Frobenius reciprocity for Deligne-Lusztig induction/restriction

Question

I am currently trying to understand the properties of Deligne-Lusztig induction, following Carter's Finite groups of Lie type and Digne-Michel's Representations of finite groups of Lie type. I am reasonably satisfied with the construction, but I am having difficulty understanding the proofs of the properties of the Deligne-Lusztig induction functor (it is worth noting that these two books have very different approaches to this theory, and as I am bouncing back-and-forth to fill in gaps, I may be missing something obvious). I am specifically interested in the simple case of inducing from a torus to a Borel, so I will try to simplify definitions and statements where possible.

In Digne-Michel (Definition 11.1), the DL functor $R_{T \subset B}^G$ is the generalised induction functor associated to the $G^F$ -module-$T^F$ afforded by $H^*_{c}(L^{-1}(U))$. Here $B=TU$ is the Levi decomposition of the Borel subgroup $B$, $L : G \to G$ is the Lang map and $H^*_c$ is $l$-adic cohomology with compact support. If (like me) you have difficulty visualising this representation, we can concretely realise the DL functor on the level of characters by the formula $$R_{T \subset B}^G(\theta)(g) = \frac{1}{|T^F|} \sum_{t \in T^F} \theta(t^{-1}) \mathcal{L}((g, t), L^{-1}(U))$$ where $\mathcal{L}((g,t), L^{-1}(U))$ denotes the Leftschetz number of the right-left multiplication action of $(g,t)$ on $L^{-1}(U)$. This is Proposition 11.2 in Digne-Michel.

The authors then remark that there is an adjoint functor $^* R^G_{T \subset B}$, called DL restriction, which can be given explicitly by the formula $$^* R^G_{T \subset B}(\psi)(t) = \frac{1}{|G^F|} \sum_{g \in G^F} \psi(g^{-1}) \mathcal{L}((g,t), L^{-1}(U)).$$

Question 1: Why does $R^G_{T \subset B}$ admit an adjoint functor? This is not (to my knowledge) elaborated upon at all in either of the books I am following.

Question 2: I guessed that one can prove $R^G_{T \subset B}$ admits an adjoint functor by using some abstract category theoretic condition, and then simply define $^* R^G_{T \subset B}$ as the adjoint. If this is the approach, then we get Frobenius reciprocity for free, but I don't know where the above formula comes from. If, on the other, we define $^* R^G_{T \subset B}$ by the above formula (which certainly seems like a very reasonable guess for an adjoint), how does one prove Frobenius reciprocity? Which of these is the easier/standard approach?

Question 3: Regardless of how one defines $^* R^G_{T \subset B}$, is $(R^G_{T \subset B}, ^* R^G_{T \subset B})$ a bi-adjoint pair? I know that standard induction agrees with co-induction for finite groups, but I have very little intuition for DL induction and I cannot make a guess either way.

Answer 1

Answer to Questions 1 and 2: At least in Digne--Michel the functor ${}^\ast R_{T \subseteq B}^G$ is rigorously defined. The reference is [Digne--Michel, p.47]. What they do is the following: If $G$, $H$ are any two finite groups and $M$ is a $G$-module-$H^{\rm opp}$, then we have the functor $R_H^G \colon E \mapsto M \otimes_{\mathbb{C}[H]} E$. This allows to define the DL-induction $R_{T \subseteq B}^G$.

Then, the dual $H$-module-$G^{\rm opp}$ $M^\ast = {\rm Hom}(M,\mathbb{C})$ defines the adjoint functor. This can be verified using the Hom-$\otimes$-adjunction. Finally, [Digne--Michel, Prop. 4.5] applies to give the traces of both, $R_H^G$ and ${}^\ast R_H^G$, which answers your Q2.

Answer 2

Deligne-Lusztig induction is not a functor between module categories (at the level of characters it sends characters to virtual characters, and not in general to actual characters).

What is true is that it is a functor between derived categories, and as such has an adjoint (the same functor is both left and right adjoint).