Duflo’s conjecture for the branching problem to the Iwasawa $$AN$$ A N -subgroup of a semisimple Lie group

Abstract

The purpose of this paper is to prove Duflo’s conjecture for \((G,\pi ,\,AN)\) where \(G\) is a real connected 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 deduced from the following general theorem which we prove in this paper: let \(G\) be a real connected semisimple Lie group with Lie algebra \({\mathfrak {g}}\) . 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. Here \(\text {p} : {\mathfrak {g}}^* \longrightarrow ({\mathfrak {a}}\oplus {\mathfrak {n}})^*\) is the natural projection.