Canonical form and stationary subalgebras of points of general position for simple linear Lie groups

A Restriction Theorem for Modules Having a Spherical Submodule

“…From [E, Table 1 ] we know that the generic stabilizer is of type G2 and hence (see for example [Po2] or [LV]) (V, L) has generically closed orbits. Moreover, dim VM = 1 [E,Table 1]. One the other hand, (A(¡, G2) is not a spherical pair, as follows from Kramer's table [VK].…”

Invariants d'un sous-groupe unipotent maximal d'un groupe semi-simple

“…-D'après le corollaire 2 du théorème 2, on a : dim V < 1 4-2 dim U. Par suite, si le rang de G est au moins 2, on a : dim V < dim G et on utilise alors la classification de [10]. Si le rang de G est 1, alors G ^ SL^ (C) et dim V < 3 donc V est isomorphe à E(c^) ou E(2o7i).…”

On complex weakly commutative homogeneous spaces

“…First we consider the case where the group ρ(L ) is simple. The results in [8] and restrictions listed in Table 3 show that ρ(L ) is one of the groups listed in Table 4 or 3 SL(7), E 6 (π 1 ).…”

“…It remains to consider the case m = 1. We show that in this case for L = SL(5)×SL(2) we have C(n) L = C, thus proving the commutativity of the homogeneous space G/L for a semisimple group L. According to a result in [8], the generic stabilizer for the action SL(5) : n is one-dimensional and consists of nilpotent elements. Moreover, the generic stabilizer for the action L : v is a semidirect product of the subalgebra sl(2) (such that the projection of this subalgebra to both factors of the algebra l is nontrivial) and a four-dimensional nilpotent radical.…”

“…Since l 2 β = 0, it remains to consider the cases (see [8]) listed in Table 5. In this table, the column marked l 2 contains the image of the algebra l 2 under the representation ϕ 2 , and the column marked l 2 β → l 2 contains the linear algebra that is the image of the algebra l 2 β under the simplest representation l 2 .…”