Relativistic elliptic matrix tops and finite Fourier transformations

Abstract

We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix there is a natural symmetry between the spectral parameter $z$ and relativistic parameter $\eta$. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows to consider $z$ and $\eta$ on an equal footing. Depending on the type of integrable reduction any of the parameters can be chosen to be the spectral one. Then another one is the relativistic deformation parameter. As a by-product we describe the model of $N^2$ interacting $GL(M)$ matrix tops and/or $M^2$ interacting $GL(N)$ matrix tops depending on a choice of the spectral parameter.