  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 $\mathfrak q_{\Lambda}$ such that
$Sy(\mathfrak q)=Sy(\mathfrak q_{\Lambda})=Y(\mathfrak q_{\Lambda})$ where
$Y(\mathfrak q_{\Lambda})$ denotes the algebra of invariant functions on
$\mathfrak q_{\Lambda}^*$. An adapted pair for $\mathfrak q_{\Lambda}$ is a
pair $(h,\,\eta)\in \mathfrak q_{\Lambda}\times\mathfrak q_{\Lambda}^*$ such
that $\eta$ is regular and $(ad\,h)\eta=-\eta$. In a previous paper of A.
Joseph (2008) adapted pairs for every truncated biparabolic subalgebra
$\mathfrak q_{\Lambda}$ of a simple Lie algebra $\mathfrak g$ of type $A$ were
constructed and then provide Weierstrass sections for $Y(\mathfrak
q_{\Lambda})$ in $\mathfrak q_{\Lambda}^*$. These latter are linear
subvarieties $\eta+V$ of $\mathfrak q_{\Lambda}^*$ such that the restriction
map induces an algebra isomorphism of $Y(\mathfrak q_{\Lambda})$ onto the
algebra of regular functions on $\eta+V$. Here we show that for each of the
adapted pairs $(h,\,\eta)$ constructed in the paper mentioned above one can
express $\eta$ 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 $\pi$ 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.
