By Yuval Z Flicker

The world of automorphic representations is a usual continuation of experiences in quantity conception and modular types. A guideline is a reciprocity legislations bearing on the endless dimensional automorphic representations with finite dimensional Galois representations. easy kinfolk at the Galois part replicate deep relatives at the automorphic facet, referred to 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 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 services. comparability of hint formulae is simplified utilizing "regular" features and the "lifting" is acknowledged and proved by way of 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 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) mounted by means of transpose-inverse-bar. particularly, the multiplicity one theorem is new and up to date. There are purposes to development of Galois representations by way of specific decomposition of the cohomology of Shimura different types of U(3) utilizing Deligne's (proven) conjecture at the mounted aspect formulation.

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**Example text**

We begin with a description of these classes. Let F be a local or global field of characteristic 0. Fix an algebraic closure F of F. Let G be a reductive group defined over F and G = G ( F ) the group of F-rational points of G. Denote by a an automorphism of G defined over F. The elements 8, 5' of G are called a-conjugate if there is h in G with 6' — h6a(h~l). They are called stably a-conjugate if there is h in G(F) with 8' — hSa(h~1). The term (stable) conjugacy (no mention of a) is employed if a is the trivial automorphism.

Then $(ka,f°dg)= f f(Int(x)(ka))dx JG/Za(k

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.