Coadjoint orbits and reductive dual pairs
Suppose \((\tilde G,\tilde G')\) is a reductive dual pair of subgroups of the metaplectic group \(\tilde Sp(2n,{\mathbb{R}})\) with \(\tilde G\) compact. In an earlier paper (see the preceding review), the author established an association between the unitary highest weight modules of a classical simple Lie group, such as \(\tilde G'\), and the coadjoint orbits - that is, he showed how to quantize the orbits. But it is known about reductive dual pairs that the restriction of the oscillator representation of \(Sp(2n,{\mathbb{R}})\) to \(\tilde G\tilde G'\), decomposes as \(\sum \tau_ i \otimes \tau_ i'\) where the correspondence \(\tau_ i\leftrightarrow \tau_ i'\) is a bijection between the unitary highest weight modules of the two groups. Thus, we have two realizations of the unitary highest weight modules of \(\tilde G'\). How are the two connected ? The main result of this paper is a theorem which describes the bijection \(\tau_ i\leftrightarrow \tau_ i'\) as a correspondence between quantizations of orbits \(\pi({\mathcal O})\leftrightarrow \pi'({\mathcal O}')\).
scientific article; zbMATH DE number 4010814
scientific article