Mark G. Davidson

Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let D=G/K be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let DR=J∩D⊂D be its real form in a formally real Euclidean Jordan algebra J⊂V; DR=H/L is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal–Bargmann transform from a unitary G-space of holomorphic functions on D to an L2-space on DR. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to D of the spherical functions on DR and find their expansion in terms of the L-spherical polynomials on D, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L2-space, the measure being the Harish–Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on D. Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.

Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2, R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L2 (R+,tα dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character Ψ↦ e−i(2n+α+1)Ψ; under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.

Differential recursion relations for Laguerre functions on Hermitian matrices

In our previous papers [1, 2] we studied Laguerre functions and polynomials on symmetric cones Ω = H/L. The Laguerre functions ℓ n ν, n ∈ Λ, form an orthogonal basis in L 2(Ω, dμν) L and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations (πν, ℋν) of the autormorphism group G corresponding to a tube domain T(Ω). In this article, we consider the case where Ω is the space of positive definite Hermitian matrices and G = SU(n, n). We describe the Lie algebraic realization of πν acting in L 2 (Ω, dμν) and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.

The Cayley-Hamilton and Frobenius theorems via the Laplace transform

The Cayley–Hamilton theorem on the characteristic polynomial of a matrix A and Frobenius’ theorem on minimal polynomial of A are deduced from the familiar Laplace transform formula L(eAt)=(sI−A)−1. This formula is extended to a formal power series ring over an algebraically closed field of characteristic 0, so that the argument applies in the more general setting of matrices over a field of characteristic 0.

Gradient-type differential operators and unitary highest weight representations of SU(p,q)

Abstract Let C p + q be equipped with a hermitian form of signature ( p , q ) and let SU ( p , q ) denote the subgroup of the corresponding invariance group U ( p , q ) consisting of matrices with determinant 1. To certain highest weights λ, we associate a first-order group invariant linear differential operator D λ ; whose kernel contains a unitary highest weight representation with highest weight λ. The Fock model realization of unitary highest weight representations of U ( p , q ) is the fundamental tool used to implement this construction. The operator D λ is shown to be equivalent to an operator D λ which acts on Hol ( G K , H λ ) , the space of holomorphic vector valued functions defined on G K . We identify a set Λ 1 of highest weights such that Ker( D λ ) is a proper subspace of Hol ( G K , H λ ) and show that those λ in Λ 1 correspond to points occurring at the far right of the discrete set in the classification scheme of Enright, Howe, and Wallach. First-order differential equations arising from this proper containment are explicitly derived from the operator D λ . We illustrate the fundamental nature of these first-order equations by deriving from them a system which completely determines the irreducible spaces for ladder representations of SU ( p , q ).

Linear Systems of Differential Equations

In previous chapters, we have discussed ordinary differential equations in a single unknown function, y(t). These are adequate to model real-world systems as they evolve in time, provided that only one state, that is, the number y(t), is needed to describe the system. For instance, we might be interested in the temperature of an object, the concentration of a pollutant in a lake, or the displacement of a weight attached to a spring. In each of these cases, the system we wish to describe is adequately represented by a single function of time. However, a single ordinary differential equation is inadequate for describing the evolution over time of a system with interdependent subsystems, each with its own state.

Putzer's Algorithm for e At via the Laplace Transform

Summary A method due to E. J. Putzer computes the matrix exponential eAt for an n × n matrix A without transforming A to Jordan canonical form. A variation of Putzers algorithm is presented. This approach is based on an algorithmically produced formula for the resolvent matrix (sI - A) -1 that is combined with simple Laplace transform formulas to give a formula, similar to Putzers, for eAt.

Generating functions associated to highest weight representations

A bounded linear map Θ defined on an L 2-space with values in a reproducing kernel Hilbert space is necessarily given as an integral operator with kernel K Θ. We discuss in general how the adjoint of K Θ is the generating function associated to a basis of the domain space. Our primary applications are the highest weight representations for a Hermitian group G modeled by the geometric realization. We obtain new formulas relating the generating function to the action of an Abelian subalgebra 𝔭 −⊂𝔤ℂ, where 𝔤 ℂ is the complexification of the Lie algebra of G.

Power Series Methods

Thus far in our study of linear differential equations, we have imposed severe restrictions on the coefficient functions in order to find solution methods. Two special classes of note are the constant coefficient and Cauchy–Euler differential equations. The Laplace transform method was also useful in solving some differential equations where the coefficients were linear. Outside of special cases such as these, linear second order differential equations with variable coefficients can be very difficult to solve.

Second Order Constant Coefficient Linear Differential Equations

This chapter begins our study of second order linear differential equations, which are equations of the form