4 1. EIGENFUNCTIONS OF THE HYPERBOLIC LAPLACE OPERATOR

d[u, v] =

1

2i

(u Δv − v Δu) dμ , (1.10c)

{v, u} = −{u, v} . (1.10d)

So [u, v] is a closed 1-form on U if u, v ∈ Es(U) for an open U ⊂ H.

The bracket with qs gives for elements of Es a substitute for Cauchy’s theorem:

Theorem 1.1. Let C be a piecewise smooth simple closed curve in H and u

an element of Es(U), where U ⊂ H is some open set containing C and its interior.

Then for w ∈ H C we have

(1.11)

C

[u, qs( · , w)] =

πi u(w) if w is inside C,

0 if w is outside C,

where the curve C is traversed in the positive direction.

See Theorem 2.1 in [4].

2. Principal series

All the coeﬃcient modules used in the cohomology groups mentioned in the

introduction are spaces of vectors in the principal series representation associated

to the spectral parameter s. The standard realizations of the principal series rep-

resentation use spaces of functions on the boundary ∂H of the hyperbolic plane.

With the Poisson transform we can also use a realization in Es.

We write Vs to denote “the” principal series representation when we do not

want to specify precisely the space under consideration. Spaces

Vs∞

and

Vsω

of

smooth and analytic vectors are identified with the appropriate superscript.

In [4] we treat the material in this section in more depth. In particular, we

study the various models more systematically. Each of the models of Vs has its

advantages and disadvantages.

2.1. Models of the principal series on the boundary of the hyperbolic

plane. We list some standard models of the principal series.

• Line model. In the introduction we already mentioned the well known model

of Vs, consisting of functions on R with the transformation behavior

(2.1) ϕ

2s

a

c

b

d

(x) = |cx +

d|−2sϕ

ax + b

cx + d

under

a

c

b

d

∈ G. To get a sensible result at x = −

d

c

, we need to require that ϕ

behaves well as |x| → ∞. By Vs∞, the space of smooth vectors in Vs we denote the

space of ϕ ∈ C∞(R) that have an expansion

(2.2) ϕ(t) ∼

|t|−2s

∞

n=0

cn

t−n

as |t| → ∞. Similarly, the space

Vsω

of analytic vectors consists of the ϕ ∈

Cω(R)

(real-analytic functions on R) for which the series appearing on the right-hand side

of (2.2) converges to ϕ(x) for |x| ≥ x0 for some x0. Analogously, we define

Vsp,

p ∈ N, as the space of ϕ ∈

Cp(R)

satisfying (2.2) with the asymptotic expansion

replaced by a Taylor expansion of order p.

We call this the line model of Vs. It is well known and has a simple transforma-

tion formula (2.1) that reminds us of the transformation behavior of holomorphic