ISOTROPY REPRESENTATIONS, EIGENVALUES OF A CASIMIR ELEMENT, AND COMMUTATIVE LIE SUBALGEBRAS

Abstract

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