ABSTRACT
We first recall the Kostant linear convexity theorem [Theorem 8.2]Ko73. Let G be a noncompact connected semisimple Lie group with Lie algebra g $ \mathfrak g $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_1.tif"/> . Let g = k ⊕ p $ \mathfrak g = \mathfrak k \oplus \mathfrak p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_2.tif"/> be a fixed Cartan decomposition of g $ \mathfrak g $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_3.tif"/> . Let K be the connected subgroup of G with Lie algebra k $ \mathfrak k $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_4.tif"/> . Note that p $ \mathfrak p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_5.tif"/> is the orthogonal complement of k $ \mathfrak k $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_6.tif"/> in g $ \mathfrak g $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_7.tif"/> with respect to the Killing form. The Killing form is negative definite on k $ \mathfrak k $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_8.tif"/> and positive definite on p $ \mathfrak p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_9.tif"/> and let a ⊂ p $ \mathfrak a \subset \mathfrak p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_10.tif"/> be a maximal abelian subspace in p $ \mathfrak p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_11.tif"/> . Let π : p → a $ \pi : \mathfrak p \rightarrow \mathfrak a $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_12.tif"/> be the orthogonal projection of p $ \mathfrak p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_13.tif"/> on a $ \mathfrak a $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math7_14.tif"/> .
