Introduction to semigroup theory by John M. Howie

By John M. Howie

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P. van den Ban 5 The Discrete Series Flensted-Jensen’s Duality The idea of passing to the dual Riemannian form Xd plays an important role in the theory of the discrete series of X. We shall restrict ourselves to giving a short account of some of the main ideas involved. For simplicity of the exposition we make the mild assumption in this section that G is the analytic subgroup with Lie algebra g of a complex Lie group G C with Lie algebra gC . Let G d , K d and H d be the analytic subgroups of G C with Lie algebras gd , kd and hd , respectively.

P. van den Ban Proof. It suffices to prove the equivariance for f and g smooth. In that case the function f, g ξ : x → f (x), g(x) Hξ belongs to C ∞ (P : 1 : 1), which may be identified with the space of smooth sections of the density bundle over P\G. 5). To see that the pairing is equivariant, we note that, for x ∈ G, πξ,λ (x) f, πξ,−λ¯ (x)g ξ equals the pullback of f, g ξ under the diffeomorphism Pg → Pgx. The integration of densities is invariant under diffeomorphisms. 1. 2), see [28], Sect.

By using the method of complexification of the previous subsection, it can be shown that dim b is independent of b, though in general there are several, but finitely many, H -conjugacy classes of Cartan subspaces. The number dim b is called the rank of X. , invariant under the involution θ . 2, q = {(X, −X ) | X ∈ g}. For each Cartan subalgebra j ⊂ g, the space b j := {(X, −X ) | X ∈ j} is a Cartan subspace of q. Moreover, the map j → b j establishes a bijection between the collection of all Cartan subalgebras of g onto the collection of Cartan subspaces of q; it induces a bijection from the finite set of G-conjugacy classes of Cartan subalgebras of g onto the set of H -conjugacy classes of Cartan subspaces of q.

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