The Dynkin index and conformally invariant systems associated to parabolic subalgebras of Heisenberg type

Abstract

Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to parabolic subalgebras of Heisenberg type. When they constructed such systems of operators, two constants, which play a role for the construction, were defined as the constants of proportionality between two expressions. In this paper we give concrete and uniform expressions for these constants. To do so we introduce a new constant inspired by a formula on the Dynkin index of a finite dimensional representation of a complex simple Lie algebra.