By Prof. Charles Rockland (auth.)

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T1 o g (x + y + 1/2 [x,y]) Hn with the map corresponding to g T1). 22a) The corresponding equality holds for the associated Lie algebra representations. 22b) Q(g~) = Ug Q(~) Ug I for every % c s l m 2n . 11) . has the symmetric representation = Z alj Ds i D s J + Z bijsis j + ~ cljSiDsj + ZcijDsj si Hence, by Def. 23) T~ij (Ug Dsl -- u l)(ug . . 14 . QED Remarks: I. 23) up to multi- plication by a complex number of modulus that, no matter how we choose the Uglg2 = e (gl,g2) U g l U g 2 where of modulus 1 .

11) . has a unique symmetric representation P e Diff~ ~ n ) is uniquely determined by QED p. 16. 14 . 10) . To prove n on G acts on be the vector ]R n @ this we need some ]R2n Clearly determined homogeneous of A(~,~) = Z qisi + S1]R 2n via g~ ~ ~ ~ g-1 . space of all linear differential of the form are constants. , Z bit i . bi condition element lemmas. Let Let a linear map Z ais i + Z biDsi L r Diff~ n , where uniquely by its coefficients a i, determines ai,b i . 13) defines every Q : S 1jR 2n a vector % ~ Diff 1jR n si ~ ) si ti ~ ) Ds i space isomorphism.

57) ~(Xo, Eo ) (~, D~, D r) is hypoelliptic < > ~(Xo, ~o ) (s Ds I) and ~(Xo, to ) (~, Ds both fail to have 0 as an eigenvalue . 28. Suppose our original operator P had its principal symbol and sub-principal part fully homogeneou~ with respect to ~ rather than just with respect to as would be the case if, for example, operator. to ~. 37)) homogeneous of degree functions 0 (with respect to ~) v l, .... ,v n also could be used as local parameters N(~)(Xo,~O ) and given by (Xo,-~o) . 1 , then u l, ....