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Symmetry And Group

Automorphic Representations of Low Rank Groups by Yuval Z Flicker

By Yuval Z Flicker

The realm of automorphic representations is a ordinary continuation of reviews in quantity idea and modular types. A guideline is a reciprocity legislations pertaining to the limitless dimensional automorphic representations with finite dimensional Galois representations. uncomplicated family members at the Galois facet mirror deep kinfolk at the automorphic facet, referred to as "liftings". This e-book concentrates on preliminary examples: the symmetric sq. lifting from SL(2) to PGL(3), reflecting the three-d illustration of PGL(2) in SL(3); and basechange from the unitary crew U(3, E/F) to GL(3, E), [E : F] = 2. The publication develops the means of comparability of twisted and stabilized hint formulae and considers the "Fundamental Lemma" on orbital integrals of round capabilities. comparability of hint formulae is simplified utilizing "regular" capabilities and the "lifting" is acknowledged and proved via personality kinfolk. this enables an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), an evidence of multiplicity one theorem and tension theorem for SL(2) and for U(3), a decision of the self-contragredient representations of PGL(3) and people on GL(3, E) mounted by means of transpose-inverse-bar. specifically, the multiplicity one theorem is new and up to date. There are purposes to building of Galois representations by way of specific decomposition of the cohomology of Shimura types of U(3) utilizing Deligne's (proven) conjecture at the mounted element formulation.

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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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