By R.K. Lazarsfeld
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Extra info for Positivity in Algebraic Geometry (Draft for Parts 1 and 2)
Lecomte and V. Ovsienko, Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization, preprint [L-Sm] T. P. Smith, Primitive ideals and nilpotent orbits in type G 2 , J. Algebra, 114 (1988), 81-105 [L-Sm-St] T. P. T. Stafford, The minimal nilpotent orbit, the Joseph ideal and differential operators, J. Algebra, 116 (1988), 480-501 [L-St] T. T. Stafford, Rings of differential operators on classical rings of invariants, Memoirs of the AMS, 81 no. 412 (1989) [Se] J.
The correction is uniquely determined, but its nature is mysterious to us. This paper can largely be read independently of [A-B1] and [A-B2J, as the symbols in fact only motivate the construction of the differential operators. Once we figure out the correct formula for Do, we give a selfcontained proof that Do E V~l (0). The more abstract and general results we prove in Section 2 for differential operators on cones of highest weight vectors then give in particular the main properties of our operators Dx: (i) the operators Dx commute, (ii) the operators Dx generate a maximal commutative sub algebra of V(O), and (iii) Ix and Dx are adjoint operators on R( 0) with respect to a (unique) positive definite Hermitian inner product (·1·) on R(O) such that (111) = l.
To start off, we put Bo = Qo. Now we proceed by induction and define Bp+1 by the relation (25) where 9 E Rp(X) and h E Rp+1 (X). This relation is exactly the condition that multiplication by fv is adjoint to Dv. We need to check that Bp+1 is well-defined. Clearly the functions fvg span Rp+1 (X). Also there exists a complex scalar cp+1 such that cp+lQp+1(h,fvg) = Bp(Dv(h), g). 1. So Bp+l = Cp+1Qp+1. Thus Bp+l is well-defined. ) Our hypothesis that some Dv is non-zero on Rp+1 (X) ensures that cp+1 =I O.