Related papers: Equivariant Operators between some Modules of the …
We analyze the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated K\"othe sequence spaces. We establish relationships with spaces of multipliers and apply these results…
In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with…
Let $V$ be a vector space over a field $F$, $V^*$ its dual space and $L(V)$ the algebra of all linear operators on $V$. For an operator $a\in L(V)$ let $a*$ be its adjoint acting on $V*$, and for a subset $R$ of $L(V)$ let $R"$ be its…
We compute the K-theory of the C*-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G.…
We introduce a simplified (coarse) version of pseudo-differential calculus for operators of order zero on complete Riemannian manifolds. This calculus works for the usual Hormander (1,0) class of operators, as well as for…
This paper presents the connections between univariate and bivariate Hermite polynomials and associated differential equations with specific representations of Lie algebra sl(2,R) whose Cartan sub-algebras coincide the associated…
We show that the adjoint equivariant derived category of $D$-modules on a reductive Lie algebra $\mathfrak{g}$ carries an orthogonal decomposition in to blocks indexed by cuspidal data (in the sense of Lusztig). Each block admits a monadic…
We develop an elementary method to compute spaces of equivariant maps from a homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant…
Given a differential operator defined in terms of left-invariant vector fields on a Lie group, we prove that the local condition defining maximal hypoellipticity is equivalent to a global estimate if the operator is left invariant. As a…
The Lie algebra of vector fields on $R^m$ acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to $sl_{m+1}$, and its affine subalgebra is a maximal parabolic…
In this paper, we employ the framework of localization algebras to compute the equivariant K-homology class of the Euler characteristic operator, a central object in studying equivariant index theory on manifolds. This approach provides a…
In this paper, for a vertex operator algebra $V$ with an automorphism $g$ of order $T,$ an admissible $V$-module $M$ and a fixed nonnegative rational number $n\in\frac{1}{T}\Bbb{Z}_{+},$ we construct an $A_{g,n}(V)$-bimodule $\AA_{g,n}(M)$…
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…
We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal…
The universal enveloping algebra U(g) of a Lie algebra g acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or "quantum group") is a deformation of a universal…
Let $ n\geq 2, A=(a_{ij})_{i,j=1}^{n}$ be a real symmetric matrix, $a=(a_i)_{i=1}^{n}\in \Bbb R^n.$ Consider the differential operator $D_A = \sum_{i,j=1}^n a_{ij}{\partial^2 \over \partial x_i \partial x_j}+ \sum_{i=1}^n a_i{\partial \over…
(This short article is a continuation of a longer, review work, in the same volume of Proceedings, by Ashtekar, Marolf and Mour\~ao [gr-qc/9403042]. All the details and other results are to be found in joint papers of the author with Abhay…
We present a method for calculating the results of operation of differential operators operating on components of vector in generalized coordinates not restricted to orthogonal one. For this we use the relationships between covariant,…
Let $\mathcal{X} \subset \mathbb{P}_k^d$ be Drinfeld's halfspace over a finite field $k$ and let $\mathcal{E}$ be a homogeneous vector bundle on $\mathbb{P}_k^d$. The paper deals with two different descriptions of the space of global…