BEGINNINGS OF REPRESENTATION THEORY 15 for these lectures, a module obtained by taking finite parts of completed unitary SL2(R)-modules. For the Dn + the modules are formal power series ψ = k 0 ak(τ i)kdτ ⊗n/2 . We think of these as associated to GR-modules arising from the open orbit H. The Lie algebra sl2(C), thought of as vector fields on P1, operates on ψ above by the Lie derivative, and SO(2, C) operates by linear fractional transformations. Associated to the closed SO(2, C) orbit i are formal Laurent series γ = l 1 bl i)l ∂τ ⊗n/2 dz. This is also a (sl2(C), SO(2, C))-module. The pairing between SO(2, C)-finite vec- tors, i.e., finite power and Laurent series, is ψ, γ = Resi(ψ, γ). There are also representations associated to the closed SL2(R) orbit and open SO(2, C) orbit that are in duality (cf. [Sch3]). There is a similar picture if one takes the other real form SU(1, 1)R of SL2(C). It is a nice exercise to work out the orbit structure and duality in this case. We shall revisit Matsuki duality in this case, but set in a general context, in Lecture 2. Why we work over Q. Setting = C/Λ we say that and are isomorphic if there is a linear mapping α : C C with α(Λ) = Λ . This is equivalent to and being biholomorphic as compact Riemann surfaces. Normalizing the lattices as above the condition is τ = + b + c , a b c d SL2(Z). Thus the equivalence classes of compact Riemann surfaces of genus one is identified with the quotient space SL2(Z)\H. For many purposes a weaker notion of equivalence is more useful. We say that and are isogeneous if the condition α(Λ) = Λ is replaced by α(Λ) Λ . Then Λ /α(Λ) is a finite group and there is an unramified covering map . More generally, we may say that if there is a diagram of isogenies ✼✼ ✼✼✼ ✼✼ . Identifying each of the universal covers with the same C, we have Λ Λ , Λ Λ and then Λ Q = Λ Q = Λ Q. The converse is true, which suggests one reason for working over Q.
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