On the coadjoint orbits of the unitriangular group
This paper is a continuation of another work of the author [J. Algebra 176, No. 3, 959-1000 (1995; Zbl 0837.20050)]. Let \(U_n(K)\) be the group of all \(n\times n\) unipotent upper triangular matrices with coefficients in an algebraically closed field \(K\). Let \(u_n(K)\) denote the Lie algebra of all \(n\times n\) nilpotent upper triangular matrices and let \(u_n(K)^*\) be the dual space of \(u_n(K)\). The group \(U_n(K)\) acts on \(u_n(K)^*\) via the coadjoint representation: \((x\cdot f)(a)=f(xax^{-1})\) for \(x\in U_n(K)\), \(f\in u_n(K)^*\), \(a\in u_n(K)\). Then \(u_n(K)^*\) decomposes as the disjoint union \(u_n(K)^*=\bigcup V_D(\varphi)\) of some of its (basic) subvarieties. The main goal of this paper is to describe a certain decomposition of a basic subvariety \(V_D(\varphi)\) into \(U_n(K)\)-invariant subvarieties.
scientific article; zbMATH DE number 869795