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Part of the book series: Progress in Mathematics ((PM,volume 204))

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Abstract

Most of the content of Section 5.1 is fairly standard. We recall the definition of a Tits system, also called a BN-pair, and then collect its various general properties in Theorem 5.1.3, which we will need subsequently, including the Bruhat decomposition. We also recall the definition of a topological Tits system. Its definition is modeled so that the associated “flag varieties” have aCW-complex structure with Bruhat cells as the cells (cf. Theorem 5.1.5). It is shown that for a Tits system(GBNS)Gis an amalgamated product of its minimal parabolic subgroups {Ps} s Es andN.Conversely, given a finite family of groupsB{P.}.EsandNsatisfying a number of properties, it is shown that(GBNS)is a Tits system, whereGis the amalgamated product ofB{P.}andN(cf. Theorem 5.1.8). This result, due to Tits, will be used in the construction of Kac-Moody groups in the next chapter.

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Kumar, S. (2002). Tits Systems. In: Kac-Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, vol 204. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0105-2_5

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  • DOI: https://doi.org/10.1007/978-1-4612-0105-2_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6614-3

  • Online ISBN: 978-1-4612-0105-2

  • eBook Packages: Springer Book Archive

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