arXiv subjects
Publications and source records attributed to Dehbia Achab.
In the recent paper [A14], a geometric realization to minimal representations of simple real Lie groups of non Hermitian type is given, based on the geometric setting introduced in [A11]. We give in this paper a geometric realization to minimal representations of simple real Lie groups of Hermitian type.
Some minimal representations of SL(n,R) can be realized on a Hilbert space of holomorphic functions. This is the analogue of the Brylinski-Kostant model. They can also be realized on a Hilbert space of homogeneous functions on ${\bboard R}^n$. This is the analogue of the Kobayashi-Orsted model. We will describe the two realizations and a transformation which maps one model to the other. It can be seen as an analogue of the classical Bargmann transform.
The minimal representations of Sp(r,R) can be realized on a Hilbert space of holomorphic functions. This is the analogue of the Brylinski-Kostant model. It can also be realized on a Hilbert space of L^2 functions on R^r. This is the Schröodinger model. We will describe the two realizations and a transformation which maps one model to the other. It involves the classical Bargmann transform and can be seen as its analogue.
In the recent paper [AF12], we introduced an analysis of the Brylinski-Kostant model for spherical minimal representations for simple real Lie groups of non Hermitian type. We generalize here that analysis and give a unified geometric realization to a family of unitary irreducible representations of such groups.
We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $Ξ$ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\goth g$, and furthermore a real form ${\goth g}_{\bboard R}$. The connected and simply connected Lie group $G_{\bboard R}$ with ${\rm Lie}(G_{\bboard R})={\goth g}_{\bboard R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $Ξ$
This paper deals with the analytic continuation of holomorphic automorphic forms on a Lie group $G$. We prove that for any discrete subgroup $Γ$ of $G$ there always exists a non-trivial holomorphic automorphic form, i.e., there exists a $Γ$-spherical unitary highest weight representation of $G$. Holomorphic automorphic forms have the property that they analytically extend to holomorphic functions on a complex Ol'shanski\uı semigroup $S\subeq G_\C$. As an application we prove that the bounded holomorphic functions on $Γ\bs S\subseteq Γ\bs G_\C$ separate the points.