Tangent cones to Schubert varieties in types An, Bn and Cn

We study tangent cones to Schubert subvarieties of the flag variety of a complex reductive group G. Let T be a maximal torus of G, B be a Borel subgroup of G containing T, Φ be the root system of G with respect to T, W   be the Weyl group of Φ, and F=G/BF=G/B be the flag variety. We prove that if every irreducible component of Φ is of type BnBn or CnCn, and w1w1, w2w2 are two distinct involutions in W  , then the tangent cones at the point p=eBp=eB to the corresponding Schubert subvarieties Xw1Xw1, Xw2Xw2 of FF do not coincide as subschemes of the tangent space TpFTpF. We also show that if every irreducible component of Φ is of type AnAn or CnCn, then the reduced tangent cones to Xw1Xw1 and Xw2Xw2 do not coincide as subvarieties of TpFTpF.