Timeline for Frobenius reciprocity for Deligne-Lusztig induction/restriction
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 13, 2021 at 8:55 | comment | added | Martin Skilleter | @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. | |
| Jul 13, 2021 at 7:42 | comment | added | AlexIvanov | @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)? | |
| Jul 12, 2021 at 10:00 | answer | added | Jeremy Rickard | timeline score: 6 | |
| Jul 11, 2021 at 2:48 | vote | accept | Martin Skilleter | ||
| Jul 10, 2021 at 18:09 | answer | added | AlexIvanov | timeline score: 3 | |
| Jul 9, 2021 at 15:27 | comment | added | Ben Wieland | 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 $\mathbb F_{q^n}$. | |
| Jul 9, 2021 at 1:28 | comment | added | Martin Skilleter | 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. | |
| Jul 8, 2021 at 17:37 | comment | added | Ben Wieland | 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). | |
| Jul 8, 2021 at 3:03 | history | asked | Martin Skilleter | CC BY-SA 4.0 |