Conormal Varieties on the Cominuscule Grassmannian

Abstract

Let G be a simply connected, almost simple group over an algebraically closed field k, and P a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification φ : T * G/P → X(u), where X(u) is a Schubert variety corresponding to the loop group LG. Let N * X(w) ⊂ T * G/P be the conormal variety of some Schubert variety X(w) in G/P ; hence we obtain that the closure of φ(N * X(w)) in X(u) is a B-stable compactification of N * X(w). We further show that this compactification is a Schubert subvariety of X(u) if and only if X(w 0 w) ⊂ G/P is smooth, where w 0 is the longest element in the Weyl group of G. This result is applied to compute the conormal fibre at the zero matrix in any determinantal variety.

Mather classes and conormal spaces of Schubert varieties in cominuscule spaces

“…In [LS17], Lakshmibai and Singh identified certain conormal spaces as open subsets of affine Schubert varieties. It would interesting to obtain localization formulae for the conormal spaces using localization for affine Schubert varieties.…”

Conormal Varieties on the Cominuscule Grassmannian - II

“…Let a 1 , • • • , a k be the entries in the first column of T, written in increasing order, i.e., top to bottom; and b 1 , • • • , b n−k the entries in the second column, also written be parabolic subgroups of LG corresponding to the subsets D\{α 0 }, D\{α d }, and D\{α 0 , α d } respectively. Following [LS17], there exists an embedding φ : T * X X w → LG /P such that φ(T * X X w ) is an open subset of some Schubert subvariety of LG /P. Further, we can identify the structure map π and the Springer map µ as the restriction to φ(T * X X w ) of the quotient maps π d : LG /P → LG /G d and π 0 : LG /P → LG /G0 respectively.…”