Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T23:17:14.571Z Has data issue: false hasContentIssue false

ON CENTRALISERS AND NORMALISERS FOR GROUPS

Published online by Cambridge University Press:  02 August 2012

BORIS ŠIROLA*
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (email: sirola@math.hr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 𝕂 be a field, char(𝕂)≠2, and G a subgroup of GL(n,𝕂). Suppose gg is a 𝕂-linear antiautomorphism of G, and then define G1={gGgg=I}. For C being the centraliser 𝒞G (G1) , or any subgroup of the centre 𝒵(G) , define G(C) ={gGggC}. We show that G(C) is a subgroup of G, and study its structure. When C=𝒞G (G1) , we have that G(C) =𝒩G (G1) , the normaliser of G1 in G. Suppose 𝕂 is algebraically closed, 𝒞G (G1) consists of scalar matrices and G1 is a connected subgroup of an affine group G. Under the latter assumptions, 𝒩G (G1) is a self-normalising subgroup of G. This holds for a number of interesting pairs (G,G1); in particular, for those that we call parabolic pairs. As well, for a certain specific setting we generalise a standard result about centres of Borel subgroups.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[Bor]Borel, A., Linear Algebraic Groups, second enlarged edition, Graduate Texts in Mathematics, 126 (Springer, New York, 1997).Google Scholar
[Bou]Bourbaki, N., Lie Groups and Lie Algebras (Springer, Berlin, 1989), Ch. 1–3.Google Scholar
[BK]Brylinski, R. and Kostant, B., ‘Nilpotent orbits, normality, and Hamiltonian group actions’, J. Amer. Math. Soc. 7 (1994), 269298.Google Scholar
[Ko1]Kobayashi, T., ‘Discrete decomposability of the restriction of A 𝔮(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties’, Invent. Math. 131 (1998), 229256.CrossRefGoogle Scholar
[Ko2]Kobayashi, T., ‘Discretely decomposable restrictions of unitary representations of reductive Lie groups—examples and conjectures’, in: Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto (1997), Advanced Studies in Pure Mathematics, 26 (Math. Soc. Japan, Tokyo, 2000), pp. 99127.Google Scholar
[Ko3]Kobayashi, T., ‘Restrictions of generalized Verma modules to symmetric pairs’, Transform. Groups 17 (2012), 523546.CrossRefGoogle Scholar
[Kob]Koblitz, N., A Course in Number Theory and Cryptography, 2nd edn, Graduate Texts in Mathematics, 114 (Springer, Berlin, 1994).Google Scholar
[Ks]Kostant, B., ‘A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel–Weil theorem’, in: Noncommutative Harmonic Analysis, Progress in Mathematics 220 (Birkhäuser, Boston, 2004), pp. 291353.CrossRefGoogle Scholar
[LS]Levasseur, T. and Smith, S. P., ‘Primitive ideals and nilpotent orbits in type G 2’, J. Algebra 114 (1988), 81105.CrossRefGoogle Scholar
[O1]Osinovskaya, A. A., ‘Restrictions of irreducible representations of classical algebraic groups to root A 1-subgroups’, Comm. Algebra 31 (2003), 23572379.Google Scholar
[O2]Osinovskaya, A. A., ‘On the restrictions of modular irreducible representations of algebraic groups of type A n to naturally embedded subgroups of type A 2’, J. Group Theory 8 (2005), 4392.CrossRefGoogle Scholar
[S1]Širola, B., ‘Pairs of semisimple Lie algebras and their maximal reductive subalgebras’, Algebr. Represent. Theory 11 (2008), 233250.CrossRefGoogle Scholar
[S2]Širola, B., ‘Pairs of Lie algebras and their self-normalizing reductive subalgebras’, J. Lie Theory 19 (2009), 735766.Google Scholar
[S3]Širola, B., ‘Normalizers and self-normalizing subgroups’, Glas. Mat. Ser. III 46 (2011), 385414.CrossRefGoogle Scholar
[S4]Širola, B., ‘Normalizers and self-normalizing subgroups II’, Cent. Eur. J. Math. 9 (2011), 13171332.Google Scholar