and because $\mathrm{Inf}\alpha$ restricts to $\alpha$ on the common underlying sets, the naturality of $\mathrm{\Theta }$ in its first variable gives for $\phi \in {\mathrm{Hom}}_G(R_L^G{V}^{\mathrm{\prime }},X)$ $(\mathrm{\Lambda }\circ \mathrm{\Theta })(\phi \circ R_L^G(\alpha ))=\mathrm{\Theta }(\phi )\circ \mathrm{Inf}\alpha {\mid }_V=(\mathrm{\Theta }(\phi ){\mid }_V)\circ \alpha =(\mathrm{\Lambda }\circ \mathrm{\Theta })(\phi )\circ \alpha \mathrm{.}$ Naturality in $X$ is the same computation with postcomposition by a $G$-linear $\beta :X\to X^{\mathrm{\prime }}$: $\mathrm{\Theta }$ is natural in its second variable and $\beta$ maps $X^U$ into $(X^{\mathrm{\prime }}{)}^U$, so both composites send $\phi$ to $\beta \circ \mathrm{\Lambda }(\mathrm{\Theta }(\phi ))$. Hence $\mathrm{\Lambda }\circ \mathrm{\Theta }$ is a natural bijection in both variables.

Adjunction. By step 3.1 the assignment $\phi \mapsto \mathrm{\Lambda }(\mathrm{\Theta }(\phi ))$ is a bijection ${\mathrm{Hom}}_G(R_L^{G}V,X)\to {\mathrm{Hom}}_L(V,{}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G}X)$ natural in $V$ and in $X$, with $\mathrm{\Lambda },\mathrm{\Theta }$ as constructed from the definition of the functors and the published adjunction; this is exactly the assertion that $R_L^G$ is left adjoint to ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^G$, and by [L5] it applies to every split parabolic subgroup of $G$, standard or conjugate. ∎