Suppose $P=L\ltimes U$ is an $F$-stable parabolic subgroup of a finite group of Lie type $G$, with $F$-stable Levi complement $L$. Here $F$ is a Frobenius endomorphism, and $G^F$ is the subgroup of fixed points under $F$, etc. Let $\Lambda$ be a commutative ring of some appropriate characteristic. The parabolic induction and restriction functors can defined as
$$
R\_{L\subset P}^G:\Lambda L^F\text{-mod}\to\Lambda G^F\text{-mod}:M\mapsto \Lambda G^F/U^F\otimes\_{\Lambda L^F}M
$$
and
$$
^\ast R^G\_{L\subset P}\colon \Lambda G^F\text{-mod}\to\Lambda L^F\text{-mod}:N\mapsto \hom\_{\Lambda G^F}(\Lambda G^F/U^F,N).
$$


In certain situations it is more convenient to change which one is a tensor functor, and which is a hom functor, as follows:
$$
R\_{L\subset P}^G=\hom\_{\Lambda L^F}(\Lambda U^F/G^F,-)
$$
and
$$
^\ast R^G\_{L\subset P}=\Lambda U^F/G^F\otimes\_{\Lambda G^F}-.
$$


On the level of elements, is there an explicit isomorphism between the two functors for induction, and for restriction?



Translating parabolic induction as $\Lambda G^F/U^F\otimes_{\Lambda L^F}-$ to $\hom_{\Lambda L^F}(\Lambda U^F/G^F,-)$?
I think it's easiest to break this up into the composition of three simpler isomorphisms.


First, for any ring $A$, if $M\_A$ is a right $A$-module and $\_AX$ is a left $A$-module, there is a natural map
$$M\otimes\_AX\to\text{Hom}\_A\left(\text{Hom}\_A(M,A),X\right)$$
given by
$$m\otimes x\mapsto[\varphi\mapsto \varphi(m)x],$$
which is an isomorphism if $M$ is finitely generated projective as a right $A$-module.


This applies when $A=\Lambda L^F$ and $M=\Lambda[G^F/U^F]$, with no restriction on $\Lambda$.


Second, if $H$ is a finite group, and $Y\_{\Lambda H}$ is a right $\Lambda H$-module, there is a natural isomorphism 
$$\text{Hom}\_{\Lambda H}(Y,\Lambda H)\to\text{Hom}\_\Lambda(Y,\Lambda)$$
given by composing with the $\Lambda$-module homomorphism $\Lambda H\to\Lambda$ given by $\sum\_h\lambda\_hh\mapsto\lambda\_1$, with the inverse map
$$\text{Hom}\_\Lambda(Y,\Lambda)\to\text{Hom}\_{\Lambda H}(Y,\Lambda H)$$ given by
$$\theta\mapsto[y\mapsto\sum\_h\theta(yh^{-1})h].$$


Third, 
$$\Lambda[U^F\backslash G^F]\cong\text{Hom}\_\Lambda\left(\Lambda[G^G/U^F],\Lambda\right)$$
as $\Lambda L^F$-$\Lambda G^F$-bimodules by the map
$$U^Fg\mapsto (gU^F)^\ast,$$
where $\left\{(gU^F)^\ast\right\}$ is the dual basis of the standard basis $\left\{gU^F\right\}$.


Combining these three isomorphisms, we get a chain of isomorphisms
$$\begin{align}
\Lambda[G^F/U^F]\otimes\_{\Lambda L^F}X &\to
\text{Hom}\_{\Lambda L^F}\left(\text{Hom}\_{\Lambda L^F}(\Lambda[G^F/U^F],\Lambda L^F),X\right)\\
&\to
\text{Hom}\_{\Lambda L^F}\left(\text{Hom}\_{\Lambda}(\Lambda[G^F/U^F],\Lambda),X\right)\\
&\to
\text{Hom}\_{\Lambda L^F}\left(\Lambda[U^F\backslash G^F],X\right)
\end{align}$$
given by
$$\begin{align}
gU^F\otimes x&\mapsto [\varphi\mapsto\varphi(gU^F)x]\\
&\mapsto [\theta\mapsto \sum\_{h\in L^F}\theta(gh^{-1}U^F)hx]\\
&\mapsto [U^Fg'\mapsto\sum\_{h\in L^F}(g'U^F)^\ast(gh^{-1}U^F)hx\\
&= \left[U^Fg'\mapsto
\begin{cases}0&\mbox{ if } (g')^{-1}gU^F\not\subseteq P^F\\
h'x&\mbox{ if } (g')^{-1}gU^F=h'U^F\mbox{ where }h'\in L^F
\end{cases}\right].
\end{align}$$


A similar calculation works for parabolic restriction, except that there we need $|U^F|$ to be invertible in $\Lambda$, so that $\Lambda[U^F\backslash G^F]$ is a projective right $\Lambda G^F$-module.


