Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Abstract

Abstract. -Let g be a finite dimensional complex reductive Lie algebra and S(g) its symmetric algebra. The nilpotent bicone of g is the subset of elements (x, y) of g×g whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of S(g) ⊗ C S(g) generated by the 2-order polarizations of invariants of S(g). The main result of this note is that the nilpotent bicone is a complete intersection of dimension 3(bg − rk g), where bg and rk g are the dimension of Borel subalgebras and the rank of g respectively. This affirmatively answers a conjecture of KraftWallach concerning the nullcone [KrW2]. In addition, we introduce and study in this note the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by M. Mustaţǎ in [Mu]. At last, we give applications of our results to invariant theory.

A remark on Mishchenko–Fomenko algebras and regular sequences

“…Denote by ̟ 1 and ̟ 2 the first and second projections from g × g to g, Note that the scheme N is not reduced [3]. Since the algebra C[g × g] is Cohen-Macaylay, and since the elements p i , i = 1, .…”

“…It is based on geometric properties of the nilpotent bicone (cf. Definition 2.1) introduced and studied in [3]. We recall in Section 2 the main results of [3] on the nilpotent bicone.…”

“…Definition 2.1) introduced and studied in [3]. We recall in Section 2 the main results of [3] on the nilpotent bicone. As a consequence we get Theorem 1.2 for ξ nilpotent and regular.…”

Very nilpotent basis and n-tuples in Borel subalgebras

“…Finally, the nilpotent bicone, studied in [2], is the affine subscheme N ⊂ g × g defined by the polarized polynomials p i (x + t y) = 0, ∀t ∈ k. Its underlying set consists of pairs whose any linear combination is nilpotent. It is a non-reduced complete intersection, which contains C nil (g) and which has G.(n × n) as an irreducible component.…”

The null-cone and cohomology of vector bundles on flag varieties

“…Such actions and their generalizations have been considered by various authors; see, for example, [CM,KW1,KW2,LMP]. Our interest in the question was motivated by applications in the study of ordinary deformations of Galois representations.…”