A slice theorem for truncated parabolics of index one and the Bézout equation
The paper contains a confirmation of a conjecture stated by the author in the paper [``Parabolic actions in type \(A\) and their eigenslices.'' Transform. Groups 12, No. 3, 515--547 (2007; Zbl 1232.17020)]. The conjecture (called slice conjecture) states that for a parabolic subalgebra \({\mathfrak p}\) of a simple Lie algebra \({\mathfrak g}\), the coadjoint orbits in the canonical truncation \({\mathfrak p}_E\) admits a slice. In the same paper, this conjecture was established for parabolics in type \(A\) which are invariant under the Dynkin diagram involution. In the paper under review, the slice conjecture is confirmed for parabolics of index one in \({\mathfrak {sl}}(n)\). The technique developed leads to another algorithm of solving the Bézout equation \(xp-yq=-1\), with \(p,q\) coprime. The method seems to be less efficient than the Euclidean algorithm, but it is a remarkable connection between the combinatorics of algebraic Lie algebras and elementary number theory.
scientific article; zbMATH DE number 5149357
scientific article