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.