1. THE CLASSICAL THEORY: PART I 9
If we choose for our reference point i ∈ H (= [ i
1
] ∈
P1),
then we have the identifi-
cations
H
∼
= SL2(R)/ SO(2)
P1∼
=
SL2(C)/B
where (this is a little exercise)
SO(2) =
a b
b a
:
a2
+
b2
= 1 =
cos θ − sin θ
sin θ cos θ
B =
a b
c d
: i(a − d) = −b − c .
The Lie algebras are (here k = Q, R or C)
sl2(k) =
a b
c −a
: a, b ∈ k
so(2) =
0 −a
a 0
: a ∈ R
b =
a −b
b −a
: a, b ∈ C .
Remark. From a Hodge-theoretic perspective the above identifications of the
period domain H and its compact dual
ˇ
H are the most convenient. From a group-
theoretic perspective, it is frequently more convenient to set
ζ =
τ − i
τ + i
, Im τ 0 ⇔ |ζ| 1
and identify H with the unit disc Δ ⊂ C ⊂
P1.
When this is done, SL2(R) becomes
the other real form
SU(1, 1)R = g =
a b
c d
∈ SL2(C) :
t¯Hg
g = H
of SL2(R), where here H =
1 0
0 −1
. Then
H i ↔ 0 ∈ Δ
SO(2) ↔
eiθ 0
0 e−iθ
B ↔
a 0
b a−1
.
Thus, for the Δ model SO(2) becomes a “standard” maximal torus and B is a
“standard” Borel subgroup.
We now think of H as the parameter space for the family of PHS’s of weight
one and with dim V = 2. Over H there is the natural Hodge bundle
V1,0
→ H
with fibres
Vτ,0 1
:=
Vτ1,0
= line in VC.
Under the inclusion H →
P1,
the Hodge bundle is the restriction of the tautological
line bundle OP1 (−1). Both
V1,0
and OP1 (−1) are examples of homogeneous vector
bundles.
