On the unitary cohomology of semisimple groups
Abstract
We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group SO∘(n,1). For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.
Related
- Sharp Hausdorff--Young inequalities for solvable and semisimple Lie groups
- Spherical density for real semisimple groups
- Wehrl-type entropy problem for compact connected semisimple Lie groups
- Haar measure for totally positive split real semisimple quantum groups of classical type
- On the complexification of $ q $-deformations of compact semisimple Lie groups
BibTeX
@misc{bader2026unitary,
title = {{On the unitary cohomology of semisimple groups}},
author = {Uri Bader and Michael Glasner and Yuval Gorfine and Roman Sauer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08652},
eprint = {2609.08652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08652},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the unitary cohomology of semisimple groups
AU - Uri Bader
AU - Michael Glasner
AU - Yuval Gorfine
AU - Roman Sauer
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08652
UR - https://arxiv.org/abs/2609.08652
AB - We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group SO^∘(n,1). For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.
ER -