A simple expression for the Casimir operator on a Lie group

Abstract

The expression for the Casimir operator for a real semisimple Lie group G in terms of coordinates given by the Iwasawa decomposition G = K A N G = KAN reduces on G/N to the difference of an elliptic operator with constant coefficients on A and an invariant operator on M. This result immediately identifies the principal series of induced representations with representations defined on the eigenspaces of the restriction of the Casimir operator to G/N.