The Spectrum of Certain Invariant Differential Operators Associated to a Hermitian Symmetric Space

Abstract

Let G / K be an irreducible Hermitian symmetric space of rank n and let 9 = t + p+ + p_ be the usual decomposition of 9 = Lie(G)c. Let us write P, V, and W = P ® V respectively, for the algebra of holomorphic polynomials, the algebra of constant coefficient holomorphic differential operators, and the "Weyl algebra" of polynomial coefficient holomorphic differential operators on 1'-, and regard all three as K -modules in the usual way.Let I = WK be the algebra of K-invariant differential operators on p_., and let A be the set {oX = (Ab" . ,An) E zn I Ai 2: ... 2: An 2: a}.Then, as we show in the next section, A naturally parametrizes both the irreducible K-submodules P).. of P, as well as a certain distinguished (vector space) basis {D.d of I. The problem we consider is to determine, for A, J.l E A, the scalar eigenvalue cl'(A) by which DI' acts on P>..Our main result, proved in Section 1, is the following characterization of these eigenvalues. For A = (Ai,' .. ,An), let us write IAI for Ai + ... + An; and set Am = {A E A IIAI:::; m}. Also, put P = (Pl,···,Pn) where Pi = d( n -2i + 1)/2 and d is as in the next section.*

The Capelli identity for Grassmann manifolds

“…We refer the reader to [7] and [2] for a discussion of the background of this identity. In the early 90s, the identity was generalized considerably by B. Kostant and the author ( [13], [14], [19]) and by Howe and Umeda [10] to the context of Jordan algebras and multiplicity free actions. Since then there has been considerable interest in generalizations of Capelli-type identities, we refer the reader to [4], [11], [12], [15], [16], [17], [18] and the references therein.…”

“…Moreover, this point of view leads to a much larger class of identities in the context of Jordan algebras ( [13], [14], [19]). …”

On a q-analog of a Sahi result

“…y ν | C[Matn] λ is a scalar operator since C[Mat n ] λ is a simple K-module and y ν is K-invariant. Sahi arranges studying an explicit formula for these scalars [11], [5,Proposition 3.3]:…”

The Asymptotic Behavior of Certain Birth Processes

“…The probabilities P m, n are closely related to the more general interpolation polynomials P λ (x; ρ) defined in [10]. It would be interesting to obtain a statistical interpretation for the general P λ (x; ρ) which might shed light on some of their properties.…”