BEGINNINGS OF REPRESENTATION THEORY 13

The proof is by tracing through the isomorphism. To see why should be true

we make the following observation: Under the action of

(

a b

c d

)it

∈ SL2(R), s(τ)

transforms to

a b

c d

τ

1

=

aτ + b

cτ + d

= (cτ + d)

aτ+b

cτ+d

1

;

i.e., s(τ) transforms by (cτ + d)−1. On the other hand, using ad − bc = 1 we find

that

d

aτ + b

cτ + d

=

dτ

(cτ + d)2

.

Thus s(τ)2 and dτ transform the same way under SL2(R), and consequently their

ratio is a constant function on H.

Beginnings of representation theory3

In these lectures we shall be primarily concerned with infinite dimensional rep-

resentations of real, semi-simple Lie groups and with finite dimensional representa-

tions of reductive Q-algebraic groups. Leaving aside some matters of terminology

and definitions for the moment we shall briefly describe the basic examples of the

former in the present framework.

Denote by Γ(H,

Vn,0)

the space of global holomorphic sections over H of the

nth

tensor power of the Hodge bundle, and by dμ(τ) the SL2(R) invariant area form

dx ∧

dy/y2

on H.

Definition. For n 2 we set

Dn

+

= ψ ∈ Γ(H,

Vn,0)

:

H

ψ(τ)

2dμ(τ)

∞ .

There is an obvious natural action of SL2(R) on Γ(H,

Vn,0)

that preserves the

pointwise norms, and it is a basic result [Kn2] that the map

SL2(R) → Aut(Dn

+

)

gives an irreducible, unitary representation of SL2(R).

As noted above there is a holomorphic trivialization of V1,0 → H given by the

non-zero section

σ(τ) =

τ

1

.

Then using the definition of the Hodge norm and ignoring the factor of 2,

σ(τ)

2

= y.

Writing

ψ(τ) = fψ(τ)σ(τ)

we have

H

ψ(τ)

2dμ(τ)

=

i

2

|fψ(τ)|2(Im τ)n−2dτ

∧ d¯ τ.

Thus we may describe Dn

+

as

f ∈ Γ(H, OH) : |fψ(x +

iy)|2yn−2dx

∧ dy ∞ .

4A

general reference for this is [Ke1].