Invariant differential operators on nonreductive homogeneous spaces

复杂度为1的齐性空间上不变微分算子的代数与解析性质

Algebraic and analytic properties of invariant differential operators on a homogeneous space of complexity $1$

本文研究齐性空间$SL_3(\mathbb R)/A$上不变微分算子的性质，明确了其非交换代数的生成元与关系，刻画了代数中心并证明对称元本质自伴，还首次发现了非中心的本质自伴不变微分算子。

On the Schmidt--Kolchin conjecture on differentially homogeneous polynomials. Applications to (twisted) jet differentials on projective spaces

The main goal of this paper is to prove the Schmidt--Kolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next, we establish a one-to-one correspondance between differentially homogeneous polynomials in \((N+1)\) variables, and twisted jet differentials on projective spaces. As a by-product of our study of differentially homogeneous polynomials, we are thus able to understand explicitly twisted jet differentials on projective spaces.

Global symbolic calculus of pseudo-differential operators on homogeneous vector bundles

A symbolic calculus for a pseudo-differential operators acting on sections of a homogeneous vector bundle over a compact homogeneous space $G/H$ with compact $G$ and $H$ is developed. We realize the symbol of a pseudo-differential operator as a linear operator acting on corresponding irreducible unitary representations of $H$ valued in the algebra $C^\infty(G)$ of smooth functions. We write down how left invariant vector fields of $SU(2)$ act on the sections of homogeneous vector bundles associated to the fibration $\mathbb{T}\hookrightarrow SU(2)\rightarrow\mathbb{C}P^1$, which is known as th…

Algebras of invariant differential operators on unit sphere bundles over two-point homogeneous Riemannian spaces

Let $G$ be the identity component of the isometry group for an arbitrary curved two-point homogeneous space $M$. We consider algebras of $G$-invariant differential operators on bundles of unit spheres over $M$. The generators of this algebra and the corresponding relations for them are found. The connection of these generators with two-body problem on two-point homogeneous spaces is discussed.

The Hua operators on homogeneous line bundle on Bounded Symmetric Domains of Tube Type

Let D be a bounded symmetric domain of tube type . We show that the image of the Poisson transform on the degenerate principal series representation attached to the Shilov boundary of D is characterized by a covariant differential operator on a homogeneous line bundle on D.

Invariant differential operators on spherical homogeneous spaces with overgroups

We investigate the structure of the ring ${\mathbb D}_G(X)$ of $G$-invariant differential operators on a reductive spherical homogeneous space $X=G/H$ with an overgroup $\widetilde{G}$. We consider three natural subalgebras of ${\mathbb D}_G(X)$ which are polynomial algebras with explicit generators, namely the subalgebra ${\mathbb D}_{\widetilde{G}}(X)$ of $\widetilde{G}$-invariant differential operators on $X$ and two other subalgebras coming from the centers of the enveloping algebras of $\mathfrak g$ and $\mathfrak k$, where $K$ is a maximal proper subgroup of $G$ containing $H$. We show t…