Index © (Hilbert space direct sum), 5 -, 49 \A\ (A a linear map), 129 M A , 86 MP, 52 , n , 20 ll(c d)llp, ll(c d)llA, 88 |[a]k (slash operator), 34 |r«r (slash operator), 34, 43 A, 68 A+, 68 Ak(N,u), 201 A, 55 A ^ (idelesof E), 311 A 5 , 55 a.e. (almost every), 55 A*, 57 A* , 57 A 1 , 87 Afin, 55 an(S), 16, 199 absolute value, 52 Ad(p), 153, 167 a d p O ( y ) = [X,Y], 152, 162 adeles, 55 adjoint, 131 of a function, 144 representation on 0c 153 admissible representation, 153 antiholomorphic discrete series, 190 archimedean, 52 arctan, 63 B (Borel subgroup), 3, 60 BG (Borel subsets of G), 70 BT, 252 B, 123 BT, 252 B(R) + , 14, 63 Banach space, 131 Beta function B(n,m), 289 Borel measure, 70 Borel subgroup, 60 Bruhat decomposition, 69 C(G) (cont. functions on G), 109 Co(X) (cont. functions vanishing at oo), 109 CC(G(A),UJ), 5 G c (G,x), 134 CC(X), 49 Cartan decomposition p-adic, 208 real, 66, 68, 93 Cartan subalgebra, 162 Casimir element, 167 Cauchy-Schwarz inequality, 131 central character, 138 character value for 7Tk, 302 Chebyshev polynomial, 355 x(£iie2,si,S2) (character of B(R)), 157 X7rk, 190 class group, 306 class number, 306 closed graph theorem, 131 closed linear transformation, 131 commensurable, 20 compact neighborhood, 49 complementary series, 186 conductor of a character, 11, 196 of an order, 303 congruence subgroup, 8 conjugate-linear functional, 131 constant term, 4 countable additivity, 70 critical strip, 124 cusp holomorphic at, 16 of T, 12 vanishing at, 17 cusp form, 18, 19 cuspidal function, 4 cuspidal representation, 5 D (subset of SL 2 (R)), 99 (subset of H), 19, 99 Do, 105, 261 373

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