The integrality of an adapted pair

Abstract: Let $\mathfrak a$ be an algebraic Lie algebra. An adapted pair for $\mathfrak a$ is pair $(h,\eta)$ consisting of an ad-semisimple element of $h \in \mathfrak a$ and a regular element of $\eta \in \mathfrak a^*$ satisfying $(ad \ h)\eta=-\eta$. An adapted pair $(h,\eta)$ is said to satisfy integrality if $ad \ h$ has integer eigenvalues on $\mathfrak a$. Integrality is shown to hold for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple Lie algebra; but may fail in general. Call $\mathfrak a$ regular if there are no proper semi-invariant polynomial functions on $\mathfrak a^*$ and if the subalgebra of invariant functions is polynomial. In this case there are no known counter-examples to integrality. It is shown that if $\mathfrak a$ is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra $\mathfrak g$ which is regular and admits an adapted pair $(h,\eta)$, then the eigenvalues of $ad \ h$ on $\mathfrak a$ lie in $\frac{1}{m}\mathbb Z$, where $m$ is a coefficient of a simple root in the highest root of $\mathfrak g$. Let $\mathfrak a$ be a regular Lie algebra admitting an adapted pair $(h,\eta)$. Let $\mathfrak a_\mathbb Z$ be the subalgebra spanned by the eigensubspaces of $ad \ h$ with integer eigenvalue. It is shown that the canonical truncation of $\mathfrak a_\mathbb Z$ is regular. Sufficient knowledge of the relation between the generators for the invariant polynomial functions on $\mathfrak a^*$ and on $\mathfrak a^*_\mathbb Z$ can then lead to establishing the integrality of $(h,\eta)$. This method is used to show the integrality of an adapted pair for a truncated parabolic subalgebra of a simple Lie algebra of type $C$.