SELBERG TRACE FORMULA 3

component for each cusp. For simplicity, let us also temporarily assume there

2

feC

is just one cusp at infinity. Then L . =

E(-,^+ir)dr.1

The spectrum of A thus consists of the continuum (T-,OD)together with a

discrete set of eigenvalues {0 = AQ X^ A2 ••• fn}. Following a standard

1 2 1

notation, we will write: X. = s.(l-s.) = - r + r., with s. = ^ + ir.. The

J j J ^ j J z J

eigenvalue \~ = 0 obviously corresponds to the trivial representation in

2 1

L (r\G). Eigenvalues A. G (0, j) correspond to complementary series

2

irreducibles in L (r\G), and hence are called complementary series

eigenvalues. For such eigenvalues, r. is pure imaginary: r^ = i and in the

complementary series r. = it. with t- G (0, 7z). Let M be the number (possibly

J J J

zero) of complementary series eigenvalues: so t^ = ^ t^ t2 ••• t« ^.

Now consider the geodesic flow G on T\G. As is well-known, G is given

f

e1/2

0 1

by right translation by at = _./« • ^y a closed geodesic 7 of T\h

one means both a closed orbit of G and its projection to T\h (it should be

clear from context which is meant). Each closed geodesic 7 corresponds to a

conjugacy class 7 of hyperbolic elements of V (diagonalizable over K). To

simplify notation, we will usually also confuse 7, 7 and elements 7 in 7,

leaving it to the context to make the meaning clear. Ve will write L for the

length of the closed geodesic 7. Equivalently, elements of 7 are conjugate to

if :-w.]*-v-

Each closed geodesic 7 determines a period orbit measure / * on C,(r\G):

p (f) = f. Here, C, denotes the bounded continuous functions, and f is

T

r

L

short for ' f(7(t)) dt, 7(t) being the natural parametrization of the orbit

Jo

7. Equidistribution theory is concerned with the weak limits of the \k as

L —» OD. To study them, it is very convenient to form the sums fp(f,T) =