Another Characterization of Parabolic Subgroups

Let G be a reductive algebraic group defined over an algebraically closed field k. Let H be a closed connected subgroup of G containing a maximal torus T of G. In [13] it was shown (at least in characteristic zero) that the parabolic subgroups of G can be characterized among all such subgroups H by a certain finiteness property of the induction functor (-)Iz and its derived functors Lk,G(-). This theme is continued in the present paper, where it is shown that the parabolic subgroups can be characterized by yet another familiar property of the induction functor, at least in characteristic zero. We also obtain several results which are independent of the characteristic by added hypotheses on H, or by using the restriction functor instead of induction. After describing the relevant property below, Section 2 begins by studying induction from a special type of subgroup H. Namely, H is the semidirect product of T with U,, where U, is the unipotent radical of a standard parabolic subgroup P,. This type of subgroup was also found to be useful in [ 133. Section 3 contains the main result, involving the restriction functor. One first reduces the question to the case when L, c H c P, for some parabolic P, with Levi factor L,, then proves the result with this added hypothesis on H. In Section 4 we work with the induction functor (-) 1 g. If L,c H E P, holds, we obtain the desired result, but we do not have at this time a way to reduce the question to this case. However, if the characteristic is 0 we can reduce the question to the case of a solvable subgroup H, where the above condition is automatic (taking I to be empty). In the appendix we apply these results and those of [13] to a special case which was first studied in [6, 141. The appendix also contains a short 214 0021~8693/91 $3.00