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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