Nets of standard subspaces on non-compactly causal symmetric spaces

TY - CHAP

T1 - Nets of standard subspaces on non-compactly causal symmetric spaces

AU - Frahm, Jan

AU - Neeb, Karl-Hermann

AU - Ólafsson, Gestur

PY - 2025/3

Y1 - 2025/3

N2 - Let G be a connected simple linear Lie group and H⊂G a symmetric subgroup such that the corresponding symmetric space G∕H is non-compactly causal. We show that any irreducible unitary representation of G leads naturally to a net of standard subspaces on G∕H that is isotone, covariant and has the Reeh–Schlieder and the Bisognano–Wichmann property. We also show that this result extends to the universal covering group of SL 2(ℝ), which has some interesting application to intersections of standard subspaces associated to representations of such groups. For this, a detailed study of hyperfunction and distribution vectors is needed. In particular, we show that every H-finite hyperfunction vector is in fact a distribution vector.

AB - Let G be a connected simple linear Lie group and H⊂G a symmetric subgroup such that the corresponding symmetric space G∕H is non-compactly causal. We show that any irreducible unitary representation of G leads naturally to a net of standard subspaces on G∕H that is isotone, covariant and has the Reeh–Schlieder and the Bisognano–Wichmann property. We also show that this result extends to the universal covering group of SL 2(ℝ), which has some interesting application to intersections of standard subspaces associated to representations of such groups. For this, a detailed study of hyperfunction and distribution vectors is needed. In particular, we show that every H-finite hyperfunction vector is in fact a distribution vector.

UR - http://www.scopus.com/inward/record.url?scp=85218051712&partnerID=8YFLogxK

U2 - 10.1007/978-981-97-7662-7_5

DO - 10.1007/978-981-97-7662-7_5

M3 - Book chapter

SN - 978-9819784486

VL - 2

T3 - Progress in Mathematics

SP - 115

EP - 195

BT - Symmetry in Geometry and Analysis

A2 - Pevzner, Michael

A2 - Sekiguchi, Hideko

PB - Birkhäuser Verlag

ER -