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.