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By Lemma 3.4, Z H ( S ′ ) is a Levi subgroup of a parabolic subgroup of P ′ of H.
Then Z H K ′ ( T ) is a Levi subgroup of a proper parabolic subgroup P of H K ′.
There exists a unique H ( R n ) –conjugacy class of (a) Couples ( L, P ) where P is a minimal parabolic R n –subgroup scheme of H and L is a Levi subgroup of P such that L is a loop reductive group scheme.
We can assume without loss of generality that E is obtained from an L -torsor F. Then Q is isomorphic8 to the twist F P, and F L is a Levi subgroup of the R n -group F P. Since Levi subgroups of F P are conjugate under R u ( F P ) ( R n ) [41, XXVI, 1.8], it follows that F L is R n –isomorphic to M. The group scheme F L carries then a maximal torus and the claim is proved.
(2) That Z G ( S ) is a Levi subgroups of G follows from Lemma 3.4.
Z G ( S ) is a Levi subgroup of G and Z G ( S ) = Z G ( s ).
Then there exists a parabolic subgroup P ⊂ H such that Z H ( S ) is a Levi subgroup of P. See [41, XXVI, cor. 6.2].
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Other gay and lesbian entrepreneurs have helped turn Blue Ridge's main street into a hub of gourmet restaurants and boutique stores popular with visitors, creating an island of prosperity in a region hit by the decline of logging and the closure of a Levi's denim factory.
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