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 (τ − 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 XΛ = C/Λ we say that XΛ and XΛ are isomorphic if there is a linear mapping α : C ∼ − C with α(Λ) = Λ . This is equivalent to XΛ and XΛ being biholomorphic as compact Riemann surfaces. Normalizing the lattices as above the condition is τ = aτ + b cτ + 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 XΛ and XΛ 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 XΛ → XΛ . More generally, we may say that XΛ ∼ XΛ if there is a diagram of isogenies XΛ ✼✼ ✼✼✼ ✼✼ ✡ ✡ ✡ ✡ ✡ ✡ XΛ XΛ . 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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2013 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.