Generalized transvectants-Rankin-Cohen brackets (Q1870303)

In this paper the authors introduce bilinear differential operators on the space of tensor densities on \(\mathbb{R}^n\). These operators generalize the \(\roman{sl}_2\)-invariant differential operators in one-dimensional case, called transvectants, discovered by P. Gordan (in 1887). The main result of the paper is the following theorem: Let \({\mathcal{F}}_\lambda\) be the space of tensor densities of degree \(\lambda\) on \(\mathbb{R}^n\). Then: (i) For every \(k=0, 1, 2, \dots ,\) there is a \(o(p+1,q+1)\)-invariant bilinear differential operator of order \(k\) \[ B_{2k}:{\mathcal F}_\lambda\otimes{\mathcal F}_\mu \longrightarrow {\mathcal F}_{\lambda+\mu+\frac{2k}{n}}, \] unique up to a constant provided neither of \(\lambda\) and \(\mu\) belongs to the set \[ \left\{0,-\frac 1n ,-\frac 2n , \dots , \frac{2-2k}{n} \right\} \cup \left\{\frac{n-2}{2n} , \frac{n-4}{2n} , \dots , \frac{n-2k}{2n} \right\}. \] (ii) There is no other bilinear \(o(p+1,q+1)\)-invariant differential operator on tensor densities, for generic values of \(\lambda\) or \(\mu\). (Here \(n=p+q\).)