WEYL GROUP-ACTIONS ON LAGRANGIAN CYCLES AND ROSSMANNS FORMULA

Abstract

Let GR be a connected Lie group, gR its Lie algebra, and gR* the vector space dual of gR. Each coadjoint orbit, i.e., GR-orbit in gR*, has an intrinsically defined GR-invariant symplectic structure, and thus carries a distinguished GR-invariant measure. Kirillov’s character formula — in those cases when it applies — expresses the irreducible unitary characters of GR as Fourier transforms of the distinguished measures on coadjoint orbits, which are then lifted from the Lie algebra to the group via the exponential map. Kirillov [Ki] originally established the formula for nilpotent groups, and Auslander-Kostant [AK] for a large class of solvable groups.