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Let W~',C = S P ( ~ ~ Then, ). since ~ ( 6 =~ -) Hence w1';(6,) a(,. -- u-lr(6)u is a positive involu- Sym(V) be as above. Let SEW^' and let A we have A' = -A and c d ( ~ and ) the converse if easily seen. We have therefore verified that W~'~(G,,) = d ( 7 ) . Let w(7) be the annihilator of d ( ~ ) we , have proven that defines a monomorphism ~ o i ( $ ~ ( ~ ) / w ( rinto ) ) Pol(V(Q) whose image is d ( X - , p , r ) . We can refine this picture slightly by noting that if N ( e V p , 3 ) = g 2 ( V ) / W 3 ( p ) then is a commutative diagram for all subspace containing W(T) for all T.

662) = N(6)T(61,62) we have l ' ( 6 ) ~l(6) = N(6)T = TN(6) and formula (2) is verified. Next, notice that T(o(aI). ~ ( 6 ~ = ) ) ~ ( T ( 6 ~ . 82) Hence, which easily implies 1. xhare distinct > 0 , i = l . ,h. A is totally positive if x,(K) E isomorphisms of & into 81 we say We now distinguish two subsets of quaternions. The totally positiw quaternions. b) where -a and -b arc totally positive. The totally indefinite quaternions, 5-, are those 9 a . b ) for which a,-b arc totally positive. , o is the only positive involution; but for BC, a is not a positive involution.

1 ~ y g ]on p. 197. Perhaps t h e e a s i e s t method S t e i n and S. Wainger [ s w ~ ] , van d e r Corput, d e s c r i b e d i n A. Zygmund's book I n o u r c a s e , t h e p r o o f s s i m p l i f y , and we reproduce t h e s e f o r completeness. u *f i s t h e p r o d u c t of t h e 5 0 t r a n s f o r m s of t h e f a c t o r s , we may r e s o l v e t h e problem by e s t i m a t i n g (T I-. a q) = a o~(P,Q) - Since and 2 exp(i[sp + s q] s - a s 2) . a E R+, (p,q) E R~ 2 (p,q) E R 9 45(-~,-q) = [ + ; ( P , ~ ) I - i n o u r i n v e s t i g a t i o n s we may suppose t h a t q = 1.

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