Let G be a semi-simple algebraic group over C, B a Borel subgroup of G, T a maximal torus in B and P a parabolic in G containing B. In a previous work [7], the authors classified the singular T -fixed points of an irreducible T -stable subvariety X of the generalized flag variety G/P . It turns out that under the restriction that G doesn't contain any G2-factors, the key geometric invariant determining the singular T -fixed points of X is the linear span Θx(X) of the reduced tangent cone to X at a T -fixed point x. The goal of this paper is to describe this invariant at the maximal singular T -fixed points when X a Schubert variety in G/B and G doesn't contain any G2-factors. We first decribe Θx(X) solely in terms of Peterson translates, which were the main tool in [7]. Then, taking a further look at the Peterson translates (with the G2-restriction),we are able to describe Θx(X) in terms of its isotropy submodule and the Bruhat graph of X at x. This refinement gives a purely root theoretic description, which should be useful for computations. Finally, still with the G2-restriction, these considerations lead us to a non-recursive algorithm for X's singular locus solely involving only the root system of (G, T ) and the Bruhat graph of X.
IntroductionLet G be a semi-simple algebraic group over an arbitrary algebraically closed field k, and suppose T ⊂ B ⊂ P are respectively a maximal torus, a Borel subgroup and an arbitrary standard parabolic in G. Each G/P , including G/B, is a projective G-variety with only finitely many B-orbits. Every B-orbit contains a unique T -fixed point x ∈ (G/P ) T , and these cells define an affine paving of G/P . If x ∈ (G/P ) T , then the closure of the B-orbit Bx is called the Schubert variety in G/P associated to x. This Schubert variety will be denoted throughout by X(x). We will use the well known fact that the T -fixed points in G/B are in one to one correspondence with the elements of the Weyl group W = N G (T )/T , so we don't distinguish between elements of W and fixed points in G/B.Schubert varieties are in general singular, and it's an old problem, inspired by a classical paper [8] of Chevalley, to describe their singular loci (or, equivalently, their smooth points). A related problem, with interesting consequences in representation theory, is to determine The first author was partially supported by the Natural Sciences and Engineering Research Council of Canada The second author was partially supported by the SNF (Schweizerischer Nationalfonds) 2 the locus of rationally smooth points of a Schubert variety (cf.[9]). In fact, if G is defined over C and simply laced (i.e. every simple factor is of type A, D or E), then all rationally smooth points of any Schubert variety in G/P are in fact smooth (see [7]). In this paper, we consider the singular locus of a Schubert variety in an arbitrary G/P , where G does not contain any G 2 -factors. Our results are an outgrowth of [7], where we used Peterson translates (defined below) to characterize the T -fixed points in the sing...