1549380232-Automorphic_Forms_and_Applications__Sarnak_

362 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS

II Finite Upper Half Plane II

space= Hq =

{z=x+v;5y I x,yElFq,y#O}

where 0 # a^2 , for any a E lFq

group G = GL(2,1Fq)

{(~ ~ ) I ad - be # 0}

group action gz = ~;1~, z E Hq

origin= VO

subgroup K fixing origin =

K= { ( ~ ~ ) E G} ~1Fq(V<5)*

H~G/K

pseudo-distance d(z, w) = 1 1;;,t;~:~,

Im(x + yv;5) = y, N z = (x^2 - y^2 o)

Xa = Xq(O, a), graph - vertices z E Hq

edge between z and w if d(z,w) =a

6. = Aa - ( q + 1) I, if a # 0 or 46

Aa = adjacency operator for Xa

spherical function

1 ~K

h'lf(z) = TKT 2=I< X'lf (k z), 7r E G ,

i.e., 7r occurs in IndX.l, d'lf = deg 7r

spherical transform

of f: K\G/K ----7 C

f(7r) = 2=zEHa J( z)h'lf (z)

inversion

f(z) = 1b1 2=7rEGK d'lf f(7r)h1f (z )

horocycle transform

F(y) = 2=xEFq f(x + yv;5)

not invertible

Table 2. Part 2. Basic geometry of the finite upper half plane.

For the finite upper half plane one can build up spherical functions from char-
acters of inequivalent irreducible unitary representations 7r of G = GL(2, lFq) oc-
curring in the left regular representation of G on functions in L^2 ( G / K). This is the
same thing as saying that 7r occurs in the induced representation IndX.1. See Terras
[82], Ch. 16, for the definition of induced representation. We use fJK to denote the
set of 7r occurring in I ndX. l. Then the spherical function h'lf corresponding to 7r is
obtained by averaging the character x'lf = Trace(7r) over K as in formula (4 .7) in
Lecture 1. See Terras [82] for more information on these finite spherical functions
and the representations of GL(2,1Fq)·