International Congress of Mathematicians

588 Michael Harristhen its local component n^ is tempered at every finite prime v. Clozel in [Cll]already showed this to be true for almost all unramified v. Generalizing a methoddeveloped by Lubotzky, Phillips, and Sarnak for the 2-sphere, Clozel [C12] uses theversion of the Generalized Ramanujan Conjecture proved in [HT] to obtain effectiveconstructions of families of equidistributed points on odd-dimensional spheres.With (0.1) out of the way, we can propose the following improvement of (1.3):Problem 1. Show that0-TI,E(YY)WE V —+ On,E v (Yl v ). (1-10)For n = 2 this was established by Carayol assuming standard conjectures onthe semisimplicity of Frobenius. Theorem 1.9 shows that it holds up to semisimplification.The techniques of [HT], like the earlier work of Kottwitz treating unramifiedplaces, is based on a comparison of trace formulas, and cannot detect thedifference between two representations with the same semisimplification. Assumingsemisimplicity of Frobenius, the equality (1.10) follows easily from Theorem 1.9and Deligne's conjecture, apparently inaccessible, on the purity of the monodromyweight filtration.Compatibility with functoriality. Given cuspidal automorphic representationsuj of GL(Hì)E, for i = 1,2, ...,r, and a homomorphism p : GL(n{) x ••• xGL(n r )—¥GL(N) of algebraic groups, Langlands functoriality predicts the existenceof an automorphic representation p»(IIi ®... E r ), not necessarily cuspidal, ofGL(N)E, such that, for almost all unramified places v of E,ON,E V (P*(Y1I ...n r )„) = po (®l = ia ni: E v (flì,vj)- (1A1)In recent years this has been proved for general number fields E in several importantspecial cases: the tensor products GL(2) x GL(2)^GL(4) (Ramakrishnan)and GL(2) x GL(3)^GL(6) (Kim-Shahidi), and the symmetric powers Sym 3 :GL(2)^GL(4) (Kim-Shahidi) and Sym 4 : GL(2)^GL(5) (Kim). It has been verifiedin all four cases that (1.11) holds for all v.Construction of supercuspidal representations by "backwards lifting".The unitary representation n £ Ao(n,F) is isomorphic to its contragredientif and only if the local factor L(s,n x n) has a pole at s = 0, which is necessarilysimple.The local factor can be decomposed as a product:L(s,n x n) = L(s,n,Sym 2 )L(s,n, A 2 ), (1A2)where the two terms on the right are defined for unramified n by Langlands and ingeneral by Shahidi. Only one of the factors on the right has a pole. Using the classA good (n,E) of automorphic representations, Henniart has shown that it is the firstfactor (resp. the second factor) if and only if a(n) is orthogonal (resp. symplectic);