1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. THREE SYMMETRIC SPACES 359

In Figure 2 we show t he finite upper h alf plane graph for q = 3 (the octahedron)

and one of t he 3 different finite upper half plane graphs for q = 5 (drawn on a

dodecahedron).

x (2 ,1)

3

x 5 (2,1)

Figure 2. Some finite upper h a lf plane g raphs for q = 3 and 5. The graph for

q = 5 is shown on a dodecahedron whose edges are indicated by dotted lines

while the edges of the graph Xs(2, 1) are given by solid lines.

In Figures 3 and 4, we illustrate some of the geometry of the symmetric spaces.
Geodesics in the Poincare upper half pla ne are curves which minimize the Poincare
distance ds from Table 2. It is not h ard to see that the points in H on the y-axis
form a geodesic and thus so do the images of this curve under fractional linear
transformations from G = SL(2, IR) since ds is G-invariant. These are semi-circles
and half-lines orthogonal to the real axis. These are the straight lines of a non-
Euclidean geometry.
Geodesics in the tree are paths { Xn In E Z} which are infinite in both di-
rections (i .e., Xn is connected to Xn+l by an edge). We define geodesics in the
finite upper half plane to b e images of the analogue of the y-axis under fractional
linear transformation by elements of G L(2, lF q). The three types of geodesics are
illustrated in Figure 3.
Horocycles are curves orthogonal to the geodesics. The horocycles are pic-
tured in Figure 4. In the Poincare upper half plane t hey are lines parallel to the
x-axis and their images by fractional linear transformation from SL(2,IR). For the
finite upper half pla ne we m ake the an alogous definition of horocycles.
In the tree, horocycles are obtained by fixing a half geodesic or chain C =
[O,oo] = {xn In E Z, n ~ O}. Then the chain connecting a point x in the tree to
infinity along C is called [x, oo]. If x and y are points of T , then [x, oo] n [y, oo] =
[z, oo]. We say x and y are equivalent if d(x, z) = d(y, z). Horocycles with respect
to C are the equivalence classes for t his equivalence relation.
Next we co nsider spherical functions on the symmetric spaces. These are
K -invariant eigenfun ctions of the Laplacian(s) which a re normalized to h ave the