UNITARY REPRESENTATIO N THEORY FOR SOLVABLE LIE GROUPS CHAPTER III The Coadjoint Representatio n 0. Introduction: Let S be a solvable (real) Lie algebra . In this chapter we shall develop an inductive method for studying the coadjoint representatio n ad of S on S . We shall focus on a single problem , which a r i s e s as follows: Let 0 € S 0 defines an alternating bilinea r for m B , on S vi a the formul a B (x, y) = 4 ([x, y]) for all x, y € S. Becaus e B, is an alternating form , we can choose a 0 b a s i s x,, v., x_, y_ , . . . , x , y , zn, . . • , z for S so that 1 1 2 2 m m 1 n B . (x.,y.) = 6.. , 1 i, j m , / i J ij - ~ B . (x.,x.) = 0 , 1 i, j m , 0 i j - - B , ( y . , y . ) = 0 , l i , j m , and B, (z.,x) = 0, 1 i n, for all x e S. 0 1 ~ ~ The subspace R of S spanned by z , z , • • • , z is called the radica l 1 c n of B, . R is p r e c i s e l y {z e S: B, (z,S) = 0} . Let us call a subspace V of S self-orthogonal (or (0) self- orthogonal, should it be n e c e s s a r y to be so explicit) if B (V, V) = 0. R is self-orthogonal , and the subspace spanned by x , x ? , ••• , x and R is als o self-orthogonal.

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