Dual pairs in the Pin-group and duality for the corresponding spinorial representation. (English) Zbl 1511.22019
Summary: In this paper, we give a complete picture of Howe correspondence for the setting \((O(E, b), Pin(E, b),\Pi)\), where \(O(E, b)\) is a real orthogonal group, \(Pin(E, b)\) is the two-fold Pin-covering of \(O(E, b)\), and \(\Pi\) is the spinorial representation of \(Pin(E, b)\). More precisely, for a dual pair \((G, G^{\prime})\) in \(O(E, b)\), we determine explicitly the nature of its preimages \((\tilde{G}, \tilde{G}^{\prime})\) in \(Pin(E, b)\), and prove that apart from some exceptions, \((\tilde{G}, \tilde{G}^{\prime})\) is always a dual pair in \(Pin(E, b)\); then we establish the Howe correspondence for \(\Pi\) with respect to \((\tilde{G}, \tilde{G}^{\prime})\).
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 20G05 | Representation theory for linear algebraic groups |
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