Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras

Let g be a finite dimensional Lie algebra. We say that g is quasi-reductive if there is a linear form f on g such that the stabilizer of f modulo the center of g is a reductive Lie algebra whose center consists of semisimple elements. If g is reductive, then g is quasi-reductive itself, and also every Borel or Levi subalgebra of g. However the parabolic subalgebras of g are not always quasi-reductive (except in types A or C, see [P05]). Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is g. The classification of quasi-reductive biparabolic subalgebras in the classical case has been achieved recently in unpublished work of Duflo et al. ([DKT]). In this paper, we investigate the quasi-reductivity of biparabolic subalgebras of reductive Lie algebras. As a main result, we complete the classification of quasi-reductive parabolic subalgebras in the reductive Lie algebras.

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