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arXiv:1712.05552 [math.RT]AbstractReferencesReviewsResources

Unipotent representations of real classical groups

Jia-jun Ma, Binyong Sun, Chen-Bo Zhu

Published 2017-12-15Version 1

Let $\mathbf G$ be a complex orthogonal or complex symplectic group, and let $G$ be a real form of $\mathbf G$, namely $G$ is a real orthogonal group, a real symplectic group, a quaternionic orthogonal group, or a quaternionic symplectic group. For a fixed parity $\mathbb p\in \mathbb Z/2\mathbb Z$, we define a set $\mathrm{Nil}^{\mathbb p}_{\mathbf G}(\mathfrak g)$ of nilpotent $\mathbf G$-orbits in $\mathfrak g$ (the Lie algebra of $\mathbf G$). When $\mathbb p$ is the parity of the dimension of the standard module of $\mathbf G$, this is the set of the stably trivial special nilpotent orbits, which includes all rigid special nilpotent orbits. For each $\mathcal O \in \mathrm{Nil}^{\mathbb p}_{\mathbf G}(\mathfrak g)$, we construct all unipotent representations of $G$ (or its metaplectic cover when $G$ is a real symplectic group and $\mathbb p$ is odd) attached to $\mathcal O$ via the method of theta lifting and show in particular that they are unitary.

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