Abstract
In this paper we consider a homogeneous holomorphic line bundle over an elliptic adjoint orbit of a real semisimple Lie group, and set a continuous representation of the Lie group on a certain complex vector subspace of the complex vector space of holomorphic cross-sections of the line bundle. Then, we demonstrate that the representation is irreducible unitary.
Highlights
1.1 A Geometrical Realization of Irreducible Unitary RepresentationsFirst of all, let us recall the definition of irreducible unitary representation
Let us recall the definition of irreducible unitary representation
Let GC be a connected complex semisimple Lie group, let G be a connected closed subgroup of GC such that g is a real form of gC, and let T be a non-zero elliptic element of g
Summary
Let us recall the definition of irreducible unitary representation. Let G be a Lie group, let H = (H, ⟨ · , · ⟩) be a complex Hilbert space, and let ρ : G → GL(H), g → ρ(g), be a group homomorphism, where GL(H) denotes the general linear group of H and it does not matter whether ρ is continuous here. Φmax(u) = 1 for all u ∈ U+ ∩ GQ−, ∫ |⟨ρ(g)φmax, φmax⟩|2dμ(g) = 64π3 < ∞, and the representation space H of ρ corresponds to the complex Hilbert space of square integrable holomorphic 1-forms ω on the open unit disk in C This T is a non-zero elliptic element of g = su(2) and Define holomorphic homomorphisms χ+ : Q− → C∗ and χ− : Q− → C∗ by χ+. Let us set tR := {H ∈ tC | ξ(H) ∈ R for all ξ ∈ Ξ}, fix a fundamental root system {ξa}na=1 ⊂ Ξ, and denote the dual basis of {ξa}na=1 by {Wa}na=1 It follows that iT ∈ tR = spanR{Wa}na=1, itR = t ⊂ k ⊂ g.
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