Littlewood-Richardson Coefficient, Springer Fibers and the Annihilator Varieties of Induced Representations

TL;DR

This paper connects Littlewood-Richardson coefficients with the geometry of Springer fibers and annihilator varieties in the context of induced representations of GL(n,C), providing new geometric criteria for orbit inclusion.

Contribution

It establishes a geometric criterion linking Littlewood-Richardson coefficients to orbit inclusions and describes the structure of intersections relevant to representation theory.

Findings

Non-vanishing Littlewood-Richardson coefficients characterize orbit inclusions.

The geometry of the intersection of nilpotent orbits with parabolic subalgebras is elucidated.

Results have implications for understanding Whittaker supports and annihilator varieties.

Abstract

For $G=GL(n,\mathbb{C})$ and a parabolic subgroup $P=LN$ with a two-block Levi subgroup $L=GL(n_1)\times GL(n_2)$, the space $G\cdot (\mathcal{\mathcal{O}}+\mathfrak{n})$, where $\mathcal{O}$ is a nilpotent orbit of $\mathfrak{l}$, is a union of nilpotent orbits of $\mathfrak{g}$. In the first part of our main theorem, we use the geometric Sakate equivalence to prove that $\mathcal{O'}\subset G\cdot (\mathcal{\mathcal{O}}+\mathfrak{n})$ if and only if some Littlewood-Richardson coefficients do not vanish. The second part of our main theorem describes the geometry of the space $\mathcal{O}\cap\mathfrak{p}$, which is an important space to study for the Whittaker supports and annihilator varieties of representations of $G$.