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
=
+ b
+ 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
+ b
+ d
=

(cτ + d)2
.
Thus s(τ)2 and 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
τ.
Thus we may describe Dn
+
as
f Γ(H, OH) : |fψ(x +
iy)|2yn−2dx
dy .
4A
general reference for this is [Ke1].
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