- Conormal Varieties on the Cominuscule Grassmannian - II

Let \(X_w\) be a Schubert subvariety of a cominuscule Grassmannian \(X\), and let \(\mu:T^*X\rightarrow\mathcal N\) be the Springer map from the cotangent bundle of \(X\) to the nilpotent cone \(\mathcal N\). In this paper, we construct a resolution of singularities for the conormal variety \(T^*_XX_w\) of \(X_w\) in \(X\). When \(X\) is the usual, symplectic, or orthogonal Grassmannian, we also compute a system of equations defining \(T^*_XX_w\) as a subvariety of the cotangent bundle \(T^*X\) set-theoretically. This further yields a system of defining equations for the corresponding orbital varieties \(\mu(T^*_XX_w)\). Inspired by the defining system of equations, we conjecture a type-independent equality, namely \(T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu(T^*_XX_w))\). We show that the set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian \(X\) of type A, B, C, D. - Conormal Varieties on the Cominuscule Grassmannian with V. Lakshmibai

Let \(G\) be a simply connected, almost simple group over an algebraically closed field \(\mathbf k\), and \(P\) a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification \(\phi:T^*G/P\rightarrow X(u)\), where \(X(u)\) is a Schubert variety corresponding to the loop group \(LG\). Let \(N^*X(w)\subset T^*G/P\) be the conormal variety of some Schubert variety \(X(w)\) in \(G/P\); hence we obtain that the closure of \(\phi(N^*X(w))\) 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 \(X(w_0w)\subset G/P\) 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 \(0\) matrix in any determinantal variety. - Finite Groups Generated in Low Real Codimension with Ivan Martino

To appear in Linear Algebra and its Applications

We study the intersection lattice of the arrangement \(\mathcal{A}^G\) of subspaces fixed by subgroups of a finite linear group \(G\). When \(G\) is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of \(G\). We generalize the notion of finite reflection groups. We say that a group \(G\) is generated (resp. strictly generated) in codimension \(k\) if it is generated by its elements that fix point-wise a subspace of codimension at most \(k\) (resp. precisely \(k\)).

If \(G\) is generated in codimension two, we show that the intersection lattice of \(\mathcal{A}^G\) is atomic. We prove that the alternating subgroup \(\mathsf{Alt}(W)\) of a reflection group \(W\) is strictly generated in codimension two; moreover, the subspace arrangement of \(\mathsf{Alt}(W)\) is the truncation at rank two of the reflection arrangement \(\mathcal{A}^W\).

Further, we compute the intersection lattice of all finite subgroups of \(GL_3(\mathbb{R})\), and moreover, we emphasize the groups that are ``minimally generated in real codimension two", i.e, groups that are strictly generated in codimension two but have no real reflection representations. We also provide several examples of groups generated in higher codimension. - Cotangent Bundles of Partial Flag Varieties and Conormal Varieties of their Schubert Divisors with V. Lakshmibai

To appear in Transformation Groups

Let \(P\) be a parabolic subgroup in \(G=SL_n(\mathbf k)\), for \(\mathbf k\) an algebraically closed field. We show that there is a \(G\)-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle \(T^*G/P\). Restricting this identification to the conormal variety \(N^*X(w)\) of a Schubert divisor in \(G/P\), we show that there is a compactification of \(N^*X(w)\) as an affine Schubert variety. It follows that \(N^*X(w)\) is normal, Cohen-Macaulay, and Frobenius split. - Cotangent Bundle to the Flag Variety with V. Lakshmibai and C.S. Seshadri

Published in Transformation Groups

We show that there is a \(SL_n\)-stable closed subset of an affine Schubert variety in the infinite dimensional Flag variety (associated to the Kac-Moody group \(\widehat{SL_n}(\mathbb C)\)) which is a natural compactification of the cotangent bundle to the finite-dimensional Flag variety \(SL_n/B\).