Lie Groups pp 227-242 | Cite as

The Borel Subgroup

  • Daniel Bump
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)


The Borel subgroup B of a (noncompact) Lie group G is a maximal closed and connected solvable subgroup. We will give several applications of the Borel subgroup in this chapter and the next. In this chapter, we will begin with the Iwasawa decomposition, an important decomposition involving the Borel subgroup. We will also show how invariant vectors with respect to the Borel subgroup give a convenient method of decomposing a representation into irreducibles. We will restrict ourselves here to complex analytic groups such as \(\mathrm{GL}(n, \mathbb{C})\) obtained by complexifying a compact Lie group. A more general Iwasawa decomposition will be found later in  Chap. 29.


Borel Subgroup Maximal Compact Subgroup Open Orbit Iwasawa Decomposition Affine Algebraic Group 
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  1. 132.
    D. Mumford, J. Fogarty, and F. Kirwan. Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, third edition, 1994.Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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