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Extra info for 2-complete subgroups of a conjugately biprimitively finite group with the primary minimal condition

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The collection of facets, also called the Coxeter complex of , is denoted by P( ). It is equipped with a natural action by the Weyl group W of . If X ∈ aq , we denote its class by CX ∈ P( ). For C ∈ P( ), we put (C) = {α ∈ C = {α ∈ | α > 0 on C} , | α = 0 on C} . Then = − (C) ∪ C ∪ (C) (disjoint union). 1) Let S be the intersection of the root hyperplanes ker α, α ∈ C . Then, clearly, the set D = {X ∈ S | ∀α∈ (C) α(X) > 0} contains C, hence is a nonempty open subset of S and therefore spans S. 1) it follows that D ∈ P( ).

Later, in [34] a proof based on a generalization of the mentioned functional equation was given. The Plancherel Theorem for a Reductive Symmetric Space 41 The meromorphic continuation is absolutely crucial for the development of the theory, since the set ia∗Pq (where the π P,ξ,λ are unitary) is not contained in the region Reλ + ρ P , α < 0 (α ∈ (P)). By meromorphic continuation one still has that j (P : ξ : λ : η)(v) = ηv , showing that j (P : ξ : λ) = j (P : ξ : λ : ·) defines an injective homomorphism V (P, ξ ) ֒→ C −∞ (P : ξ : λ) H , for regular λ.

4) 30 E. P. van den Ban where dist(x, e) ¯ denotes the Riemannian distance in Xd between x and the origin d e¯ = eK . Note that dist(x, e) ¯ = |X | for X ∈ pd and x = exp X K d . 4). 2 Let Re( ) be dominant with respect to the roots of apd in nd . Then the Poisson transform P ≃ (a) is a topological linear isomorphism B(L ) → E (Xd ), and ≃ (b) restricts to a topological linear isomorphism D′ (L ) → E ∗ (Xd ). For Xd of rank one, part (a) of the theorem is due to Helgason [61]. In [62] he conjectured part (a) to be true in general, and established it on the level of K d -finite functions.

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2-complete subgroups of a conjugately biprimitively finite group with the primary minimal condition by Schlepkin A.K.


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