Generalized flag geometries associated with (2k + 1)-graded Lie algebras

In this paper, we present the construction of a geometric object, called a generalized flag geometry, $(X^{+};X^{-})$, corresponding to a (2k +1)-graded Lie algebra $g=g_k\oplus \cdots \oplus g_{-k}$. We prove that $(X^+;X^-) can be realized inside the space of inner filtrations of g and we use this realization to construct “algebraic bundles” on $X^{+}$ and $X^{-}$ and some sections of these bundles. Thanks to these constructions, we can give a realization of $g$ as a Lie algebra of polynomial maps on the positive part of $g$, $n_1^{+}:=g_1\oplus \cdots \oplus g_k$, and a trivialization in $n_1^{+}$ of the action of the group of automorphisms of $g$ by "birational"maps.