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A REMARK ON SCHUBERT CELLS AND DUALITY OF ORBITS ON FLAG MANIFOLDS 32. Schubert cells in the category of K C -B double cosetsThe principal idea of our considerations in [GM] was that C(S) 0 will be essentiallyindependent of neither S nor the flag manifold X = G C /P . To justify it, we needto build bridges between C(S) for different S and for it we need to see connectionsbetween different K C -orbits. It turns out that Schubert cells are very efficient tool forsuch considerations as in Section 2 and Section 8 in [GM]. They give a possibilityto obtain an important information about general C(S) from a consideration ofsimplest S. Here we refine connections between K C -orbits and Schubert cells andgive more examples of applications.For a simple root α in the root system with respect to the order defined by B, wecan define a parabolic subgroupof G C such that dim C P α = dim C B + 1.P α = B ∪ Bw α BLemma 1. Let S 1 be a K C -B double coset. Then we have:(i) If dim C S 1 P α = dim C S 1 , then S1 cl P α = S1 cl .(ii) If dim C S 1 P α = dim C S 1 + 1, then there exists a K C -B double coset S 2 suchthat S1 cl P α = S cl2 .Proof. Though this lemma follows easily from [M2] Lemma 3, we will give a prooffor the sake of completeness. Write S 1 = K C gB. Then we have a natural bijection(g −1 K C g ∩ P α )\P α /B ∼ = K C \K C gP α /B = K C \S 1 P α /Bby the map x ↦→ gx.(i) If dim C S 1 P α = dim C S 1 , then (g −1 K C g ∩ P α )B/B is Zariski open in P α /B =P 1 (C) and hence it is dense. So we haveS cl1 = (K C gB) cl ⊃ S 1 P α ⊃ S 1and therefore S1 cl = S1 cl P α .(ii) Suppose dim C S 1 P α = dim C S 1 + 1. Then there exists a p ∈ P α such that(g −1 K C g ∩ P α )pB/B is Zariski open in P α /B = P 1 (C) since the number of K C -Bdouble cosets in G C is finite. If we write S 2 = K C gpB, then we have(S 2 ) cl ⊃ S 1 P α ⊃ S 2and therefore S cl2 = S cl1 P α . q.e.d.Theorem 1. Let S 1 be a K C -B double coset in G C and w an element of W . Thenwe have:(i) S1 cl (BwB) cl = S2 cl for some K C -B double coset S 2 .(ii) (minimal expression) There exists a w ′ ∈ W such that w ′ < w (Bruhat order),l(w ′ ) = dim C S 2 − dim C S 1 and thatS cl1 (Bw ′ B) cl = S cl2 .Here l(w ′ ) = dim C Bw ′ B − dim C B is the length of w ′ .