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
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.
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