On Semisimple Classes and Component Groups in Type D

In adjoint simple algebraic groups H of type D we show that for every semisimple element s , its centralizer splits over its identity component, i.e. C H ( s ) = C H ( s ) ∘ ⋊ A ˇ for some complement A ˇ with strong stability properties. We derive several consequences about the action of automorphis...

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Bibliographic Details
Published inVietnam journal of mathematics Vol. 52; no. 2; pp. 435 - 444
Main Authors Cabanes, Marc, Späth, Britta
Format Journal Article
LanguageEnglish
Published Singapore Springer Nature Singapore 01.04.2024
Springer Nature B.V
Subjects
Automorphisms
Group theory
Mathematics
Mathematics and Statistics
Original Article
20C33
20G40
Projective special orthogonal group
Semisimple classes
20D06
Component group
20C15
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Summary:In adjoint simple algebraic groups H of type D we show that for every semisimple element s , its centralizer splits over its identity component, i.e. C H ( s ) = C H ( s ) ∘ ⋊ A ˇ for some complement A ˇ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups D l , sc ( q ) and 2 D l , sc ( q ) and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conjecture for the prime 3, see (B. Späth: 2023).
ISSN:2305-221X
2305-2228
DOI:10.1007/s10013-023-00642-2