...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...

...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)...

...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...

...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...