Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6414327 | Journal of Algebra | 2016 | 67 Pages |
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 qâ 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 qÎ such that Sy(q)=Sy(qÎ)=Y(qÎ) where Y(qÎ) denotes the algebra of invariant functions on (qÎ)â. An adapted pair for qÎ is a pair (h,η)âqÎÃ(qÎ)â such that η is regular in (qÎ)â and (adh)(η)=âη. In Joseph (2008) [9] adapted pairs for every truncated biparabolic subalgebra qÎ of a simple Lie algebra g of type A were constructed and then provide Weierstrass sections for Y(qÎ) in (qÎ)â. These Weierstrass sections are linear subvarieties η+V of (qÎ)â such that the restriction map induces an algebra isomorphism of Y(qÎ) 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 gâ under the restriction map gââqâ. 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 Ï 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.