2013/06/03 by Fauquant-Millet, Florence, Joseph, Anthony
#17B35 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1306.0529
Let \mathfrak g be a simple Lie algebra over an algebraically closed field \bf k of characteristic zero and \bf G its adjoint group. Let \mathfrak q be a biparabolic subalgebra of \mathfrak g. The algebra Sy(\mathfrak q) of semi-invariants on \mathfrak q^* is polynomial in most cases, in particular when \mathfrak g is simple of type A or C. On the other hand \mathfrak q admits a canonical truncation \mathfrak qΛ such that Sy(\mathfrak q)=Sy(\mathfrak qΛ)=Y(\mathfrak qΛ) where Y(\mathfrak qΛ) denotes the algebra of invariant functions on \mathfrak qΛ^*. An adapted pair for \mathfrak qΛ is a pair (h, η)∈ \mathfrak qΛ×\mathfrak qΛ^* such that η is regular and (ad h)η=-η. In a previous paper of A. Joseph (2008) adapted pairs for every truncated biparabolic subalgebra \mathfrak qΛ of a simple Lie algebra \mathfrak g of type A were constructed and then provide Weierstrass sections for Y(\mathfrak qΛ) in \mathfrak qΛ^*. These latter are linear subvarieties η+V of \mathfrak qΛ^* such that the restriction map induces an algebra isomorphism of Y(\mathfrak qΛ) onto the algebra of regular functions on η+V. Here we show that for each of the adapted pairs (h, η) constructed in the paper mentioned above one can express η as the image of a regular nilpotent element y of \mathfrak g^* under the restriction to \mathfrak q. Since y must be a \bf G translate of the standard regular nilpotent element defined in terms of the already chosen set π of simple roots, one may attach to y a unique element of the Weyl group. Ultimately one can then hope to be able to describe adapted pairs (in general) through the Weyl group.