Stabilizers and orbits of first level vectors in modules for the special linear groups

Abstract

Abstract.Under some restrictions on the highest weight, the stabilizers of certain vectors in irreducible modules for the special linear groups with a rational action are determined. We consider infinitesimally irreducible modules whose highest weights have all coefficients at least 2 when expressed as a linear combination of the fundamental dominant weights and vectors whose nonzero weight components have weights that differ from the highest weight by a single simple root. 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.