Invariant differential operators are linear combinations of symmetric positive ones (Q1318144)

Let \(E\) be a line bundle over the noncompact Hermitian symmetric space \(G/K\), and \(D_ G(E)\) the algebra of \(G\)-equivariant differential operators on \(E\). This paper proves that the algebra \(D_ G(E)\) has a basis consisting of symmetric and positive elements. More precisely, to each constituent \(Z\) of \(S({\mathfrak p})\), where \(\mathfrak p\) is the complexification of the tangent space of \(G/K\) at \(eK\) and \(S({\mathfrak p})\) denotes its symmetric algebra, a symmetric and positive operator \(D_ Z\) in \(D_ G(E)\) can be attached in a canonical way, and under this correspondence there are \(Z_ 1, \dots, Z_ r\) \((r = \text{rank } G/K)\) such that \(D_{Z_ 1}, \dots, D_{Z_ r}\) are algebraically independent and generate \(D_ G(E)\) as \(\mathbb{C}\)-algebra. For the special case of classical Hermitian symmetric spaces, G. Shimura poved the same result, but his algebraic approach is different from that of this paper.