Duflo's conjecture for the branching to the Iwasawa $AN$-subgroup

TL;DR

This paper proves Duflo's conjecture for discrete series representations of Hermitian type simple Lie groups when restricted to the maximal exponential solvable subgroup, linking orbit holomorphicity to open coadjoint orbits.

Contribution

It establishes the equivalence between holomorphicity of strongly elliptic coadjoint orbits and the openness of their projections onto the $AN$-subgroup, confirming Duflo's conjecture in this setting.

Findings

Proves Duflo's conjecture for Hermitian type simple Lie groups.

Characterizes holomorphic coadjoint orbits via their projections.

Links orbit properties to the structure of the $AN$-subgroup.

Abstract

The purpose of this paper is to prove Duflo's conjecture for $(G,\pi, AN)$ where $G$ is a simple Lie group of Hermitian type and $\pi$ is a discrete series of $G$ and $AN$ is the maximal exponential solvable subgroup for an Iwasawa decomposition $G=KAN$. This is essentially reduced from the following general theorem we prove in this paper: let $G$ be a connected semisimple Lie group . Then a strongly elliptic $G$-coadjoint orbit $\mathcal{O}$ is holomorphic if and only if $\text{p}(\mathcal{O})$ is an open $AN$-coadjoint orbit, where $\text{p} : \mathfrak{g}^* \longrightarrow (\mathfrak{a}\oplus\mathfrak{n})^*$ is the natural projection.