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Extra resources for Algebra and analysis: problems and solutions
It is clear from the above discussions that this horospherical decomposition depends on the basepoint x0 = K . In fact, the subgroups A P , M P and the Langlands decomposition of P all depend on the basepoint x0 = K . In the following, we will identify N P × A P × X P with X , and for a point (n, a, z) ∈ N P × A P × X P , the image ν0 (n, a, z) ∈ X will be denoted by either (n, a, z) or naz for simplicity, unless a different basepoint is used. The K -conjugation on parabolic subgroups transports the Langlands decomposition, and the K -action on X preserves the horospherical decomposition.
Brieﬂy, it follows from (1) M Q = N P A P M P (K ∩ M Q ), P = N P A P M P ; and (2) N P ⊂ N P , A P ⊂ A P , and M P = M P . 12 Summary and comments. In this section, we have deﬁned parabolic subgroups by ﬁrst introducing the standard ones associated with a choice of a positive chamber a+ of a maximal abelian subalgebra a. Two important concepts are the Langlands decomposition of parabolic subgroups and the induced horospherical decomposition of X . To understand these results, it is helpful to keep the example of G = SL(n, R) in mind.
In the Tits metric, for any H ∈ P and g ∈ G − P, g H ∈ g P , and the distance d(H, g H ) is independent of g. On the other hand, it is reasonable to expect that if g is closer to the identity element, then g P should be closer to P . This is important for applications to compactiﬁcations of the symmetric space X . Brieﬂy, it will be seen that the boundary of compactiﬁcations of X is often a cell complex related to or parametrized by the Tits building (G), and a topology is needed to measure the closeness of these boundary cells (or boundary components).
Algebra and analysis: problems and solutions by Lefort G.