Projective representations of almost unimodular groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918139069988864 |
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| author | Guinto, Aldo Garcia |
| author_facet | Guinto, Aldo Garcia |
| contents | Given an almost unimodular $G$, so that the Plancherel weight $φ_G$ on the group von Neumann algebra $L(G)$ is almost periodic, we show that the basic construction for the inclusion $L(G)^{φ_G} \leq L(G)$ is isomorphic to a twisted group von Neumann algebra of $G \times Δ_G(G)\hat{\ }$ with a continuous 2-cocycle, where $Δ_G$ is the modular function. We show that when $G$ is second countable and admits a Borel 2-cocycle, $G$ is almost unimodular if and only if the central extension $\mathbb{T} \rtimes_{(1,ω)} G$ is almost unimodular. Using this result and the connection between $ω$-projective representations of $G$ and the representations of $\mathbb{T} \rtimes_{(1,ω)} G$, we show that the formal degrees of irreducible and factorial square integrable projective representations behaved similarly to their representations counterparts and obtain the Atiyah--Schmid formula in the setting of second countable almost unimodular groups with a 2-cocycle twist and a finite covolume subgroup, which uses the Murray--von Neumann dimension for certain Hilbert space modules over the twisted group von Neumann algebra with its twisted Plancherel weight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective representations of almost unimodular groups Guinto, Aldo Garcia Operator Algebras Given an almost unimodular $G$, so that the Plancherel weight $φ_G$ on the group von Neumann algebra $L(G)$ is almost periodic, we show that the basic construction for the inclusion $L(G)^{φ_G} \leq L(G)$ is isomorphic to a twisted group von Neumann algebra of $G \times Δ_G(G)\hat{\ }$ with a continuous 2-cocycle, where $Δ_G$ is the modular function. We show that when $G$ is second countable and admits a Borel 2-cocycle, $G$ is almost unimodular if and only if the central extension $\mathbb{T} \rtimes_{(1,ω)} G$ is almost unimodular. Using this result and the connection between $ω$-projective representations of $G$ and the representations of $\mathbb{T} \rtimes_{(1,ω)} G$, we show that the formal degrees of irreducible and factorial square integrable projective representations behaved similarly to their representations counterparts and obtain the Atiyah--Schmid formula in the setting of second countable almost unimodular groups with a 2-cocycle twist and a finite covolume subgroup, which uses the Murray--von Neumann dimension for certain Hilbert space modules over the twisted group von Neumann algebra with its twisted Plancherel weight. |
| title | Projective representations of almost unimodular groups |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2509.09065 |