The Index of Representations Associated with Stabilisers

indq 6 indqξ for any ξ ∈ q . It is not always true that indq = indqξ, see Example 1.1 below. However, it was conjectured by ELASHVILI that if q = g is semisimple, then this equality always holds. It easily seen that it suffices to prove the equality indg = indgξ only for the nilpotent elements ξ ∈ g≃ g∗. The conjecture was recently proved by CHARBONNEL [3]. A proof for the classical Lie algebras, with weaker assumptions on the ground field, was found independently by the second author [14]. One can consider two types of problems connected with Eq. (0.1). First, to find properties of v that guarantee the equality of the indices. Second, to describe representations such that (0.1) turns into equality for each v ∈V . We begin with pointing out two simple sufficient conditions. If either qv is reductive or dimqv·v is maximal, then Eq. (0.1) turns into equality. Let Q be a connected algebraic group with Lie algebra q. Given a representation ρ : Q → GL(V ) (or (Q : V ) for short), we say that (Q : V ) has good index behaviour (GIB), if ind(q,V∗) = ind(qv,(V/q·v)) for each v ∈ V . We prove that most of sufficiently large reducible representations have GIB. Namely,