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6 S. GINDIKIN AND T. MATSUKIand hencexS ∩ S ′ 3 ≠ ϕfor some K C -B double coset S 3 such that S ′ 3 ⊂ S ′ 0(Bw −1 B) cl . Hence S 3 (BwB) cl ⊃ S 0and therefore dim C S 3 ≥ dim C G C − l(w) > dim C S. So we haveS ′ 3 ∩ S ′ = ϕbecause S ′ is the union of G R -B double cosets S ′ 4 satisfying S 4 ⊂ S. Hence we haveand thereforeRemark 4.xS ⊄ S ′x /∈ C(S).q.e.d.(i) The condition l(w) = codim C S − 1 does “not always” implycodim C S cl (BwB) cl = 1.Counter examples exist already for G R = SU(2, 1).(ii) The construction of the domain Ω(J(S 1 )) is essentially equivalent to theconstruction of “Schubert domain” in [HW]. Unfortunately, their basic definitionneeds a correction and after this correction their proof of Corollary 3.2 correspondingto our Corollary is not complete. We can see that the proof of Corollary using theresults in Section 2 is extremely simple. Let us explain the connection between thesetwo constructions introducing notations in [HW].Take a Borel subgroup B 0 of G C so that G R B 0 is closed in G C . A Borel subgroupB of G C is called an “Iwasawa Borel subgroup” ifB = g 0 B 0 g −10 for some g 0 ∈ G R .Let Z = G C /Q be a flag manifold. Then we can take Q so that Q ⊃ B 0 . EverySchubert cell Y in Z for B is written asY = (Bg 0 wQ) cl = (g 0 B 0 wQ) clwith some w ∈ W . Let C 0 be a closed K C -Q double coset (do not miss with C 0 ofSection 1!). The “incidence divisor” H Y is written asH Y = {g | gC 0 ∩ Y ≠ ϕ} = Y C −10 = (g 0 B 0 wQ) cl C −10 = (C 0 (Qw −1 B) cl g −10 ) −1 .In this point, in [HW], it is written: “If codim Y ≤ dim C 0 + 1, then H Yhypersurface in G C .” It is wrong as we remarked in (i).But if codim H Y = 1, thenH −1Y= C 0 (Qw −1 B 0 ) cl g0 −1 = T j g0−1for some j ∈ J ′ = J(C 1 ) (where C 1 is the dense K C -B 0 double coset in C 0 ) andg 0 ∈ G R by our notation.So their definition of y(D) should be corrected toy(D) = {Y = (g 0 B 0 wQ) cl | codim H Y = 1}.is a