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10 1. THE CLASSICAL THEORY: PART I In general, given • a homogeneous space Y = A/B where A is a Lie group and B ⊂ A is a closed subgroup, and • a linear representation r : B → Aut E where E is a complex vector space, there is an associated homogeneous vector bundle E := A ×B E Y = A/B where A×B E is the trivial vector bundle A×E factored by the equivalence relation (a, e) ∼ (ab, r(b−1)e) where a ∈ A, e ∈ E, b ∈ B. The group A acts on E by a·(a , e) = (aa , e) and there is an A-equivariant action on E → Y . There is an evident notion of a morphism of homogeneous vector bundles then E → Y is trivial as a homogeneous vector bundle if, and only if, r : B → Aut(E) is the restriction to B of a representation of A. Example. Let τ0 ∈ H ⊂ P1 be the reference point. For the standard linear representation of SL2(C) on VC, the Borel subgroup B is the stability group of the flag (0) ⊂ Vτ1,0 0 ⊂ VC. It follows that there is over P1 an exact sequence of SL2(C)-homogeneous vector bundles 0 → OP1(−1) → V → OP1(1) → 0 where V = P1 × VC with g ∈ SL2(C) acting on V by g · ([z],v) = ([gz],gv). The restriction to H of this sequence is an exact sequence of SL2(R)-homogeneous bun- dles 0 → V1,0 → V → V0,1 → 0. The bundle V1,0 is given by the representation cos θ − sin θ sin θ cos θ → eiθ of SO(2). Using the form Q the quotient bundle V/V1,0 := V0,1 is identified with the dual bundle V1,0∗. The canonical line bundle is ωP1 ∼ OP1(−2). Thus ωH ∼ (V1,0)⊗2. Proof. For the Grassmannian Y = Gr(n, E) of n-planes P in a vector space E there is the standard GL(E)-equivariant isomorphism TP Y ∼ Hom(P, E/P ). In the case above where E = C2 and z = [ z0 z1 ] ∈ P1 we have TzP1 ∼ Vz1,0∗ ⊗ VC/Vz1,0