Summary: Let \(G\) be a simply connected, almost simple group over an algebraically closed field \(\mathbf{k}\) of characteristic \(p\), where either \(p = 0\) or \(p\) is an odd prime which is also a good prime for \(G\). Let \(P\) be a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification \(\phi : T^\ast G /P \rightarrow X(u)\), where \(X(u)\) is a Schubert variety in a partial affine flag variety associated with the loop group \(G \left(\mathbf{k}[t,t^{-1}]\right)\). Let \(N^\ast X(w) \subseteq T^\ast G /P\) be the conormal variety of some Schubert variety \(X(w)\) in \(G / P\); hence we obtain that the closure of \(\phi (N^\ast X(w))\) in \(X(u)\) is a \(B\)-stable compactification of \(N^\ast X(w)\). We further show that this compactification is a Schubert subvariety of \(X(u)\) if and only if \(X (w_0 w) \subseteq G /P\) 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.
For the entire collection see [Zbl 1481.17001].