Let g be a complex reductive Lie algebra. We investigate a formalism for producing a natural model for representations of sl(2) × g which decompose in such a way so as to produce interesting correspondences between representations of sl(2) and of g. Our formalism generalizes the standard dual pair SL(2)× O(2n). We use it to obtain a formally unitary representation of sl(2,R)×u(3) that essential...

Theta correspondence θ over R was established by Howe in ( [5]). In [10], we proved that θ preserves unitarity under certain restrictions, generalizing the result of Jian-Shu Li ( [16]). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual p...

The orbit method conjectures a close relationship between the set of irreducible unitary representations of a Lie group G, and admissible coadjoint orbits in the dual of the Lie algebra. We define admissibility for nilpotent coadjoint orbits of p-adic reductive Lie groups, and compute the set of admissible orbits for a range of examples. We find that for unitary, symplectic, orthogonal, general...

We determine the unitary dual of the geometric graded Hecke algebras with unequal parameters which appear in Lusztig’s classification of unipotent representations for exceptional p-adic groups. The largest such algebra is of type F4. Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent represent...

Let G be a connected and simply connected nilpotent Lie group with Lie algebra g and unitary dual Ĝ. The moment map for π ∈ Ĝ sends smooth vectors in the representation space of π to g∗. The closure of the image of the moment map for π is called its moment set. N. Wildberger has proved that the moment set for π coincides with the closure of the convex hull of the corresponding coadjoint orbit. ...

Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate. This is generally understood in terms competition between scrambling effects unitary dynamics and disentangling measurements. We find that, surprisingly, EPTs are possible even absence dynamics, where they best arising from alone. motivates us in...

We consider one dimensional block cellular automata, where the local update rules are given by Yang-Baxter maps, which set theoretical solutions of equations. show that such systems superintegrable: they possess an exponentially large conserved charges, charge densities propagate ballistically on chain. For these quantities we observe a complete absence "operator spreading". In addition, models...