G-stable Pieces and Lusztig’s Dimension Estimates

We use G-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig’s dimension estimates [L2, section 4]. This is a generalization of [HL]. In Lemma 1.1 and Proposition 1.2, we assume that G is arbitrary connected algebraic group and G̃ is an algebraic group with identity component G. Lemma 1.1. Let g ∈ G̃. Define i : G̃ → G̃ by i(h) = ghgıhı. For any closed subgroup A of ZG with gAgı = A, set HA = {h ∈ G; i(h) ∈ A}. Then (1) HA is an algebraic group and i|HA : HA → A is a morphism of algebraic groups. (2) i(A) = i(HA) . (3) dim(HA) = dim(ZG(g)) + dim(A)− dim(ZA(g)). If h, h ∈ HA, then i(hh) = ghhgı(h)ıhı = (ghgıhı)h(ghgı(h)ı)hı = i(h)hi(h)hı = i(h)i(h) ∈ A and hh ∈ HA. If h ∈ HA, then i(hı) = hıi(h)ıh = i(h)ı ∈ A and hı ∈ HA. Part (1) is proved. Now i(A) is a connected subgroup of i(HA). Define δ : A → A by δ(z) = gzgı. Then dim(i(A)) = dim(A)− dim(A). Define σ : A → A by σ(z) = δ(z)δ(z) · · · z, where m is the order of the automorphism δ. Then σ is a group homomorphism and i(HA) ⊂ {z ∈ Z; σ(z) = 1}. Notice that σ(A) = {t; t ∈ A} is of dimension dim(A). Thus dim(i(HA)) 6 dim(A)− dim(σ(A)) 6 dim(A)− dim(σ(A )) = dim(A)− dim(A). Therefore, dim(i(A)) = dim(i(HA)) = dim(A) − dim(A ). Part (2) is proved. The author is partially supported by NSF grant DMS-0700589.