Unipotent representations of real classical groups

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

Real-time representations of the output gap

Methods are described for the appropriate use of data obtained and analysed in real time to represent the output gap. The methods employ cointegrating VAR techniques to model real-time measures and realizations of output series jointly. The model is used to mitigate the impact of data revisions; to generate appropriate forecasts that can deliver economically meaningful output trends and that can take into account the end-of-sample problems encountered in measuring these trends; and to calculate probability forecasts that convey in a clear way the uncertainties associated with the gap measures. The methods are applied to data for the United States 1965q4–2004q4, and the improvements over standard methods are illustrated

Images of Real Representations of SLn(Zp)SL_n(Z_p)

In this paper, we investigate abstract homomorphism from the special linear group over complete discrete valuation rings with finite residue field, such as the ring of p-adic integers, into the general linear group over the reals. We find the minimal dimension in which such a representation has infinite image. For positive characteristic rings, this minimum is infinity.Comment: 10 pages, corrected typos and simplified the proof