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Recently I have been trying to understand Deigne-Lustzig induction in the case of $G = \text{Sl}(2,\mathbb{F}_p).$

In this case the appropriate Deligne Lustzig variety is given by $X:xy^q-y^qx = 1,$ aka the Drinfeld curve. The action of $G$ on affine space fixes $X$ and in addition commutes with the action of the $q+1$ roots of unity, $\mu_{q+1},$ by scaling. In almost all expositions I have read on Deligne Lustzig induction, $\mu_{q+1}$ is noncanonically identified with a nonsplit torus of $G.$ However for the action of $T$ we choose does not seem to be the action of $G$ restricted to $T.$

My question is rather vague. If we are only interested in the action of $T$ after identifying it with the roots of unity, why is it important to mention it at all? Perhaps this generalizes in some way that explains this but it is not clear in the case of $ \text{Sl}(2,\mathbb{F}_p).$ Thanks.

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up vote 5 down vote accepted

It would be helpful to know what sources you are consulting, since the rank 1 case is only a warm-up to the general Deligne-Lusztig theory. Their 1976 Annals paper (followed by a considerable amount of detailed work by Lusztig and others) treats arbitrary finite groups of Lie type and their complex characters in a novel way, but at the cost of using the full structure theory of reductive groups over a finite field together with sophisticated algebraic geometry. Two different textbook accounts of the basic theory were given: Roger Carter's large 1985 book Finite Groups of Lie Type and the more narrowly focused 1991 book Representations of Finite Groups of Lie Type by Francois Digne and Jean Michel.

Contrasting with all of these approaches is the recent elementary introduction by Cedric Bonnafe Representations of $SL_2(\mathbb{F}_q)$. Here the Drinfeld curve provides the concrete geometric setting for the "induction" construction which later got generalized by Deligne-Lusztig to extend traditional Harish-Chandra induction. In this concrete setting the variety is described independently of the ambient algebraic group, so the actions involved are down-to-earth, including the commuting action of a finite cyclic group. Implicitly that cyclic group is the group of rational points of an algebraic torus, but the action here ignores that interpretation at first.

In the general theory the Deligne-Lusztig variety is realized in terms of the algebraic group, so that the finite group of Lie type together with a twisted version of a maximal torus act in a natural way on the $\ell$-adic cohomology of the variety. Here all the steps become less concrete and intuitive. But Bonnafe adds to his account of the rank 1 case a brief introduction to Deligne-Lusztig induction in general. The transition is not elementary, but does point to the essential role played by the various twisted tori (relative to the Weyl group and Frobenius map) in the study of different "series" of irreducible characters of the finite group. In effect the complex characters of these finite tori are essential data needed to parametrize the characters of a more complicated finite group of Lie type.

P.S. I realize I haven't exactly answered the (admittedly vague) question raised here, but it's really necessary to grapple with some of the ideas in the general construction. What I'd emphasize is that the Deligne-Lusztig variety acquires two natural commuting actions, by the finite group $G^F$ on the left and by the finite torus $T^F$ on the right. Then the Euler characteristic of $\ell$-adic cohomology for the direct product group is the basic object of study (affording a generalized character), with a finer decomposition in terms of the complex characters of the finite torus, etc. How to get to this point from the Drinfeld curve in rank 1? It takes at least a minor miracle.

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For what it is worth, Bonnafe's book -- mentioned in Jim's answer -- seems to me to be a nice one; I think one can benefit from studying it. But I agree with Jim's comment that it is not an easy road from rank 1 to the general story. –  George McNinch Dec 1 '11 at 13:04
    
Thanks for the insight! I'll take a closer look at Bonnafe's book. –  Lalit Jain Dec 1 '11 at 19:15
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