An algorithm is presented that computes explicit generators for the ring of differential operators on an orbifold, the quotient of a complex vector space by a finite group action. The algorithm also describes the relations among these generators. The algorithm presented in this paper is based on Schwarz’s study of a map carrying invariant operators to operators on the orbifold and on an algorit...

on the complex upper half-plane H, provably invariant under the linear fractional action of SL2(R). Even in this relatively simple case, it is oppressive to verify invariance directly. Worse, it is misguided to think in terms of such verification. The more relevant issue is the origin of the expression. We do not want to verify invariance of expressions in coordinates, but, rather, to systemati...

For two positive integers m and n, we let Pn be the open convex cone in R consisting of positive definite n × n real symmetric matrices and let R be the set of all m×n real matrices. In this article, we investigate differential operators on the non-reductive manifold Pn × R (m,n) that are invariant under the natural action of the semidirect product group GL(n,R)⋉R on the MinkowskiEuclid space P...

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finitedimensional represent...

Let D = d/dX . We develop a theory of combinatorial differential operators of the form Ω(X, D) where Ω(X, T ) is a species of structures built on two sorts, X and T , of underlying elements. These operators act on species, F (X), instead of functions. We show how to compose these operators, how to compute their adjoints and their counterparts in the context of underlying symmetric functions and...

Motivated by the notion of reduced enveloping algebra, we introduce central reductions of universal enveloping Dalgebras of restricted Lie algebroids. In the case of a tangent Lie algebroid the central reductions are sheaves of twisted crystalline differential operators. For a nilpotent p-character of a classical semisimple Lie algebra, we discover a connection between the reduced enveloping al...

We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that t...

We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and Green’s operators, we employ the algebra of integro-differential operators. The operations implemented for regular boundary problems include computing Green’s ...