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.