Semi-group compactifications of Algebraic Groups. arXiv:2404.09878
Preprint, arXiv:2404.09878 [math.GR] (2024).
Summary: We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing representations, and that the matrix coefficients are dense within the algebra of weakly almost periodic functions over the group. In our proof, we employ methods from semi-group theory. We establish that algebraic groups are compactification-centric, meaning \(sG = Gs\) for any element \(s\) in the weakly almost periodic compactification of the group \(G\).
MSC:
20G05 | Representation theory for linear algebraic groups |
22E50 | Representations of Lie and linear algebraic groups over local fields |
20M99 | Semigroups |
20G25 | Linear algebraic groups over local fields and their integers |
43A99 | Abstract harmonic analysis |
54H11 | Topological groups (topological aspects) |
54H13 | Topological fields, rings, etc. (topological aspects) |
54D25 | “\(P\)-minimal” and “\(P\)-closed” spaces |
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