Related papers: Unitary spherical highest weight representations

Using Jakobsen theorems, unitarizability in Hermitian Symmetric Spaces is discussed. The set of all missing highest weights is explicitly calculated and the construction of their corresponding highest weights vectors is studied.

In a recent paper ([1],[2]) we have classified explicitely all the unitary highest weight representations of non compact real forms of semisimple Lie Algebras on Hermitian symmetric space. These results are necessary in order to construct…

We classify unitary highest weight modules with a given integral infinitesimal character for the real Lie algebras $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$. We treat both regular and singular cases. For $\mathfrak{su}(p,q)$ we…

Let $G$ be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then $G$ has unitary highest weight representations. The proof of the classification of unitary highest weight representations of $G$ given by…

In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.

Let g be a semisimple Lie algebra with h a Cartan subalgebra. The orbit method attempts to assign representations of g to orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits which should lead to highest…

Let G be a semi-simple non-compact Lie group with unitary lowest/highest weight representations. We consider explicitly the relation between three types of representations of G: positive energy (unitary lowest weight)representations,…

In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the…

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair…

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of…

A class of highest weight irreducible representations of the quantum algebra U_h(gl_\infty) is constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators…

In this manuscript, we give a classification of all irreducible, unitary representations of complex spin groups.

Let $G$ be a real compact Lie group, such that $G=G^0\rtimes C_2$, with $G^0$ simple. Here $G^0$ is the connected component of $G$ containing the identity and $C_2$ is the cyclic group of order $2$. We give a criterion for whether an…

For an arbitrary unimodular Lie group $G$, we construct strongly continuous unitary representations in the Bergman space of a naturally constructed strongly pseudoconvex neighborhood of $G$ in the complexification of its underlying…

We consider the following class of unitary representations $\pi $ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau $, and that the Hilbert space…

We study the highest weight and continuous tensor product representations of q-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of the q-deformed algebra…

Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of…

We consider spherical principal series representations of the semisimple Lie group of rank one $G=SO(n, 1; \mathbb K)$, $\mathbb K=\br, \bc, \bh$. There is a family of unitarizable representations $\pi_{\nu}$ of $G$ for $\nu$ in an interval…

We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the…

In this series of papers we want to discuss the highest weight ${\frak k}_r$-finite representations of the pair $({\frak g}_r,{\frak k}_r)$ consisting of ${\frak g}_r$, a real form of a complex basic Lie superalgebra of classical type…