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34 I. Preliminaries Had we known the Principle of Functoriality, namely that discrete spectrum representations 7rz of GL(2, A) are parametrized by two dimensional representations qz : I? + GL(2,C) of a suitable Weil group r(=W F ) , we could conclude the rigidity theorem part of our global theorem about the lifting XI from C = SO(4) to PGL(4). However, this Principle is known W E , J F )induced , only for monomial representations qz = Ind(pz;WE,JE,, from characters p2 of WE,JE,= A;,/E:, where E, is a quadratic extension of F .

Let S = t S be a symmetric matrix in GL(n, C). Put g* = S t g P 1 S - l . T h e n the orthogonal group O ( S ,C) = { g E GL(n, C ) ;g = g * } controls its o w n fusion in GL(n, C). PROOF. Suppose that A , B are subsets of O ( S , C ) and g E GL(n,C) 1 satisfies gAg-' 1 = B. For each a in A we have a* = a , hence g*ag*-' = ( g a g p 1 ) * = gag-' (as b = b* for b = gag-'). Then c = g-'g* commutes with each a in A, and c*-' 1 = StcS-' 1 = Stg*tg-l,'-l - 1 = g-lStg-'S-' -1 = g-'g* = c. Let d be a square root of c, thus c = hd 2 .

If 7r1 x 7r2 is cuspidal, its Ao-lift is a packet, otherwise: quasi-packet. Each member of a stable packet occurs in the discrete spectrum of the group PGSp(2, A) with multiplicity one. The multiplicity m ( 7 r H ) of a member T H = @ T H ~ of an unstable [quasi-]packetX ~ ( 7 r lx 7r2) ( T I # 7r2) is not ("stable", or) constant over the [quasi-]packet. If 7r1 x 7r2 is cuspidal, it is m(7rH) = 1 -(1+ ( - l ) n ( " H ) ) 2 ( E (0,l)). Here n ( 7 r H ) is the number of components 7rzv of ITH (it is bounded b y the number of places v where both 7 r l u and 7raU are square integrable).