Action Integrals and discrete series

2011/08/08 by Andrés Viña, Viña, Andrés
Mathematics · #22E45 #53D50 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1108.1611

openalex publication_date 2011/08/08 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

Let G be a complex semisimple Lie group and G_{\mathbb R} a real form that contains a compact Cartan subgroup T_{\mathbb R}. Let π be a discrete series representation of G_{\mathbb R}. We present geometric interpretations in terms of concepts associated with the manifold M:=G_{\mathbb R}/T_{\mathbb R} of the constant π(g), for g∈ Z(G_{\mathbb R}). For some relevant particular cases, we prove that this constant is the action integral around a loop of Hamiltonian diffeomorphims of M. As a consequence of these interpretations, we deduce lower bounds for the cardinal of the fundamental group of some subgroups of \rm Diff(M). We also geometrically interpret the values of the infinitesimal character of the differential representation of π.

Related

• On the geometric approach to the discrete series
• A finite Linear Dependence of Discrete Series Multiplicities
• Branching laws and a duality principle, Part I
• Action integrals and infinitesimal characters
• Ellipticity and discrete series