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By Howie J.M. (ed.)

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If A is a smooth compactly supported function on H\ then for all regular semisimple 71 we put $(lufidhi)= [ h (hllh-^. /

Indeed if 5 is replaced by fiSJ^J then SJ + \SJ) is replaced by 051*0 + pMp = P[5J +*(5J)]% and the form SJ + \6J) splits if and only if 0[5J + XSJ)]lP does. If S, S' are stably cr-conjugate with regular norm, but they are not conjugate, then the forms SJ and S'J are not equivalent, and K(S') — —K(S). Thus if S = (ae)i and 5' = (uae)i, then K(S') = X(«)K(<5), X being the quadratic character of Fx trivial on NEX, E = F(Sa(S)). If NS = 7 is regular in H then Za(Sa) ~ Z H ( T ) - Indeed, if g~15a(g) = S then g~15a(S)g — Sa(S); if 5 = (ae)i then g — b\ and b~1ab = a, since Sa(S) = hi, h = Hence b~1aewtb~1we = a, namely b~xab = a, deto so that det 6 = 1 .

If ir is an admissible representation, for any / in C£°(G) the operator Tt(fdg) = fG f(g)n(g)dg has finite rank. We write tnr(fdg) for its trace. If n is irreducible but not equivalent to °7r, then tv-K(fdgxa) is zero. If n is irreducible and unramified, and fdg is spherical, then n(fdg) is a scalar multiple of the projection on the K-fLxed vector w. If, moreover, -K ~ °7r, then ir(a) acts as 1 on w, and tnr(fdgxcr) = tvir(fdg) is this scalar. Let us compute it. 4 LEMMA. Suppose that IT is unramified and t = t{rj) = t(n) is a corresponding element in T.

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An Introduction to Semigroup Theory by Howie J.M. (ed.)


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