Smooth points of T-stable varieties in G/B and the Peterson map

Abstract

Let G be a semi-simple algebraic group over C, B a Borel subgroup of G and T a maximal torus in B. A beautiful unpublished result of Dale Peterson says that if G is simply laced, then every rationally smooth point of a Schubert variety X in G/B is nonsingular in X. The purpose of this paper is to generalize this result to arbitrary T -stable subvarieties of G/B, the only restriction being that G contains no G 2 factors. A key idea in Peterson's proof is to deform the tangent space Ty(X) to X at a nonsingular T -fixed point y along the orbit of y under a root subgroup in B, which is open in a T -invariant curve C (we say a T -curve) in X. In more generality, if a T -variety X is nonsingular along the open T -orbit in a T -curve C and x ∈ C T , we may consider the limit τ C (X, x) of the tangent spaces Tz(X) as z approaches x along C. We call τ C (X, x) the Peterson translate of X at x along C. Peterson showed that a Schubert variety X in G/B, where G is semi-simple, is nonsingular at x ∈ X T as long as all τ C (X, x) coincide for all such good T -curves..Our first result generalizes this theorem to any irreducible T -variety X, provided the fixed point x is attractive under much weaker hypotheses. We then prove that if X is a T -variety in G/B where G contains no G 2 factors, then every Peterson translate τ C (X, x) is contained in the linear span Θx(X) of the reduced tangent cone to X at x. (This fails when G = G 2 .) Combining these two results leads to our characterization of the smooth T -fixed points of such X. In particular, we show that if G is simply laced, then X is nonsingular at a T -fixed point x if and only if it is rationally smooth at x and x lies on at least two good T -curves Peterson's ADE result is an immediate consequence of this. In addition, we obtain a uniform description of the nonsingular T -fixed points of a Schubert variety in G/B modulo the G 2 restriction. In particular, a Schubert variety X in such a G/B is nonsingular if and only if all the reduced tangent cones of X are linear.Finally, we also obtain versions of the above results for all algebraic homogeneous spaces G/P modulo the G 2 restrictions.

Singularities of Schubert varieties, tangent cones and Bruhat graphs

“…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.…”

“…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 singular locus of an irreducible T -stable subvariety X ⊂ G/P (a T -variety for short).Since every B-orbit meets (G/B) T , the singular locus of a Schubert variety X consists of the B-orbits of its singular T -fixed points. Here we can make use of the well known natural ordering order on W = (G/B) T : if x, y ∈ W , then x < y if and only if X(x) ⊂ X(y) but x = y.…”

“…Hence the problem of determining the singular locus of X boils down to identifying the maximal singular elements y of [e, w]. Such a y is called a maximal singularity of X.Let us now recall the notion of a Peterson translate from [7]. Suppose for the moment that X is a T -variety in G/P , and let E(X, x) denote the set of T -stable curves in X which contain the point x ∈ X T .…”

“…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.…”

“…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),…”