Principal series representations of direct limit groups

2005/11/01 by Joseph A. Wolf · 3 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models

paper · pdf · doi:10.1112/s0010437x05001430

Abstract

We combine the geometric realization of principal series representations on partially holomorphic cohomology spaces, with the Bott–Borel–Weil theorem for direct limits of compact Lie groups, obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm-preserving representations on a Banach space or unitary representations on a Hilbert space. We specialize the results to diagonal embedding direct limit groups. Finally we discuss the possibilities of extending the results to limits of tempered series other than the principal series.

Cited by

• Fundamentals of direct limit Lie theory
• A Bott-Borel-Weil theorem for diagonal ind-groups
• A construction of generalized Harish-Chandra modules for locally\n reductive Lie algebras

Related

• Principal Series Representations of Direct Limit Groups
• Principal Series Representations of Infinite Dimensional Lie Groups, I: Minimal Parabolic Subgroups
• Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations
• Fundamentals of direct limit Lie theory
• Fundamentals of direct limit Lie theory