2010/11/03 by Florence Fauquant-Millet, Fauquant-Millet, Florence, Anthony Joseph +1
Mathematics · #17B35 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B35
paper · pdf · doi:10.48550/arxiv.1011.0928
31 pages, 7 figures
arxiv created 2010/11/03 · openalex publication_date 2010/11/03 · arxiv updated 2010/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrak a be an algebraic Lie subalgebra of a simple Lie algebra \mathfrak g with index \mathfrak a ≤ \rank \mathfrak g. Let Y(\mathfrak a) 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 η∈ \mathfrak a^* and V ⊂ \mathfrak a^* a subspace of dimension index \mathfrak a such that restriction of function induces an isomorphism of Y(\mathfrak a) 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 ∈\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 (\ad \mathfrak a)η in \mathfrak a^*. The classical case is for \mathfrak g semisimple. 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" or a centralizer. In some of these cases (particular when the biparabolic is a Borel subalgebra) it was found that η could be taken to be the restriction of a regular nilpotent element in \mathfrak g. Moreover this calculation suggested 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 (and then \mathfrak g is of type A). 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.