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 ••• ^.
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) =
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