LECTURE 2. THREE SYl\IIMETRIC SPACES 363The spherical transform of a function fin L^2 (K\G/ K); i.e., a K -bi-invariant
function on G, is obtained by integrating (summing) f times the spherical function
over the symmetric space G / K. These transforms are invertible in all 3 cases.
The horocycle transform of f is obtained by integrating or summing f over
a horocycle in the symmetric sp ace. These horocycle transforms are invertible in 2
out of 3 cases.
The Selberg trace formula involves another subgroup r of G. This subgroupshould be discrete and for ease of discussion h ave compact quotient r \ G / K. For
G = SL(2, IR), this sadly rules out t he modular group r = SL(2, Z). For examples of
su ch r c SL(2, IR), see Svetlana Katok's book [41]. In Table 3 we give a dictionary
of trace formulas for the 3 types of symmetric spaces considered here.
Fundamental domains D for r\ GI K in the various symmetric spaces GI K are
depicted in Figure 5. Interactive fundamental domain drawers can b e found on
Helena Verrill 's webpage (http:hverrill.net).
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Figure 5. Fundamental domains for some discrete subgroups. That for
SL(2, Z)\H is on the left. That for t he 3-regular tree mod the fundamen-
tal group of ](4 is in the center. That for SL(2, lF3)\Hg is on the right.Tessellations of G / K by the action of copies of t hese fundamental domains
are quite beautiful. Instead of drawing tessellations in Figures 6, 7, 8 we give
contour maps of various functions. Tessellations of H by t he modular group
given by contour maps of modular forms are to be found on Frank Faris's web-
site (ricci.scu.edu;-ffaris).
In column 1 of Table 3, note that there are only two types of conjugacy classes
in r. The hyperbolic conjugacy classes b } correspond to 'Y wit h diagonal Jordan
form with distinct diagonal entries t and l/t having t > l. The norm of such an
element is N 'Y = t^2. The centralizer r 7 of 'Y is a cyclic group with generator the
primitive hyperboli c element 'Yo· The numbers logN"fo represent lengths of closed
geodesics in the funda mental domain f\H. The set of these numbers is the " length
sp ectrum". The length spectrum can be viewed as analogous to the set of primes
in Z.
The spectrum of the Laplacian on t he fundamental domain f \ H consists of
a sequence of negative numbers An with IAnl __, oo. And Ao = 0 corresponds
to the constant eigenfunction. We write An = sn(l - sn)· The eigenfunction 'Pn