By Roe W. Goodman

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Where Also (I) is equivalent to LD = DL . ~ l+j=n =~ >: ~ j<_n k<_j = n! 3 associated w i t h the graded v e c t o r space Let u e g Then the ~ . "Bernoulli operator" B(u) adu 1_e-a--~-d-~ d e f i n e s a l i n e a r t r a n s f o r m a t i o n on g (which is a polynomial f u n c t i o n o f u , by the n i l p o t e n c e o f adu) . Lemma I f (~) v e ~ , dR(v)f(u) Remark Formula i s the d i r e c t i o n a l f e Ca(Q) , u e Q , then d d--~t=O f ( u + t B ( u l v ) = . (=~=) asserts t h a t a t the p o i n t d e r i v a t i v e in the d i r e c t i o n u Since dR Set X = dR(u) , Y = dR(v) .

I xYl ~ Ixl + lyl + b . We do not know See also Jenkins [1]. 2 The main source f o r the results of t h i s section is the thesis of Vergne [1]. (cf. Rauch [ ~ , [2] f o r f u r t h e r developments). For the general theory of deformations of algebraic s t r u c t u r e s , c f . Nijenhuis-Richardson [ i ] ; f o r Lie algebra cohomology, cf. Jacobson [ I ] . An example of a n i l p o t e n t Lie algebra with every automorphism unipotent was given by Dyer [ I ] . This phenomenon was investigated f u r t h e r by M~ller-R~mer [ ~ , who showed that the 7 dimensional algebra ~ (0,I,0,i) of Example 2 has t h i s property, but that every n i l p o t e n t Lie algebra of dimension < 6 admits d i l a t i n g automorphisms.

We would then l i k e to use analysis on the n i l p o t e n t groups at each tangent space (convolution operators, e t c . ) to carry out local analysis on tangent spaces is t r i v i a l , groups of the vector spaces M . When the s p l i t t i n g of the so that the n i l p o t e n t groups are j u s t the a d d i t i v e TMx , t h i s is the essence of the "parametrix method", which has been enormously developed and refined in recent years (the theory o f p s e u d o - d i f f e r e n t i a l operators and t h e i r symbolic calculus).