Note that the similar question for the root systems D n , E 6 , E 7 , E 8 remains open. Now, let A be the symmetric algebra of the vector space m/m 2 , or, equivalently, the algebra of regular functions on the tangent space T p X w . Since R is generated as C-algebra by m/m 2 , it is a quotient ring R = A/I. By definition, the reduced tangent cone C red w to X w at the point p is the common zero locus in T p X w of the polynomials f ∈ I ⊆ A. Clearly, if C red w 1 = C red w 2 , then C w 1 = C w 2 . Our second main result is as follows.Theorem 1.3. Assume that every irreducible component of Φ is of type A n , n ≥ 1, or C n , n ≥ 2. Let w 1 , w 2 be involutions in the Weyl group of Φ and w 1 = w 2 . Then the reduced tangent cones C red w 1 and C red w 2 do not coincide as subvarieties of T p F. Note that the similar question for other root systems remains open. The paper is organized as follows. In the next Subsection, we introduce the main technical tool used in the proof of Theorem 1.2. Namely, to each element w ∈ W one can assign a polynomial d w in the algebra of regular functions on the Lie algebra of the maximal torus T . These polynomials are called Kostant-Kumar polynomials [KK1], [KK2], [Ku], [Bi]. In [Ku] S. Kumar showed that if w 1 and w 2 are arbitrary elements of W and d w 1 = d w 2 , then C w 1 = C w 2 . We give three equivalent definitions of Kostant-Kumar polynomials and formulate their properties needed for the sequel. In Subsection 1.3 we check that it is enough to prove Theorems 1.2 and 1.3 for irreducible root systems.In Section 2 we prove that if all irreducible components of Φ are of type B n and C n and w 1 , w 2 are distinct involutions in W , then d w 1 = d w 2 , see Propositions 2.7, 2.8. This implies that C w 1 = C w 2 and proves Theorem 1.2. The proof of Conjecture 1.1 for A n , F 4 and G 2 presented in [EI] is based on the similar argument.Section 3 contains the proof of Theorem 1.3. Namely, in Subsection 3.1 we describe connections between the tangent cones to Schubert varieties and the geometry of coadjoint orbits of the Borel subgroup B. In Subsections 3.2, 3.3, using the results of the second author about coadjoint orbits [Ig1], [Ig2], we prove Theorem 1.3, see Propositions 3.3 and 3.5. during the stay of Mikhail Ignatyev at Jacobs University, Bremen, supported by DAAD program "Forschungsaufenthalte für Hochschullehrer und Wissenschaftler", ref. no. A/13/00032. Mikhail Ignatyev thanks Professor Ivan Penkov for his hospitality and useful discussions, and DAAD for the financial support.
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