By Yuval Z Flicker

The realm of automorphic representations is a average continuation of stories in quantity thought and modular varieties. A guideline is a reciprocity legislations touching on the endless dimensional automorphic representations with finite dimensional Galois representations. uncomplicated family at the Galois aspect mirror deep family members 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 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 booklet 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 said and proved by way of personality relatives. this allows 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 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) mounted via transpose-inverse-bar. particularly, the multiplicity one theorem is new and up to date. There are functions to building of Galois representations via particular decomposition of the cohomology of Shimura sorts of U(3) utilizing Deligne's (proven) conjecture at the fastened aspect formulation.

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**Example text**

Let <5 be an element of G. The set of eigenvalues of 5a(5) is of the form {A, 1, A - 1 } . Indeed, if A is an eigenvalue of 5 a (6) then there is an eigenvector v with t(5cr(6))v = Xv. Hence X-1v = \8a{5))-1v, and A_1(

Functoriality and norms 34 is A(6a)2 = |det[l - Ad(6)a]\Lie{G/Ta)\. The twisted Weyl integration formula is then (put Sa for (ae)i) *(6a*)2da [ [f(9)dg = lj2[ JG A p JTE/NZ(E) Let us compute A(da)2 ( Xl HgSaa(g)-1)^. >G/Z (Ta) JO G explicitly. We may assume S is diag(a, b, c). X2 X 3 \ xi x5 x6 J modulo center. Thus we assume that X7 Xg XQ J x5 = 0 to fix representatives. Note that LieZc^cr) = {diag(x,0, — x)}, since -0~X = J XJ = / x9 -x6 x3 ( —x8 x 5 —X2 X7 / x - Ad^yx X1+X9 Xl X2-§X6 X4 — £ x s \ ( 1 + §)X7 — X4 2X5 (l+f)x3' X6 —^X2 Xg-§X4 Xl+Xg Recalling that £5 = 0, and noting that in L i e G / Z c ^ c ) the x\ + £9 is a single variable (alternatively, in X we could replace xg by zero and x\ by xi + xg), we conclude that A(6a)2 = '-!

Hence (3 = 1 and TT is trivial, as asserted. 1 Orthogonality relations. For any conjugacy class functions Xi x' on the elliptic set He of H put (X,x')e= = ^[WiToT'lTo]-1 [ x(h)x'(h)dh f A0(7)2x(7)x'(7)d7. /. Functoriality and norms 42 The sum ranges over a set of representatives T0 for the conjugacy classes of elliptic tori of H over F. [W(T0)] is the cardinality of the Weyl group of T0 (1 or 2). As usual, |T 0 | denotes the volume of T 0 . We write 7 ~ 7' if 7, y are conjugate. The measure dh on He/ ~ is defined by the last displayed equality.