Abstract: Let $ \mathfrak{a} $ be an algebraic Lie subalgebra of a simple Lie algebra $ \mathfrak{g} $ with index $ \mathfrak{a} $  ≤ rank $ \mathfrak{g} $ . Let $ Y\left( \mathfrak{a} \right) $ denote the algebra of $ \mathfrak{a} $ invariant polynomial functions on $ {\mathfrak{a}^*} $ . An algebraic slice for $ \mathfrak{a} $ is an affine subspace η + V with $ \eta \in {\mathfrak{a}^*} $ and $ V \subset {\mathfrak{a}^*} $ subspace of dimension index $ \mathfrak{a} $ such that restriction of function induces an isomorphism of $ Y\left( \mathfrak{a} \right) $ onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which $ h \in \mathfrak{a} $ is ad-semisimple, η is a regular element of $ {\mathfrak{a}^*} $ which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to $ \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta $ in $ {\mathfrak{a}^*} $ . The classical case is for $ \mathfrak{g} $ semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if $ \mathfrak{g} $ is of type A and $ \mathfrak{a} $ is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in $ \mathfrak{g} $ . Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let $ \mathfrak{a} $ be a truncated biparabolic of index one. (This only arises if $ \mathfrak{g} $ is of type A and $ \mathfrak{a} $ is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for $ \mathfrak{a} $ is the restriction of a particularly carefully chosen regular nilpotent element of $ \mathfrak{g} $ . A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.

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{{Explor lien
   |wiki=    Wicri/Mathematiques
   |area=    BourbakiV1
   |flux=    France
   |étape=   Analysis
   |type=    RBID
   |clé=     ISTEX:AF5C43CD093D4A9A35743848CEF2260C2D1420EE
   |texte=   Slices for biparabolics of index 1
}}