By Yuval Z Flicker

ISBN-10: 9812568034

ISBN-13: 9789812568038

The world of automorphic representations is a typical continuation of reports in quantity thought and modular kinds. A guideline is a reciprocity legislation bearing on the limitless dimensional automorphic representations with finite dimensional Galois representations. basic relatives at the Galois facet replicate deep relatives at the automorphic part, known 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 team U(3, E/F) to GL(3, E), [E : F] = 2. The ebook develops the means of comparability of twisted and stabilized hint formulae and considers the "Fundamental Lemma" on orbital integrals of round features. comparability of hint formulae is simplified utilizing "regular" capabilities and the "lifting" is acknowledged and proved through personality family members. 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 choice of the self-contragredient representations of PGL(3) and people on GL(3, E) fastened via transpose-inverse-bar. particularly, the multiplicity one theorem is new and up to date. There are functions to development of Galois representations via specific decomposition of the cohomology of Shimura kinds of U(3) utilizing Deligne's (proven) conjecture at the mounted aspect formulation.

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If A is a smooth compactly supported function on H\ then for all regular semisimple 71 we put $(lufidhi)= [ h (hllh-^. /

Indeed if 5 is replaced by fiSJ^J then SJ + \SJ) is replaced by 051*0 + pMp = P[5J +*(5J)]% and the form SJ + \6J) splits if and only if 0[5J + XSJ)]lP does. If S, S' are stably cr-conjugate with regular norm, but they are not conjugate, then the forms SJ and S'J are not equivalent, and K(S') — —K(S). Thus if S = (ae)i and 5' = (uae)i, then K(S') = X(«)K(<5), X being the quadratic character of Fx trivial on NEX, E = F(Sa(S)). 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 .

If ir is an admissible representation, for any / in C£°(G) the operator Tt(fdg) = fG f(g)n(g)dg has finite rank. We write tnr(fdg) for its trace. If n is irreducible but not equivalent to °7r, then tv-K(fdgxa) is zero. If n is irreducible and unramified, and fdg is spherical, then n(fdg) is a scalar multiple of the projection on the K-fLxed vector w. If, moreover, -K ~ °7r, then ir(a) acts as 1 on w, and tnr(fdgxcr) = tvir(fdg) is this scalar. Let us compute it. 4 LEMMA. Suppose that IT is unramified and t = t{rj) = t(n) is a corresponding element in T.

### Automorphic Representations of Low Rank Groups by Yuval Z Flicker

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