  Let O be the minimal nilpotent adjoint orbit in a classical complex
semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the
Euler dilation action $C^*$ on g. The algebra of differential operators on O is
D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \cite{jos}
and \cite{bkHam} for some results on the geometry and quantization of O.
  We construct an explicit subspace $A_{-1}\subset D(O)$ of commuting
differential operators which are Euler homogeneous of degree -1. The space
$A_{-1}$ is finite-dimensional, g-stable and carries the adjoint
representation. $A_{-1}$ consists of (for $g \neq sp(2n,C)$) non-obvious order
4 differential operators obtained by quantizing symbols we obtained previously.
These operators are "exotic" in that there is (apparently) no geometric or
algebraic theory which explains them. The algebra generated by $A_{-1}$ is a
maximal commutative subalgebra A of D(X). We find a G-equivariant algebra
isomorphism R(O) to A, $f\mapsto D_f$, such that the formula $(f|g)=({constant
term of}D_{\bar{g}} f)$ defines a positive-definite Hermitian inner product on
R(O).
  We will use these operators $D_f$ to quantize O in a subsequent paper.
