Parametrization of the conjugacy class of the special linear group

Yu. Palij

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 155-175

Résumé

Local description of the foliation of the group ${SL}(n)$ into the conjugacy classes, and also the foliation ${\mathfrak{sl}^*(n)}$ into the coadjoint orbits, requires the introducing of parameters on the conjugacy class (the coadjoint orbit). Under the assumption that the parameters are rational functions of natural coordinates (the matrix elements) on ${SL}(n)$, the problem is reduced to solving a system of linear equations. Such a system arises from the requirement of parameter invariance with respect to shifts along vector fields normal to the conjugacy class. Likewise, the problem of parameterization of coadjoint orbits in ${\mathfrak{sl}^*(n)}$ can be treated with the use of the Cartan–Weyl basis for ${\mathfrak{sl}(n)}$. The adjoint action is the differential of the conjugacy action. As a consequence, the parameters on the conjugacy classes and coadjoint orbits are related by the transformation specified by the mapping of the algebra ${\mathfrak{sl}(n)}$ into the group ${SL}(n)$. The groups ${SL}(3), {SL}(4)$ are considered as examples.