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A novel characterization of the Iwasawa decomposition . . .



This appendix is about (essential) uniqueness of the Iwasawa (or horospherical) decomposition G = KANof a semisimple Lie group G. This means: Theorem 0.1. Assume that G is a connected Lie group with simple Lie algebra g. Assume that G = KL for some closed subgroups K, L < G with K ∩ L discrete. Then up to order, the Lie algebra k of K is maximally compact, and the Lie algebra l of L is isomorphic to a + n, the Lie algebra of AN. 1. General facts on decompositions of Lie groups For a group G, a subgroup H < G and an element g ∈ G we define H g = gHg −1. Lemma 1.1. Let G be a group and H, L < G subgroups. Then the following statements are equivalent: (i) G = HL and H ∩ L = {1}. (ii) G = HL g and H ∩ L g = {1} for all g ∈ G. Proof. Clearly, we only have to show that (ii) ⇒ (i). Suppose that G = HL with H ∩ L = {1}. Then we can write g ∈ G as g = hl for some h ∈ H and l ∈ L. Observe that L g = L h and so Moreover we recor

Topics: Key words and phrases. Semisimple Lie groups, Iwasawa decomposition
Year: 2007
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