# Harmonic Analysis on Compact Solvmanifolds by J. Brezin By J. Brezin

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In later, the product formula PI' .... 7). At least w e are o n the right track. 7): Let us b e g i n with the parabolic case. There ro(B ) = p 47 is the linear variety dividing {(rl,r2+rl x) : x e ~p}. rl, then the ideals If pk is the largest power of rI Z and p+k £ are equal, and P P {(rl,r2+rlx) : x c p-kZp}. It follows that the number of o(U ) P £2p n =rO(Bp) is equal to the index of £ p in p -k ~ p ' namely k p , as asserted. Consider now the hyperbolic case. to -- U . Hence, if ~ ~ ~p, then If pk.

O for various choices of o,p {(u,0)= : =u ~ Q2}. A o and We can realize A "B , in w h i c h the n o n - n o r m a l factor P P in a more c o m p l i c a t e d w a y than W e are going to take a B p. Let Go, p as a depends generally on P does, at least in the h y p e r b o l i c case. In P the p a r a b o l i c case, the natural choice for B is the subgroup P lit if01 H e r e the situation resembles t • ~p . the real case as closely as one could hope. 4) a , a,b • ~p and a 2 - Db 2 1 0 There are two quite different forms can assume, depending on w h e t h e r or not B P lies in G ~p.

If denotes, derived subgroup G' o that for almost all dense in G ov . G be a d i s t r i b u t i o n on as before, lies in the A. { ( u=, O ) subgroup in o in - F o of Go, then the [(0=,t)-l(FonA)(0,t)(F n A ) ] n = G is (O,t)FoG ~ , _ double cosets, w e could conclude that G'o is It w o u l d follow that for almost all Since tion and in a d d i t i o n - - b e c a u s e it is a d i s t r i b u t i o n on r : u £ ~2 } The key step in the proof consists of showing t ~ IR, the i n t e r s e c t i o n Fo(~,t)Fo that also belongs to Go//F ° as o t h e r w i s e there is n o t h i n g to Suppose that this has been done.

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