Abelian Group Theory. Proc. conf. Honolulu, 1983 by R. Göbel, L. Lady, A. Mader

By R. Göbel, L. Lady, A. Mader

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Additional resources for Abelian Group Theory. Proc. conf. Honolulu, 1983

Example text

Given k 6 K, ka = sa • u, put a(h) = sa(h)s~l. This is an automorphism of order £ on ZK((SCTY). Suppose that the first l 1 cohomology set H ((a),ZK((sa) )), of the group (a) generated by a, with coefficients in the centralizer Zji((saY) of (sa)e in K, injects in H\(a),ZG((saf)). II. 1 Fundamental lemma Then, any x G G such that k'a = lnt(x)(ka) KZG(sa). PROOF. 51 is in Ka, must lie in Put k'a — s'a • v!. Then s'a = \im{k'a)qTni = Int(a;)lim(fca) 9mi =Int(a:)(s

Put air(g) = 7r(ag) (g in G). Then CT7r is an admissible irreducible representation of G on V. We say that TT is a-invariant if w is equivalent to a7r. In this case there is an invertible operator A: V —> V with ir(o-g) = Ai:{g)A~1 (g in G). Since IT is irreducible and A2 intertwines ir with itself, Schur's lemma ([BZ1]) implies that A2 is a scalar. Multiplying A by 1/vA2, we assume that A2 — 1. Then J4 is unique up to a sign. We put n(cr) = A, and define the operator n(fdg x a) = ir(fdga) = w(fdg)ir(cr) to be the map v >-> / f(g)n(g)Avdg.

Recall: p ^ 2. II. 1 Fundamental lemma For A ^ ± 1 we have $st(5o-,f°dg) A(5a)$™(5o-J°dg) = * ( # ! * , / f d / n ) . THEOREM. 49 = $st(N6,f$dh0), and This is the fundamental lemma for the symmetric square lifting from SL(2) to PGL(3) and the unit element of the Hecke algebra. A proof of the first assertion — due to Langlands, based on counting vertices on the Bruhat-Tits building associated with PGL(3) — is recorded in the paper [F2;II], §4, but it is conceptually difficult, hence not used in this work.

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