# WISHART EXPECTATION OPERATORS AND INVARIANT DIFFERENTIAL OPERATORS

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Let E(,n) denote the expectation operator of the Wishart distribution W(k,n,(SUMM)) and let E(,n)('t) denote the "conditional" expectation operator defined by.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;where (VBAR)V(VBAR) denotes the determinant of the k x k positive definite symmetric matrix V, trA is the trace of A, (SUMM) is a k x k positive definite symmetric matrix, C(,n) is a constant depending on n, and dV denotes Lebesgue measure on the space of k x k positive definite matrices, V > 0. We study the common eigenfunctions of the operators E(,n) (n (GREATERTHEQ) n(,0)) and the common eigenfunctions of the operators E(,n)('t) (n (GREATERTHEQ) n(,0)) which are shown to be identical with the common eigenfunctions of certain invariant partial differential operators. The eigenvalues of the expectation operators are determined in terms of the eigenvalues of the differential operators, and vice-versa. Explicit expressions for (lamda)(,n), the eigenvalues of E(,n), somewhat more complete than those obtained previously by Maass, are given. (lamda)(,n)('t), the eigenvalues of E(,n)('t), are characterized via a unique solution to a certain ordinary differential equation. An asymptotic formula for (lamda)(,n)('t), as t (--->) (INFIN), is proved, as well as for functions defined by integrals of the form.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;where f(V) is homogeneous function. Irreducible class 1 subspaces, with respect to the congruence transformations of G (k) are constructed from the common eigenfunctions of E(,n). An integral formula for the most general common eigenfunction of E(,n) is proved.