Classification of double flag varieties of complexity 0 and 1 (Q2866964)

Let \(G\) be a complex semisimple algebraic group and \(B\) a Borel subgroup of \(G\). The \textit{complexity} of an irreducible variety \(X\) acted upon by \(G\) is defined as the minimal codimension of \(B\)-orbits in \(X\). The complexity of \(X\), denoted \(c(X)\), is a nonnegative integer, and the varieties of complexity \(0\) are called spherical. The basic results on the complexity are due to \textit{Eh. B. Vinberg} [Funct. Anal. Appl. 20, 1--11 (1986); translation from Funkts. Anal. Prilozh. No. 1, 1--13 (1986; Zbl 0601.14038)]. If \(P\) and \(Q\) are proper parabolic subalgebras of \(G\), then \(G/P\times G/Q\) is called a \textit{double flag variety}. A general invariant-theoretic method for computing the complexity of double flag varieties has been developed by the reviewer (see [\textit{D. I. Panyushev}, Comment. Math. Helv. 68, No. 3, 455--468 (1993; Zbl 0804.14024)]).NEWLINENEWLINEIn this article, the author classifies all double flag varieties of complexity \(c\leq 1\). For the \textit{maximal} parabolic subalgebras \(P\) and \(Q\), such a classification was earlier obtained by \textit{P. Littelmann} [J. Algebra 166, No. 1, 142--157 (1994; Zbl 0823.20040)] and the reviewer (\(c=1\), loc. cit.).