Stabilizers and orbits of first level vectors in modules for the special linear groups. (Q2873310)

Let \(G\) be the algebraic group \(\mathrm{SL}_{r+1}\) defined over an algebraically closed field of characteristic \(p>2\). Let \(\omega=a_1\varpi_1+\cdots a_r\varpi_r\) be a dominant weight so that the coefficients \(a_i\) with respect to the fundamental weights \(\varpi_i\) are at least equal to two. Assume that the irreducible \(G\)-module \(M\) with highest weight \(\omega\) is infinitesimally irreducible. We are interested in the orbits and stabilizers of nonzero vectors in the span \(M_1\) of those weight spaces whose weight differs by a simple root from \(\omega\). (These vectors are the first level vectors of the title.) A detailed analysis is given of the possiblilities.NEWLINENEWLINE From the summary: ``For such vectors and modules a criterion for lying in the same orbit is obtained, and we prove that the stabilizers of vectors from different orbits are not conjugate. The orbit dimensions are also found. Furthermore, we show that these vectors do not lie in the orbit of a highest weight vector and their stabilizers are not conjugate to the stabilizer of such a vector.''