1. THE CLASSICAL THEORY: PART I 9 If we choose for our reference point i ∈ H (= [ i 1 ] ∈ P1), then we have the identifi- cations H ∼ SL2(R)/ SO(2) P1∼ SL2(C)/B where (this is a little exercise) SO(2) = a b b a : a2 + b2 = 1 = cos θ − sin θ sin θ cos θ B = a b c d : i(a − d) = −b − c . The Lie algebras are (here k = Q, R or C) sl2(k) = a b c −a : a, b ∈ k so(2) = 0 −a a 0 : a ∈ R b = a −b b −a : a, b ∈ C . Remark. From a Hodge-theoretic perspective the above identifications of the period domain H and its compact dual ˇ are the most convenient. From a group- theoretic perspective, it is frequently more convenient to set ζ = τ − i τ + i , Im τ 0 ⇔ |ζ| 1 and identify H with the unit disc Δ ⊂ C ⊂ P1. When this is done, SL2(R) becomes the other real form SU(1, 1)R = g = a b c d ∈ SL2(C) : t ¯Hg = H of SL2(R), where here H = 1 0 0 −1 . Then H i ↔ 0 ∈ Δ SO(2) ↔ eiθ 0 0 e−iθ B ↔ a 0 b a−1 . Thus, for the Δ model SO(2) becomes a “standard” maximal torus and B is a “standard” Borel subgroup. We now think of H as the parameter space for the family of PHS’s of weight one and with dim V = 2. Over H there is the natural Hodge bundle V1,0 → H with fibres V1,0 τ := V 1,0 τ = line in VC. Under the inclusion H → P1, the Hodge bundle is the restriction of the tautological line bundle OP1(−1). Both V1,0 and OP1(−1) are examples of homogeneous vector bundles.

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