Adapted pairs in type A and regular nilpotent elements

Let g be a simple Lie algebra over an algebraically closed field k of characteristic zero and G its adjoint group. A biparabolic subalgebra q of g is the intersection of two parabolic subalgebras whose sum is g. The algebra Sy(q) of semi-invariants on q⁎ of a proper biparabolic subalgebra q of g is polynomial in most cases, in particular when g is simple of type A or C. On the other hand q admits a canonical truncation qΛ such that Sy(q)=Sy(qΛ)=Y(qΛ) where Y(qΛ) denotes the algebra of invariant functions on (qΛ)⁎. An adapted pair for qΛ is a pair (h,η)∈qΛ×(qΛ)⁎ such that η is regular in (qΛ)⁎ and (adh)(η)=−η. In Joseph (2008) [9] adapted pairs for every truncated biparabolic subalgebra qΛ of a simple Lie algebra g of type A were constructed and then provide Weierstrass sections for Y(qΛ) in (qΛ)⁎. These Weierstrass sections are linear subvarieties η+V of (qΛ)⁎ such that the restriction map induces an algebra isomorphism of Y(qΛ) onto the algebra of regular functions on η+V. The main result of the present work is to show that for each of the adapted pairs (h,η) constructed in Joseph (2008) [9] one can express η (not quite uniquely) as the image of a regular nilpotent element y of g⁎ under the restriction map g⁎→q⁎. This is a significant extension of Joseph and Fauquant-Millet (2011) [12], which obtains this result in the rather special case of a truncated biparabolic of index one. Observe that y must be a G translate of the standard regular nilpotent element defined in terms of the already chosen set π of simple roots. Consequently one may attach to y a unique element of the Weyl group W of g. Ultimately one can then hope to be able to describe adapted pairs (in general, that is not only for g of type A) through the Weyl group.