On the Spaltenstein correspondence.
Let \(M(n,k)\) be the \(n\times n\)-matrices with entries in an algebraically closed field \(k\) and let \(V\) be a vector space over \(k\) of dimension \(n\). \textit{N. Spaltenstein} [Nederl. Akad. Wet., Proc., Ser. A 79, 452--456 (1976; Zbl 0343.20029)] established a bijection between the irreducible components of the space of full flags \(\mathcal F_x\) fixed by a nilpotent element \(x\in M(n,k)\) and the standard tableaux associated to the Young diagram of \(x\). The main result of the present article is to determine, when \(x\) is of hook type, for each irreducible component \(X\) of \(\mathcal F_x\), the unique Schubert cell \(\mathcal C_X\) of the full flag manifold \(\mathcal F(V)\), such that \(\mathcal C_X\cap X\) is a dense subspace of \(X\).
scientific article; zbMATH DE number 2124632