Automorphic Representations of Low Rank Groups by Yuval Z Flicker

By Yuval Z Flicker

The realm of automorphic representations is a normal continuation of reviews in quantity idea and modular varieties. A tenet is a reciprocity legislations bearing on the countless dimensional automorphic representations with finite dimensional Galois representations. uncomplicated family members 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 third-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 services. comparability of hint formulae is simplified utilizing "regular" capabilities and the "lifting" is acknowledged and proved by way of 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 explanation of multiplicity one theorem and pressure 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 sorts of U(3) utilizing Deligne's (proven) conjecture at the mounted aspect formulation.

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Example text

Given k 6 K, ka = sa • u, put a(h) = sa(h)s~l. This is an automorphism of order £ on ZK((SCTY). Suppose that the first l 1 cohomology set H ((a),ZK((sa) )), of the group (a) generated by a, with coefficients in the centralizer Zji((saY) of (sa)e in K, injects in H\(a),ZG((saf)). II. 1 Fundamental lemma Then, any x G G such that k'a = lnt(x)(ka) KZG(sa). PROOF. 51 is in Ka, must lie in Put k'a — s'a • v!. Then s'a = \im{k'a)qTni = Int(a;)lim(fca) 9mi =Int(a:)(s

Put air(g) = 7r(ag) (g in G). Then CT7r is an admissible irreducible representation of G on V. We say that TT is a-invariant if w is equivalent to a7r. In this case there is an invertible operator A: V —> V with ir(o-g) = Ai:{g)A~1 (g in G). Since IT is irreducible and A2 intertwines ir with itself, Schur's lemma ([BZ1]) implies that A2 is a scalar. Multiplying A by 1/vA2, we assume that A2 — 1. Then J4 is unique up to a sign. We put n(cr) = A, and define the operator n(fdg x a) = ir(fdga) = w(fdg)ir(cr) to be the map v >-> / f(g)n(g)Avdg.

Recall: p ^ 2. II. 1 Fundamental lemma For A ^ ± 1 we have $st(5o-,f°dg) A(5a)$™(5o-J°dg) = * ( # ! * , / f d / n ) . THEOREM. 49 = $st(N6,f$dh0), and This is the fundamental lemma for the symmetric square lifting from SL(2) to PGL(3) and the unit element of the Hecke algebra. A proof of the first assertion — due to Langlands, based on counting vertices on the Bruhat-Tits building associated with PGL(3) — is recorded in the paper [F2;II], §4, but it is conceptually difficult, hence not used in this work.

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