A Geometric Jacquet Functor

In the paper [BB1], Beilinson and Bernstein used the method of localisation to give a new proof and generalisation of Casselman’s subrepresentation theorem. The key point is to interpret n-homology in geometric terms. The object of this note is to go one step further and describe the Jacquet module functor on Harish-Chandra modules via geometry. Let GR be a real reductive linear algebraic group, and let KR be a maximal compact subgroup of GR. We use lower-case gothic letters to denote the corresponding Lie algebras, and omit the subscript “R” to denote complexifications. Thus (g,K) denotes the Harish-Chandra pair corresponding to GR. Let h be the universal Cartan of g, that is h = b/[b, b] where b is any Borel of g. We equip h with the usual choice of positive roots by declaring the roots of b to be negative. We write ρ ∈ h for half the sum of the positive roots. To any λ ∈ h we associate a character χλ of the centre Z(g) of the universal enveloping algebra U(g) via the Harish-Chandra homomorphism. Under this correspondence, the element ρ ∈ h corresponds to the trivial character χρ. For the rest of this paper, we work with λ ∈ h ∗