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...Operating Grant A3483. 12 JAMES ARTHUR Cc'(G(AF)) = CC'(G(F) X G(F)). Consider a function in this algebra of the form f(yl, Y2) = f(Y)f2(y2), Y, Y2 E G(F), for functions fi and f2 in CC(G(F)). Then R=-=(f)-=- is an operator on L2(G(F)) which maps a function 0 to the function (R(f )+k)(x)- | f (g)(R(g)W)(x)dg G(AF) | X fA(u)f2(Y)(Yu- lxy)dudy = G(F) G(F) G(F) ( /fl(xu)f2(uy)du)0(y)dy. G(F) Thus, R(f) is an...

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...hich can be written as a product of a rational function rp, p(rx) = rTp'IPjTril -)rp' P(ir2.x)with a normalized intertwining operator LOCAL TRACE FORMULA 15 Rp,'p(rx) = Rp'1p(rl,-_x) O RPIP(7r2.x) [l=-=(j)-=-, Theorem 2.1]. It is easy to show that the limit ,JM(Wx, P) = lim E (Jp',p(7rx)-JpIp(7rx+V))Op,(V)-1 v-O P'(P(M) exists [l(b), Lemma 6.2]. In general, the function JM(irx, f) = tr(jM(7rx, P)p(rxx, f)...

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...e answered, so the identity must remain conjectural for the present. (Incidentally, the notation in (7.6) is slightly at odds with that used in connection with automorphic forms. In the papers [l(h)-l=-=(k)-=-] we defined JM(JrX, f) in terms of the normalized intertwining operators RpIp(rX) instead of the unnormalized operators Jp'lp(7rx) used here. Moreover, we denoted the corresponding integral (7.4) sim...

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...operator LOCAL TRACE FORMULA 15 Rp,'p(rx) = Rp'1p(rl,-_x) O RPIP(7r2.x) [l(j), Theorem 2.1]. It is easy to show that the limit ,JM(Wx, P) = lim E (Jp',p(7rx)-JpIp(7rx+V))Op,(V)-1 v-O P'(P(M) exists [l=-=(b)-=-, Lemma 6.2]. In general, the function JM(irx, f) = tr(jM(7rx, P)p(rxx, f)) will have singularities in X. However, it can be shown that if ir belongs to ldisc(M), then JM(?r, f) is a Schwartz function...