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A Crash Course on Kleinian Groups by American Mathematical Society

By American Mathematical Society

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Let <5 be an element of G. The set of eigenvalues of 5a(5) is of the form {A, 1, A - 1 } . Indeed, if A is an eigenvalue of 5 a (6) then there is an eigenvector v with t(5cr(6))v = Xv. Hence X-1v = \8a{5))-1v, and A_1(

Functoriality and norms 34 is A(6a)2 = |det[l - Ad(6)a]\Lie{G/Ta)\. The twisted Weyl integration formula is then (put Sa for (ae)i) *(6a*)2da [ [f(9)dg = lj2[ JG A p JTE/NZ(E) Let us compute A(da)2 ( Xl HgSaa(g)-1)^. >G/Z (Ta) JO G explicitly. We may assume S is diag(a, b, c). X2 X 3 \ xi x5 x6 J modulo center. Thus we assume that X7 Xg XQ J x5 = 0 to fix representatives. Note that LieZc^cr) = {diag(x,0, — x)}, since -0~X = J XJ = / x9 -x6 x3 ( —x8 x 5 —X2 X7 / x - Ad^yx X1+X9 Xl X2-§X6 X4 — £ x s \ ( 1 + §)X7 — X4 2X5 (l+f)x3' X6 —^X2 Xg-§X4 Xl+Xg Recalling that £5 = 0, and noting that in L i e G / Z c ^ c ) the x\ + £9 is a single variable (alternatively, in X we could replace xg by zero and x\ by xi + xg), we conclude that A(6a)2 = '-!

Hence (3 = 1 and TT is trivial, as asserted. 1 Orthogonality relations. For any conjugacy class functions Xi x' on the elliptic set He of H put (X,x')e= = ^[WiToT'lTo]-1 [ x(h)x'(h)dh f A0(7)2x(7)x'(7)d7. /. Functoriality and norms 42 The sum ranges over a set of representatives T0 for the conjugacy classes of elliptic tori of H over F. [W(T0)] is the cardinality of the Weyl group of T0 (1 or 2). As usual, |T 0 | denotes the volume of T 0 . We write 7 ~ 7' if 7, y are conjugate. The measure dh on He/ ~ is defined by the last displayed equality.

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