An extension of Bruhatʹs lemma

In recent work of Lusztig and He [G. Lusztig, Parabolic character sheaves I, Mosc. Math. J. 4 (2004) 153–179; G. Lusztig, Parabolic character sheaves II, Mosc. Math. J. 4 (2004) 869–896; X. He, Unipotent variety in the group compactification, Adv. Math. 203 (2006) 109–131; X. He, The G-stable pieces of the wonderful compactification, Trans. Amer. Math. Soc., in press] certain decompositions are introduced in the wonderful compactification of an adjoint group G. To establish them a combinatorial machinery introduced by Bédard is used. The present note gives another approach to these results. We derive them in Section 3 from a result about G, an analogue of Bruhatʹs lemma proved in Section 2 (see Theorem 2.6). Basic in our approach is the elementary Lemma 1.6. The approach can also be used to deal with properties of Lusztigʹs parabolic character sheaves. We do not go into this here.