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

Abstract

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (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 ${\cal D}^k$ corresponding to different values of $\lambda$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each other for every values of $\lambda\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm \frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case $(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to two different values of the degree: $\lambda$ and $\lambda^{\prime}$, are isomorphic if and only if $\lambda+\lambda^{\prime}=1$.