By Stephen S. Gelbart

This quantity investigates the interaction among the classical concept of automorphic kinds and the trendy idea of representations of adele teams. examining very important contemporary contributions of Jacquet and Langlands, the writer offers new and formerly inaccessible effects, and systematically develops specific results and connections with the classical idea. The underlying subject is the decomposition of the commonplace illustration of the adele workforce of GL(2). an in depth facts of the distinguished hint formulation of Selberg is integrated, with a dialogue of the potential variety of applicability of this formulation. in the course of the paintings the writer emphasizes new examples and difficulties that stay open in the normal theory.

TABLE OF CONTENTS: 1. The Classical idea 2. Automorphic kinds and the Decomposition of L^{2}(PSL(2,R) three. Automorphic varieties as services at the Adele workforce of GL(2) four. The Representations of GL(2) over neighborhood and worldwide Fields five. Cusp types and Representations of the Adele staff of GL(2) 6. Hecke idea for GL(2) 7. the development of a different category of Automorphic varieties eight. Eisenstein sequence and the continual Spectrum nine. The hint formulation for GL(2) 10. Automorphic types on a Quaternion Algebra

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Therefore (see Corollary 4'), all characters in Irr (m) (G) are faithful. Put Irr(m)(r) = Irr(r) n Irr(m}(G) for a character r of G. (i) Assume that Irr(n} (G) contains a faithful character ¢. 7, there is a minimal positive integer r such that Irr(m)(¢r) =/:- 0 (obviously, r > 1). Let x E Irr(m)(¢r). Then x E Irr(¢), where{) E Irr(q,r- 1). By assumption, {)(1) = n. Now 0 < (¢{),x} = ({),¢x}, so setting 1Irr(m)(¢x)I = a, IIrr(n)(¢x)I = b, we obtain mn = (¢x)(l) = am+ bn (note that ¢x has no linear components since ¢(1) = n =/:- m = x(l)).

By the above, D ::; G(p') = nxeirri (G), Pl'x(l) ker x, a contradiction, since D is not p-nilpotent and G(p') is p-nilpotent (Theorem 22). Thus, ker'I? is solvable, and (iii) is proved. (iv) If 19 is faithful, then G/kerx 1 has an ordered Sylow tower for some x 1 E Irrud (G). Indeed, by (ii) there exists a nonfaithful x 1 such that Iker X1 I ~ Iker xi for all x E Irrui)(G). By assumption and (ii), G/kerx 1 has no irreducible character of degreed. Therefore, cd G/kerx1 ~ {1,/i, ... ,fn} is a chain with respect to divisibility.

By Theorem ll(a), N is p-nilpotent, proving (a). ) Let x E X(m). If >. (1) (Clifford and Lemma 36(a)); therefore,>. is linear by (a). Hence N' $ kerx. 7, x(l) divides m. Thus X(m) = {x E Irr1(G) Ix(l) divides m}. 7, and so r E X(m), proving (b). Let¢ E Irr1(N). Take x E Irr(¢0 ). Then x E X(p) by (b) and (x(l),m) = 1. Therefore, XN E Irr(N) (Clifford). Thus, XN = ¢. D IV Conjecture. Disconnected S-groups are solvable. For a disconnected S-group G, define 7r = LJ;~eX(m) 7r(X(l)) and R = Q1r(G); then G/ R is a 'Tr-group and 7r ~ 7r(m) (Lemma 37).