Isotropy representations, eigenvalues of a casimir element, and commutative lie subalgebras

摘要

Let h be a reductive subalgebra of a semisimple Lie algebra g and C_h ∈ U(h) be the Casimir element determined by the restriction of the Killing form on g to h. The paper studies eigenvalues of C_h on the isotropy representation m approx= g/h. Some general estimates connecting the eigenvalues and the Dynkin indices of m are given. If h is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C_h on exterior powers of m is connected with possible dimensions of commutative Lie subalgebras in m, thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C_h on Λ m. More generally, a similar picture arises if h = g~Θ, where Θ is an automorphism of finite order m and m is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.