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 sub...