Howe Correspondence and Springer Correspondence

Consider real reductive group G, as defined in [Wal88]. Let Π be an irreducible admissible representation of G with the distribution character ΘΠ, [Har51]. Denote by uΠ the lowest term in the asymptotic expansion of ΘΠ, [BV80]. This is a finite linear combination of Fourier transforms of nilpotent coadjoint orbits, uΠ = ∑ O cOμ̂O. As shown by Rossmann, [Ros95], the closure of the union of the nilpotent orbits which occur in this sum is equal to WF (Π), the wave front set of the representation Π, defined in [How81]. Furthermore there is a unique nilpotent coadjoint orbit OΠ in the complexification gC of the dual Lie algebra g of G such that the associated variety of the annihilator of the Harish-Chandra module of Π in the universal enveloping algebra U(g) of g is equal to the closure of OΠ, [BB85]. Moreover, the closure of OΠ coincides with the complexification of WF(Π), see [BV80, Theorem 4.1] and [Ros95]. Given a Cartan subalgebra of gC we have the corresponding Weyl group W. Springer correspondence associates an irreducible representation of W to each complex nilpotent coadjoint orbit, assuming the group is connected. See [Ros91] for a convenient geometric construction. We shall use this construction in Appendix A to extend the notion of Springer correspondence to cover the case when the reductive group is an orthogonal group (which is disconnected) and refer to this extended version as the “combinatorial Springer correspondence”, denote it by CSC, see (21) below. Thus CSC(OΠ) is an irreducible representation of W corresponding to the complex nilpotent coadjoint orbit OΠ. Let us be more specific and consider a real reductive dual pair (G′,G) in a symplectic group Sp(W). We shall always assume that the rank of G′ is less or equal to the rank of G. Let Π′ be an irreducible admissible representation of G̃′, the metaplectic cover of G′, and let Π be the irreducible admissible representation of G̃ which corresponds to Π′ via Howe correspondence for the pair (G′,G), [How89]. Howe correspondence is governed by a Capelli Harish-Chandra homomorphism