Jordan algebras and Capelli identities

The purpose of this paper is to establish a connection between semisimple Jordan algebras and certain invariant differential operators on symmetric spaces; and to prove an identity for such operators which generalizes the classical Capelli identity. The norm function on a simple real Jordan algebra gives rise to invariant differential operators Dm on a certain symmetric space which is a natural "conformal compactification" of the Jordan algebra. If t is the Lie algebra of a maximal torus of the symmetric space, and "7 is the Harish-Chandra isomorphism, then q,~ = 7(Din) is a polynomial on t*, and our generalized Capelli identity is an explicit formula for q,~ which we now describe. It turns out that the restricted root system of the symmetric space is one of three possible types A,~-l, D,,, or Cn where n = dim(t). We shall call these cases A, D, and C, respectively, and choose a basis "Yt, 9 9 9 "Y~ for I* such that the root system is {:k:(74' 7 j ) /2} , {• • 7 j ) /2} , or {=t=(% i -,/3)/2, =t=%} in the three cases. Let Pm be the polynomial on t* given by