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.
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