International Congress of Mathematicians

658 F. ShahidiThe main result of [34, also see 40] is thatM(s,n)f = ®vesA(s,n v ,w 0 )fv®®v(sfvmx JJ L s (is, n, n)/L s (l + is, n, n), (2.9)«=iwhere / = ® v f v is such that for each v $ S, f v is the unique K v -fixed functionin I(s,n v ) normalized by f v (e v ) = 1 and for each i, r t denotes the contragredientof ì'ì,ìJ= 1, • • • ,m, the irreducible components of the adjoint action of L M or L N.Here /„ is the F^-fixed function in the space of I(-s,wo(ir v j), normalized the sameway. Moreover /„ and /„ are identified as elements in spherical principal series.3. Generic representations and the non-eonstanttermSuppose F is a field, either local or global, and G is as before, with a Boreisubgroup B = TU over F. Fix an F-splitting {X a >}, i.e., a collection of rootvectors as a' ranges over simple roots of T in U which is invariant under the actionof FF = Gal(F/F). This then determines a map form U to EG a , ip(u) = (x a i) a i,where x a i is the «'-coordinate of« with respect to {X a >}. Yet {K Q /} be a collectionof elements in F such that