Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic subgroup with F-stable Levi decomposition P=LN. Let Z be the center of $G^F$.
$T$ being minisotropic means that $T^F$ is contained in no proper (split) parabolic subgroup (ergo, in no proper Levi). My question is: when can $T^F$ can have some noncentral intersection with a Levi?
We know that $Z \subseteq T^F \cap L^F$. Equality holds if, for example:
(a) $P=B$, a Borel (since another characterization of minisotropic is that the maximal split torus contained in $T^F$ coincides with the maximal split torus contained in $Z(G^F)$)
(b) $G^F=GL(p,k)$, p prime (since an anisotropic torus is generated by an elliptic element, and there are no intermediate extension fields over which some element could split)
(c) $G=Sp(4,k)$ and $T$ the Coxeter torus (since the Coxeter torus has order $q^2+1$, whose gcd with the order of each torus in each proper Levi is 2)
In contrast, there exist choices of $T$ and $L$ for which the intersection is noncentral:
(a) $G=Sp(4,k)$ and $T$ the torus corresponding to $w=-1$, which has order $(q+1)^2$ and is two copies of the group of norm-1 elements of a quadratic extension field (and so meets the parabolic with Levi $SL(2)\times GL(1)$ in an anisotropic torus).
(b) In $GL(4,k)$, I believe one could find a $t \in T^F\setminus Z$ which splits over an intermediate extension field and has a conjugate lying in $L^F=GL(2,k)\times GL(2,k)$.
This question arises when one tries to compute the simple restriction of a Deligne-Lusztig cuspidal character to a parabolic subgroup; when equality holds, the result is just given by the central character and the Green function $Q_T^G$.
I think this is closely related to the following question on p-adic groups (relevant for the construction of supercuspidal representations): when does an anisotropic-mod-center torus lie in a proper twisted Levi subgroup?