A slice theorem for truncated parabolics of index one and the Bezout equation

Let p be a parabolic subalgebra of sl(n) and pE its canonical truncation. In [A. Joseph, Parabolic actions in type A and their eigenslices, 7.19] it was conjectured that the coadjoint action in admits a slice. This was established [A. Joseph, Parabolic actions in type A and their eigenslices, 7.18] when p is invariant under the Dynkin diagram involution. In the present work, this is proved when the Levi factor of p has just two blocks of sizes p,q with p,q coprime. The solution gives rise to an algorithm for solving the Bezout equation rp−sq=−1. The construction involves finding an adapted pair (h,y)∈hE×(pE)reg, and here the choice of y is partly motivated by an observation [P. Tauvel, R.W.T. Yu, Sur l'indice de certaines algèbres de Lie, Ann. Inst. Fourier (Grenoble) 54 (2004) 1793–1810, 3.9] of P. Tauvel and R.W.T. Yu.