Space of linear differential operators on the real line as a module over the Lie algebra of vector fields

Abstract

Let ${\mathcal{D}}^k$ be the space of $k$-th order linear differential operators on $\mathbf{R}$: $A=a_k(x)\frac{d^k}{d{x}^k}+\cdots +a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\mathcal{D}}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $\lambda$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on ${\mathcal{D}}^k$ corresponding to different values of $\lambda$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\mathcal{D}}^3$ are isomorphic to each other for every values of $\lambda \ne 0,1,\frac{1}{2},\frac{1}{2}\pm \frac{\sqrt{2}1}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\mathcal{D}}^3\to {\mathcal{D}}^3$. In the higher order case $(k\ge 4)$ $\Diff(\bf R)$-module structures on ${\mathcal{D}}^k$ corresponding to two different values of the degree: $\lambda$ and ${\lambda }^{\prime }$, are isomorphic if and only if $\lambda +{\lambda }^{\prime }=1$.

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