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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.

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  • $\begingroup$ First of all, DL induction is a generalization of parabolic induction to the case where the parabolic doesn't exist (without field extension). But you have a Borel, so use it. Parabolic induction is the restriction from $T$ to $B$, followed by induction from $B$ to $G$. Since, as you say, these functors are biadjoints, the adjoint is simply restriction from $G$ to $B$ followed by induction from $B$ to $T$. Now prove that DL agrees with this simple definition. Formal: biadjoint to tensoring with a bimodule is tensoring with the same bimodule (assuming semisimple, finitely generated). $\endgroup$ Jul 8, 2021 at 17:37
  • $\begingroup$ I’m slightly confused by your comment: as $T \subset B$, what does “restriction from $T$ to $B$“ entail? In parabolic induction, we extend a representation of $T$ to one of $B$ by allowing $U$ to act trivially, then induce. Also, my understanding was that DL induction specialises to parabolic induction in the case that the torus is maximally split (i.e. contained in an $F$-stable Borel), but I’m not clear how this relates to field extensions. $\endgroup$ Jul 9, 2021 at 1:28
  • $\begingroup$ 1: Extension by letting $U$ act trivially is restriction from $B/U$ to $B$. It is better to think of $T$ as a quotient of $B$ than as a subgroup. 2: This is matter of language of talking about algebraic groups. Replacing $F$ with $F^n$ is equivalent to extending the field of definition from $\mathbb F_q$ to $\mathbb F_{q^n}$. Every Borel is $F^n$ stable for some $n$, and is said to be defined over $\mathbb F_{q^n}$. $\endgroup$ Jul 9, 2021 at 15:27
  • $\begingroup$ @mskilleter: In your Q3, what do you mean precisely by a bi-adjoint pair? Is it simply that each of the functors is left and right adjoint to the other one (as indicated in the answer of Jeremy Rickard below), or is it a notion from higher category theory (as defined here for example)? $\endgroup$
    – AlexIvanov
    Jul 13, 2021 at 7:42
  • $\begingroup$ @AlexIvanov that each of the functors is both a left and right adjoint to the other, so Jeremy’s answer does resolve my 3rd question. Apologies for the confusion, I wasn’t aware there was another possible interpretation. $\endgroup$ Jul 13, 2021 at 8:55

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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.

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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).

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