By Yuval Z Flicker

ISBN-10: 9812568034

ISBN-13: 9789812568038

The realm of automorphic representations is a usual continuation of reviews in quantity concept and modular kinds. A tenet is a reciprocity legislation concerning the countless dimensional automorphic representations with finite dimensional Galois representations. uncomplicated relatives at the Galois facet mirror deep relatives at the automorphic part, known as "liftings". This ebook concentrates on preliminary examples: the symmetric sq. lifting from SL(2) to PGL(3), reflecting the three-dimensional illustration of PGL(2) in SL(3); and basechange from the unitary staff 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 kin. 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 explanation of multiplicity one theorem and pressure theorem for SL(2) and for U(3), a decision of the self-contragredient representations of PGL(3) and people on GL(3, E) fastened via transpose-inverse-bar. specifically, the multiplicity one theorem is new and up to date. There are functions to building of Galois representations via specific decomposition of the cohomology of Shimura types of U(3) utilizing Deligne's (proven) conjecture at the mounted aspect formulation.

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We have that (x£,x£)e is 1 ifn is the a-invariant berg representation. Stein- P R O O F . 9. 1) for sp follows from the orthogonality relation for the trivial representation of the group of elements of reduced norm 1 in the quaternion division algebra, and the correspondence of [JL]. • To deal with -K which are not cuspidal or Steinberg, we record a special case of a twisted analogue of [K2], Theorem G. The proof in the twisted case, for an arbitrary reductive not necessarily connected p-adic group, follows closely that of [K2], and will not be given here.

If NS = 7 is regular in H then Za(Sa) ~ Z H ( T ) - Indeed, if g~15a(g) = S then g~15a(S)g — Sa(S); if 5 = (ae)i then g — b\ and b~1ab = a, since Sa(S) = hi, h = Hence b~1aewtb~1we = a, namely b~xab = a, deto so that det 6 = 1 . It is clear that Zn{^) = Zn{a). -j^a2. The norm map can be extended to classes of 5 in G which are not aregular. This is done next. 5 I d e n t i t y . We now deal with the (two) cases where all eigenvalues of Sa(S) are 1. If Sa(S) = 1 we write NS = 1 and NiS = 1. Then SJ = \8J) is symmetric, any two symmetric matrices are equivalent over F, hence for each 5' with S'a(S') = 1 there is S in G with SJ = SS'J^, so that 5 = SS'a(S~1), and the S with Sa(S) = 1 form a single stable er-conjugacy class.

2). Hence n = Ai(7Ti) by the definition of • X% and Ai. 8 Induced. 7(diag(a,&, c)) = fJ-(a/c) °f the Borel subgroup B, where fi is a character of Fx. Denote by 7r0 = Io(fJ-) and TTX = h{n) the representations of Ho, H\ normalizedly induced from the characters (So-O^Mo), (a0*b)^fi(a/b) of the upper triangular Borel subgroups. 2) show that the tr-character x% °f n = I(rj) vanishes at S unless S is diagonal (up to a-conjugacy), where xUS) = A0(7)-1(v(S)+v(S)) (S = J5J, j = N5). Similar standard computations show that the XTT* are also supported on the (conjugacy classes of) diagonal elements of Hi.

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