The oscillator correspondence of orbital integrals, for pairs of type one in the stable range
Let \((G,G') \subseteq \text{Sp}(W)\) be a real reductive dual pair in the sense of Howe. Howe duality defines a bijection \(\Pi \leftrightarrow \Pi'\) between certain sets of irreducible, admissible representations of \(G\) and \(G'\), or more precisely, of their twofold coverings in the metaplectic group. One of the fundamental questions in Howe duality theory is how various properties and invariants of \(\Pi\) and \(\Pi'\) are related. Howe has conjectured that for generic \(\Pi\) and \(\Pi'\) the wave front sets \(WF(\Pi)\) and \(WF(\Pi')\) should be related via the moment maps \(\tau:W\to{\mathfrak g}^*\), \(\tau':W\to{\mathfrak g}'{}^*\) as follows: \(WF(\Pi')=\tau'(\tau^{-1}(WF(\Pi)))\). For pairs of type I in the stable range the authors describe a correspondence of orbital integrals on \(\mathfrak g\) and \({\mathfrak g}'\) which agrees with the correspondence of orbits defined by these maps, via an explicit integral kernel operator. This allows a more precise formulation of Howe's conjecture by invoking the Fourier transform and an asymptotic expansion of characters. Under some strong assumptions (``deep stable range'') the formula \(WF(\Pi')=\tau'(\tau^{-1}(WF(\Pi)))\) was proved by the authors in [Invent. Math. 123, 349-376 (1996; Zbl 0845.22007)].
scientific article; zbMATH DE number 908679