BEGINNINGS OF REPRESENTATION THEORY 13 The proof is by tracing through the isomorphism. To see why it should be true we make the following observation: Under the action of ( a b c d ) ∈ SL2(R), s(τ) transforms to a b c d τ 1 = aτ + b cτ + d = (cτ + d) aτ+b cτ+d 1 i.e., s(τ) transforms by (cτ + d)−1. On the other hand, using ad − bc = 1 we find that d aτ + b cτ + d = dτ (cτ + d)2 . Thus s(τ)2 and dτ transform the same way under SL2(R), and consequently their ratio is a constant function on H. Beginnings of representation theory3 In these lectures we shall be primarily concerned with infinite dimensional rep- resentations of real, semi-simple Lie groups and with finite dimensional representa- tions of reductive Q-algebraic groups. Leaving aside some matters of terminology and definitions for the moment we shall briefly describe the basic examples of the former in the present framework. Denote by Γ(H, Vn,0) the space of global holomorphic sections over H of the nth tensor power of the Hodge bundle, and by dμ(τ) the SL2(R) invariant area form dx ∧ dy/y2 on H. Definition. For n 2 we set Dn + = ψ ∈ Γ(H, Vn,0) : H ψ(τ) 2 dμ(τ) ∞ . There is an obvious natural action of SL2(R) on Γ(H, Vn,0) that preserves the pointwise norms, and it is a basic result [Kn2] that the map SL2(R) → Aut(Dn + ) gives an irreducible, unitary representation of SL2(R). As noted above there is a holomorphic trivialization of V1,0 → H given by the non-zero section σ(τ) = τ 1 . Then using the definition of the Hodge norm and ignoring the factor of 2, σ(τ) 2 = y. Writing ψ(τ) = fψ(τ)σ(τ) we have H ψ(τ) 2 dμ(τ) = i 2 |fψ(τ)|2(Im τ)n−2dτ ∧ d¯ Thus we may describe Dn + as f ∈ Γ(H, OH) : |fψ(x + iy)|2yn−2dx ∧ dy ∞ . 4 A general reference for this is [Ke1].

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