International Congress of Mathematicians

656 F. ShahidiX(H)p be the group of F-rational characters of H. We set o = Hom(X(M)F,R).Then o* = X(M)p®zR = X(A)p®zR and oj = o* ®R C is the complex dual of o.When G is unramified over a place v, we let K v = G(0„). Otherwise, we shallfix a special maximal compact subgroup K v C G u for which G v = P V K V = B V K V .Yet K = ® V K V Then G = PK = BK. Yet K M = K n M.For each v, the embedding X(M.)p ^y X(M.)p v induces a mapa v = Hom(X(M) Fi ,,R) -• o.There exists a homomorphism HM : M —¥ a defined byexp(x,H M (m))= JJ_\x(m v )\vfor every \ € X(M.)p and m = (m v ). We extend H M to Hp on G by making ittrivial on N and K.Yet a denote the unique simple root of A in N. It can be identified by aunique simple root of A 0 in U. If pp is half the sum of F-roots in N, we set5 = (pp,ct)^1pp£ a*, where for each pair of non-restricted roots a and ß of T,(a,ß) = 2(a,ß)/(ß,ß) is the Killing form.Given a connected reductive algebraic group H over F, let L H be its L-group.Considering H as a group over F v , we then denote by L H V its L-group over F v .Yet L H° = L H® be the corresponding connected component of 1. We then have anatural homomorphism from L H V into L H. We let r) v : L M V —t L M be this mapfor M (cf. [4]).Let L N be the L-group of N defined naturally in [4]. Let L n be its ( complex )mLie algebra, and let r denote the adjoint action of L M on L n. Decompose r = 0 r ti=1~to its irreducible subrepresentations, indexed according to the values (a, ß) = i asß ranges among the positive roots of T. Alore precisely, Xßy £ L n lies in the spaceof i'i if and only if (a,ß) = i. Here Xßy is a root vector attached to the coroot ß v ,considered as a root of the L-group. The integer m is equal to the nilpotence classof L n. We let r i:V = r t • r/ v for each i (cf. [34,40,41]).If A denotes the set of simple roots of A 0 in U, we use 9 C A to denote thesubset generating Al. Then A = 6 U {a}. There exists a unique element WQ £ Wsuch that WQ(9) C A, while wo(ct) < 0. We will always choose a representative WQfor wo in G(F) and use WQ to denote each of its components.V2. Eisenstein series and .L-funetionsLet n = ® v n v be a cusp form on M. Given a F^M-finite function ip in thespace of ir, we extend (p to a function fi> on G as in Section 2 of [39] as well as in[17], and for s £ C, sets(9) = tp(g)exp{sà + pp,Hp(g)). (2.1)