Two geometric character formulas for reductive Lie groups
In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation π in terms of the same geometric data attached to π. When specialized to the case of a compact Lie group, one of them reduces to Kirillov’s character formula in the compact case, and the other, to an application of the Atiyah-Bott fixed point formula to the Borel-Weil realization of the representation π. To set the stage, let us first recall the Borel-Weil theorem for a connected, compact Lie group GR. For simplicity, we assume that GR is simply connected. We let gR denote its Lie algebra and g = C ⊗R gR the complexified Lie algebra. Via the adjoint action, GR operates on igR∗, the space of all R-linear functions λ : gR → iR. Every GR-orbit Ω in igR∗ – “coadjoint orbit” for short – carries a canonical GRinvariant symplectic structure. The orbit Ω is said to be integral if some, or equivalently any, λ ∈ Ω exponentiates to a character e of the isotropy subgroup (GR)λ. In that case, the one-dimensional representation spaces of the characters e, λ ∈ Ω, fit together into a GR-equivariant, real algebraic, Hermitian line bundle LΩ → Ω. The pair (Ω,LΩ → Ω) carries a unique “positive polarization”: a GR-invariant complex structure on the manifold Ω and the structure of a GR-equivariant holomorphic line bundle on LΩ, positive in the sense of algebraic geometry. The hypothesis of simple connectivity ensures that the square root √ KΩ of the canonical bundle KΩ exists as a GR-equivariant holomorphic line bundle. According to the Borel-Weil theorem, the natural action of GR on the space of holomorphic LΩ-valued half forms H0(Ω,O(K Ω ⊗ LΩ)) is irreducible; moreover, the association Ω H0(Ω,O(K Ω ⊗ LΩ)) (1.1)