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Symmetry And Group

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

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Extra info for Abelian Group Theory. Proc. conf. Honolulu, 1983

Example text

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