Gordan-Rankin-Cohen operators and weighted densities

Abstract

The two classifications of bilinear differential operators are often mixed in the literature:

(A) Between spaces of modular forms. These forms are transformed under $\text{SL}(2;Z)$ as weighted densities of (usually integer) weight and satisfy certain growth conditions so their spaces are finite-dimensional. The $\text{SL}(2;Z)$-invariant bilinear differential brackets between spaces of modular forms on the 1-dimensional manifold were discovered by Aronhold and Gordan, rediscovered and classified by Rankin and Cohen, and by Janson and Peetre.

(B) Between spaces of weighted densities. Here, for any complex weights, we classify bilinear differential operators between (infinite-dimensional) spaces of weighted densities, i.e., the $\mathrm{pgl}\left(2\right)\simeq \mathrm{sl}\left(2\right)$-invariant operators on the line. This is a new result.