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

Abstract

Let G be a connected semi-simple algebraic group defined over an algebraically closed field k, and let T⊂B⊂P be respectively a maximal torus, a Borel subgroup and a parabolic subgroup of G. Inspired by a beautiful result of Dale Peterson describing the singular locus of a Schubert variety in G/B, we characterize the T-fixed points in the singular locus of an arbitrary irreducible T-stable subvariety of G/P (a T-variety for short). Peterson’s result (cf. The Deformation Theorem, §1) says that if k=ℂ, then a Schubert variety X⊂G/B is smooth at a T-fixed point x if and only if it is smooth at every T-fixed point y>x (in the Bruhat-Chevalley order on the fixed point set XT) and all the limits τC(X,x)=limz→xTz(X)(z∈C∖CT) of the Zariski tangent spaces Tz(X) of X coincide as C varies over the set of all T-stable curves in X with CT={x,y}, where y>x. Using this, Peterson showed that if G is simply laced (and defined over ℂ), then every rationally smooth point of a Schubert variety in G/B is smooth. More generally, the deformation τC(X,x) is defined for any k-variety X with a T-action provided C is what we call good, i.e. C is a curve of the form \(C = \overline {Tz}\), where z is a smooth point of X∖XT and x∈CT. Our first main result (Theorem 1.4) says that if x∈X is an attractive fixed point, then X is smooth at x if and only if there exist at least two good C containing x such that τC(X,x)=TE(X,x), where TE(X,x) denotes the span of the tangent lines of the T-stable curves in X containing x. In addition, if X is Cohen-Macaulay at x and τC(X,x)=TE(X,x) for even one good C, then X is smooth at x. Our second main result (Theorem 1.6) says that if X is a T-variety in G/P, where G is simply laced, then τC(X,x)⊂TE(X,x) for each good C. This is not true for general G, but when G has no G2 factors, then τC(X,x) is always contained in the linear span of the reduced tangent cone to X at x. These results lead to several descriptions of the smooth fixed points of a T-variety in G/P and, in particular, they give simple proofs of Peterson’s results valid for any algebraically closed field. We also show (cf. Example 7.1) that there can exist T-stable subvarieties in G/B, where G is simply laced, which have rationally smooth T-fixed points in their singular loci.