Adapted pairs in type $A$ and regular nilpotent elements

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field $\mathbf{k}$ of characteristic zero and $\mathbf{G}$ its adjoint group. Let $\mathfrak{q}$ be a biparabolic subalgebra of $\mathfrak{g}$. The algebra $Sy(\mathfrak{q})$ of semi-invariants on ${\mathfrak{q}}^{\ast }$ 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 ${\mathfrak{q}}_{\Lambda }^{\ast }$. An adapted pair for ${\mathfrak{q}}_{\Lambda }$ is a pair $(h,\eta )\in {\mathfrak{q}}_{\Lambda }\times {\mathfrak{q}}_{\Lambda }^{\ast }$ such that $\eta$ is regular and $(adh)\eta =-\eta$. 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 ${\mathfrak{q}}_{\Lambda }^{\ast }$. These latter are linear subvarieties $\eta +V$ of ${\mathfrak{q}}_{\Lambda }^{\ast }$ 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 ${\mathfrak{g}}^{\ast }$ under the restriction to $\mathfrak{q}$. Since $y$ must be a $\mathbf{G}$ 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.

arXiv commentThis is a rewriting of the version submitted on the arXiv in June 2013