1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. THREE SYMMETRIC SPACES 361

II Poincare Upper H alf Plane I (q + 1)-regular Tree

space= H = space= T =

{z = x + iy Ix, y E JR, y > O} (q + 1)-regular connected graph

with no circuits

group G = SL(2, JR) group G =

{ ( ~ ~ ) I ad - be = 1} graph a utomorphisms of T

rotations, inversions, shifts

group action gz = ~;1~, z EH group action gx

origin =i origin = any point 0

subgroup K fixing origin subgroup K fixing origin

80(2, JR)= {g E Gjtgg =I} rotations about 0

H~G/K T~G/K

arc length ds^2 = dx'+.,du' y graph distance=# edges in path

joining 2 points x, y E T

Laplacian 6. = y^2 ( a~2 + 8 ~2 ) 6. =A - (q + l)J,

A=adjacency operator for T

spherical function spherical function

h 8 (z) = JK(Im(ka))8dk, Ps(x) = q-sd(x,O), with c(s) as in (2.1)

6.hs = s(s - l)hs hs(x) = c(s)p 8 (x) + c(l - s)P1-s(x)

spherical transform spherical transform

of f: K\G/ K-----+ <C of f : K\ G / K -----+ <C

J(s) = JH f(z)hs(z) d~gu J(s) = 2=xET f(x)hs(x)

inversion f(z) = inversion f(x) =

1 ~ 1 2-.fo ~^1. y'4q-t2

4 ,,. JRf(2 +it)h!+it(z)ttanh7rt dt J J(^2 +it)h!+it(x)(q+l)Lt2dt

-2-.fo

horocycle transform horocycle transform

ry = horocycle on T

F(y) = y-^1 /^2 JR f(x + iy)dx F(ry) = l::xE[J f(x)

invertible invertible

Table 2. Part 1. Basic geometry of the Poincare upper half plane and the

( q+ 1 )-regular tree.

A similar method works for the tree except that the analogue of the power
function p 8 (x) = q-sd(x,O) is not quite an eigenfunction for the adjacency operator
on T with eigenvalue>.= q^8 + q^1 -s. This is fixed by writing h 8 (x) = c(s)p 8 (x) +
c(l - s)P1-s(x), with

(2.1)

qs-1 _ ql-s

c(s) - if q2s-1 -f. 1.

• (q + l)(qs-1 _ q-s)'

Take limits when q^2 s-l = 1. More information on spherical functions on trees can
be found in references [15], [17], [29], [82], [87].