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It E ~ (G t t is × Gt~) map such complete easily such (h,k) I a manner the seen all T in is f i n i t e For ~ that K in the W all there h zero. al to K . k in and ~ ~ K . In in the Now a function = ~(h)v(k) forms (*) ~ the . is linear The prove unipotent particular, the space v on space of dh dk is c o m p l e t e l y K defined it is any f u n c t i o n by from P/U assumptions of as than G. 3 v . for It vanish forms (*) h E K O P . follows on T span that , the the if T operator dual of the is in T is finite ~ and actually dimension- lemma follows.

Of ~ and Then V2 contains G o is the natural represen- be the orthogonal V± I subspaces and, up to equivalence, V~/V~ on the quotient where v E VI from the above remarks . In particular, and v ~ V2 that the theorem is true for o and More precisely e(s,o)/L(s,~) are polynomials in q -S Sometimes, operators. ~ such that , L(s,~)/L(s,~) and (s,%~) that V~2 V± I and have the form <~(g)v,v> It follows such that ~ This means that there are n . Let is the representation of the coefficients . of the theorem to be true for Vl/V 2 .

4: (i) There is absolutel Y conver ent for defines a representation of the group opto be irreducible. 3) are true with e(s,~) unless ~ = X o where = e(s,~) X = i , i s an unramified quasi-character FX . of In that case: (3) For L(s,~) = L(s + ½(n-l),x) ~ , ~ in g(M) , L(s,~) = L(s + ½(n-l),x -I) . the equality z(~,s,~)Z@,n-s,~) = Z~,n-s,~)Z~,s,~) is true in the sense of the analytic Let w be the quasi-character w(a)l Replacing cter X ~ of sentation respect ~ we may assume is unitary x . are absolutely Assume of for by its tensor product ~ to = ~(a) continuation.

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