科研之友
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摘要
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 q(Lambda) such that Sy(q) = Sy(q(Lambda)) = Y(q(Lambda)) where Y(q(Lambda)) denotes the algebra of invariant functions on (q(Lambda))*. An adapted pair for q(Lambda) is a pair (h, eta) is an element of q(Lambda) X (q(Lambda))* such that eta is regular in (q(Lambda))* and (ad h)(eta) = -eta. In Joseph (2008) [9] adapted pairs for every truncated biparabolic subalgebra q(Lambda) of a simple Lie algebra g of type A were constructed and then provide Weierstrass sections for Y(q(Lambda)) in (q(Lambda))*. These Weierstrass sections are linear subvarieties eta+V of (q(Lambda))* such that the restriction map induces an algebra isomorphism of Y(q(Lambda)) onto the algebra of regular functions on eta + V. The main result of the present work is to show that for each of the adapted pairs (h, eta) constructed in Joseph (2008) [9] one can express eta (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 pi 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.
- 出版日期2016-5-1
