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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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Over the past 30 years the speculation of finite teams has built dramatically. Our realizing of finite easy teams has been greater through their category. many questions on arbitrary teams might be lowered to comparable questions about uncomplicated teams and functions of the speculation are starting to look in different branches of arithmetic.

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The normalized orbital integral F(6, fdg) = A(6) J fiaSg-1)^ (g e G/T), where A(5) = < 5- 1 / 2 (5)|det(l - Ad((5))|Lie JV| = a—bb—ca—c a b c depends only on the image of 6 = diag(a, b, c) in T/T(R) ~ X , ( T ) = Hom(G m , T) when fdg is spherical. - 1 . Hence F(6Jdg)=61/2(S) [ fK(5n)dn, JN where fK(g)= [ /(k^g^dk. X*(T) ~ { ( n i , n 2 , n 3 ) ; n* G Z}/{(n,n,n); n G Z}. )| denotes the volume of T(R) =T(1K with respect to dt. The map fdg — t > / is an isomorphism from the algebra MQ to the algebra C[T] W of finite Laurent series in t G T which are invariant under the action of the Weyl group W of f in G.

GT = where ZG(So-) = {geG;g6o-(g-1) = 8}, g^g)}, I. Functoriality and norms 26 factors through 1 __+ D(S)^H1(F,ZG(Sa)) — ff^G), where the double coset space D(S) = G\A(6)/ZG(SCT)(F) parametrizes the cr-conjugacy classes within the stable cr-conjugacy class of 6. The definitions introduced above will be used with G = PGL(3) and the (involution) outer automorphism a{g) = Jtg~lJ, and also with H = Ho = SL(2), H i = PGL(2) = SO(3) and the trivial a. If 7 G H, Z H (7) denotes the centralizer of 7 in H.

2) bijects the stable u-regular er-conjugacy classes in G with the regular conjugacy classes in H\ = SO(3,F). In each stable cr-conjugacy class of elements 6 such that 5a(S) has distinct eigenvalues there are two

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