A GEOMETRIC JACQUET FUNCTOR
M. EMERTON, D. NADLER, AND K. VILONEN

1. Introduction
In the paper [BB1], Beilinson and Bernstein used the method of localisation to givea 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∗ weassociate 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∗that is dominant, i.e. α̌(λ) 6∈ {−1, −2, . . .} for any positive coroot α̌.
A Harish-Chandra module with infinitesimal character λ is, by definition, a (g, K)module which is finitely generated over U (g) and on which Z(g) acts via the characterχλ . We can also view Harish-Chandra modules with infinitesimal character λ as modules over the ring Uλ which is the quotient of U (g) by the two-sided ideal generated by{z − χλ (z) | z ∈ Z(g)}. In light of this, we will sometimes refer to such Harish-Chandra
modules simply as (Uλ , K)-modules.
Let X be the flag manifold of g, and let Dλ be the sheaf of twisted differential
operators with twist λ. By a (Dλ , K)-module we mean a coherent Dλ -module which
is K-equivariant. Such a Dλ -module is, by necessity, regular holonomic as K acts
on X with finitely many orbits. According to Beilinson-Bernstein [BB2], we haveΓ(X, Dλ ) = Uλ , and the global sections functor(1.1)

Γ : {(Dλ , K) − modules} −→ {(Uλ , K) − modules}

is exact and essentially surjective. A section of the functor Γ is given by the localisation functor which takes a (Uλ , K)-module M to the (Dλ , K)-module Dλ ⊗Uλ M. The
localisation functor is an equivalence if λ is regular.
Let PR be a minimal parabolic subgroup of GR , let nR be the nilpotent radicalof pR , and let n̄R be the nilpotent subalgebra of gR opposite to nR . If AR denotes theDate: November 19, 2003.
1

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M. EMERTON, D. NADLER, AND K. VILONEN

maximal split subtorus of the centre of a Levi factor of PR , then GR admits the Iwasawa
decomposition GR = KR AR NR .
Given a (Uλ , K)-module M , we define the Jacquet module of M by the formula(1.2)

J(M ) = n-finite vectors in M̂ ,where M̂ is the n̄-adic completion lim M/n̄k M of M . We call the functor J which←−takes M to J(M ) the Jacquet functor. As we will explain in the next section, thisfunctor, which is covariant and preserves the infinitesimal character, is dual to theusual Jacquet functor. The module J(M ) is a (Uλ , N )-module (in the sense that it is
a Harish-Chandra module for the pair (g, N ) with infinitesimal character λ).
To define the geometric Jacquet functor, we proceed as follows. Let ν : Gm → A bea cocharacter of G which is positive on the roots in n. By composing ν with the left
action of G on X, we obtain an action of Gm on X with action map
a : Gm × X → X.

(1.3)
Consider the diagram
(1.4)

j

i

∼

Gm × X −→ A1 × X ←− {0} × X −→ X

where the maps j and i are the obvious inclusions. To a (Dλ , K)-module M, we associate∼a Dλ -module Ψ(M) by taking the nearby cycles of j∗ a∗ M along {0} × X −→ X.
Before stating our main theorem, let us note that the localisation theory discussed
above for (Uλ , K)-modules applies equally well to (Uλ , N )-modules.
1.1. Theorem. (i) The localisation functor M 7→ Dλ ⊗Uλ M takes the Jacquet functor
J to the geometric Jacquet functor Ψ, i.e.,Dλ ⊗Uλ J(M ) = Ψ(Dλ ⊗Uλ M ).
(ii) The global sections functor Γ takes the geometric Jacquet functor Ψ to the Jacquet
functor J, i.e.,Γ(Ψ(M)) = J(Γ(M)).
1.2. Remark. While the proof that we give of the main theorem is entirely algebraic,
its motivation is quite geometric. Section 5 below provides a discussion of its geometric interpretation: via the deRham functor DR the category of (Dλ , K)-modules is
equivalent to the category of λ-twisted K-equivariant perverse sheaves on X, and via
this equivalence, Ψ may be viewed as the operation of nearby cycles on these perverse
sheaves. We also discuss explicitly the case of GR = SL2 (R).
1.3. Remark. The functor Ψ takes (Dλ , K)-modules to (Dλ , N )-modules. This follows
immediately from the above theorem and the fact that by the translation principle we
may assume that λ is regular.
1.4. Remark. In this paper we could replace the “Iwasawa nilpotent subalgebra” n by
any larger nilpotent subalgebra.
1.5. Remark. The problem of giving a geometric construction of the Jacquet functor
was also considered by Casian and Collingwood. In their paper [CC], they describe a

A GEOMETRIC JACQUET FUNCTOR

3

nearby cycles construction on Harish-Chandra sheaves (with dominant integral infinitesimal character); they refer to the induced functor on K-groups (taking into accountweight filtrations) as a mixed Jacquet functor. They conjecture that this functor induces the usual Jacquet functor on Harish-Chandra modules.
The nearby cycles construction of [CC] coincides with the construction appearing in
the definition of the functor Ψ. (Loosely, they pass to the limit with respect to one
simple coroot at a time, while in the construction of Ψ, we pass to the limit with respect
to all simple coroots at once.) Thus, Theorem 1.1 establishes the conjecture of Casianand Collingwood.
2. Jacquet functors
In this section we recall some standard facts about the Jacquet functor as it is usually
defined, and discuss the relation between this functor and the functor J defined in the
Introduction. As a general reference for basic facts, we use [W], Chapter 4.
If M is a Uλ -module, then the Jacquet module of M with respect to n̄ is defined by
the formulaˇ ) = n̄-finite vectors in M ∗ ,
(2.1)
J(M
ˇ ) iswhere M ∗ is the algebraic dual of M . Of course, the infinitesimal character of J(M∗−χλ = χλ∗ , where λ = −λ + 2ρ.
2.1. Lemma. The functor Jˇ is an exact contravariant functor from (Uλ , K)-modules(or more generally, Uλ -modules that are finitely generated over U (n̄)) to (Uλ∗ , N̄ )modules.
Proof. The Lemma of Osborne guarantees that any (Uλ , K)-module is finitely generatedˇ ) is finitely generated
over U (n̄). If M is such a finitely generated module, then J(Mkˇ ) isover U (g), and for any positive integer k, the space of n̄ -torsion vectors in J(Mkˇ ) isfinite-dimensional. (It is dual to the finite-dimensional space M/n̄ M .) Thus J(Ma Harish-Chandra module for (g, N̄ ).
ˇ ) as the topological dual of the n̄-adic completion M̂ of M
One may regard J(M(equipped with its n̄-adic topology). The exactness of Jˇ is thus implied by the exactnessof forming n̄-adic completions. This in turn follows from the fact that since n̄ is nilpotent
the usual Artin-Rees Lemma holds for the non-commutative ring U (n̄).
Recall our definition of the Jacquet module J(M ) as n-finite vectors in the n̄-adic
completion M̂ of M .
2.2. Lemma. The functor J is an exact covariant functor from (Uλ , K)-modules to(Uλ , N )-modules.
ˇ )∗ of J(Mˇ ). The
Proof. First note that one may view M̂ as the algebraic dual J(Mdecomposition g = n ⊕ p̄ allows us to write U (g) = U (n) ⊗C U (p̄) as C-vector spaces.
ˇ ) is generated
If M is a (Uλ , K)-module then for a sufficiently large choice of k, J(Mkkover U (g) by the space M̌ [n̄ ] of n̄ -torsion elements in M̌ . Since U (p̄) leaves M̌ [n̄k ]ˇ ) is finitely generated over U (n). The lemma is now ainvariant, we see that J(Mconsequence of the preceding one (with n replaced by n̄).
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M. EMERTON, D. NADLER, AND K. VILONEN

We will now give a more concrete description of J(M ) which will be useful for our
purposes. Recall from the last section that we have a cocharacter ν : Gm → G which
when paired with the roots of g is positive precisely on the roots in n. Let us write h forthe semisimple element in g given by dν(t∂t ). (For any coordinate t on Gm , the Euler
vector field t∂t gives an element of the Lie algebra of Gm independent of the choice of
coordinate.) Note that the centraliser of h is precisely the Levi factor l of p, and theweights of h acting on g are integral.
For each natural number k, the quotient M/n̄k M is a finite-dimensional vector spaceon which h acts, and so may be written as a direct sum of generalised h-eigenspaces.
The surjection M/n̄k+1 M → M/n̄k M is h-equivariant, and so induces surjections of
the corresponding generalised h-eigenspaces.
As usual, we write M̂ for the n̄-adic completion lim M/n̄k M of M .
←−2.3. Lemma. If M is a (Uλ , K)-module then the generalised h-eigenspaces of M̂ are
finite dimensional. For any integer m the sum of generalised h-eigenspaces with eigenvalue greater than m is finite dimensional. Furthermore, M̂ is the direct product of its
generalised h-eigenspaces.
Proof. As the space M/n̄M is finite dimensional, it gives rise to a finite set S of generalised h-eigenvalues. The elements in n̄k M/n̄k+1 M can be obtained by multiplying
M/n̄M by n̄k , and so each eigenvalue of h on n̄k M/n̄k+1 M is a sum of an element of
the set S with an integer less than or equal to −k. This gives our conclusion.

2.4. Proposition. If M is (Uλ , K)-module, then the (Uλ , N )-module J(M ) is naturally
isomorphic to the direct sum of the generalised h-eigenspaces in M̂ .
Proof. To prove the proposition, we must show that a vector in M̂ is n-finite if andonly if it is a sum of generalised h-eigenvectors. Clearly any n-finite vector is such asum since it lies in a finite-dimensional h-invariant subspace of M̂ . Conversely, by theprevious lemma, the generalised h-eigenvalues of M̂ are bounded above, and so any
sum of generalised h-eigenvectors is n-finite.
For α ∈ C, let M̂α denote the α-generalised eigenspace of h acting on M̂ . We define
an increasing C-filtration on M̂ by the formulaY(2.2)
Fα (M̂ ) =
M̂β .
β≤α

Here and in what follows, for α, β ∈ C, we write β ≤ α to mean α − β is a non-negative
integer. Pulling back this filtration via the injection M → M̂ , we obtain a filtration on
M . The induced map on associated graded modules(2.3)

GrF• M → GrF• M̂

is an isomorphism. Indeed the proof of Lemma 2.3 shows that the filtration just constructed is cofinal with the n̄-adic filtration on M , and thus that M̂ is the completionof M with respect to this filtration.
Similarly, we grade n̄, and hence its enveloping algebra U (n̄), according to the eigenvalues of the adjoint action of h. Let U (n̄)β denote the graded piece on which h acts

A GEOMETRIC JACQUET FUNCTOR

5

with eigenvalue β. Note that for U (n̄)β to be non-zero, the eigenvalue β must be anon-positive integer. We define a filtration on U (n̄) by the formulaM
U (n̄)β .
(2.4)
F−k (U (n̄)) =β≤−k

The U (n̄)-module structure on M is compatible with the filtrations on U (n̄) and M .
2.5. Lemma. Let α be a complex number.
(i) For any k ≥ 0, the quotient Fα−k (M )/(F−k (U (n̄))Fα (M )) is finite-dimensional.
(ii) If α ∈ C is sufficiently small, then for any integer k ≥ 0, there is an equality
Fα−k (M ) = F−k (U (n̄))Fα (M ) (and so the quotient considered in (i) actually vanishes).
Proof. The Lemma of Osborne guarantees that M is finitely generated over U (n̄). Also,
the filtrations constructed on each of U (n̄) and M are cofinal with the n̄-adic filtrations.
Thus the lemma follows from an easy variant of the Artin-Rees Lemma (applied to thefiltrations F• rather than the n̄-adic filtrations).

3. Nearby cycles, V -filtration, and the geometric Jacquet functor
In this section, we recall the formalism of the V -filtration and the way it is used todefine the nearby cycles functor on D-modules. For a reference, see [K1].
As before, we write X for the flag manifold of g, and Dλ for the sheaf of twisteddifferential operators with twist λ. We denote by t a coordinate on A1 , and consider
the variety A1 × X and its subvariety {0} × X ' X. We write D̃λ for the sheaf of λtwisted differential operators on A1 × X. (Of course, the twisting only occurs along the
X-factor.) The V -filtration on D̃λ is the filtration where we declare that the variable
t is of degree one, ∂t is of degree −1, and everything in Dλ is of degree zero.
For a regular holonomic D̃λ -module M on A1 × X, the V -filtration V α M is a decreasing filtration on M, indexed by α ∈ C, compatible with the V -filtration on D̃λ .
The V-filtration on M is uniquely determined by the following properties:
there exist elements u1 , . . . , uk ∈ M and µ1 , . . . , µk ∈ C such thatX
(3.1)
V α (M) =V mi (D̃λ )ui ,α≤mi +µi

and
(3.2)

GrαV M is a generalised eigenspace of t∂t of eigenvalue α.

Here for α ∈ C, we have written GrαV for the quotient V α (M)/V >α (M).
The nearby cycles of M along {0} × X ' X are given byM(3.3)
ψ(M) = Gr0V M ⊕GrαV M.
α∈C/Z
α∈Z
/

We are now ready to define the geometric Jacquet functor. Recall from the firstsection that via the cocharacter ν : Gm → G, we have defined a Gm -action on X. This
gives us the diagram
(3.4)

a

j

i

∼

X←
− Gm × X −→ A1 × X ←− {0} × X −→ X

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M. EMERTON, D. NADLER, AND K. VILONEN

where a is the action map, and the maps j and i are the obvious inclusions. We define
the geometric Jacquet functor Ψ by the formula(3.5)

Ψ(M) = ψ(j∗ a∗ M),

for a (Dλ , K)-module M.

4. Proof of the main theorem
In this section, we give a proof of Theorem 1.1. To begin with, suppose that M
is a (Dλ , K)-module arising as the localisation of a (Uλ , K)-module M , so that M =Dλ ⊗Uλ M. Our aim is to explicitly write down the V -filtration on j∗ a∗ M on the levelof its global sections M̃ . We will make use of the diagramãp

Gm × X −→ Gm × X −
→ X,

(4.1)

where p is the projection, and the map ã is given by ã(g, x) = (g, gx) so that a = ã ◦ p.
To describe M̃ as a vector space, we use the coordinates in the middle copy of Gm ×X
in diagram (4.1), that is we take the global sections of p∗ M. This gives us(4.2)

M̃ = Γ(A1 × X, j∗ a∗ M) = C[t, t−1 ] ⊗C M.

To describe the action of(4.3)

Ũλ = Γ(Gm × X, D̃λ ) = C[t, t−1 , t∂t ] ⊗C Uλ ,we use the coordinates in the first copy of Gm × X in diagram (4.1). It suffices towrite down the action of the global vector field t∂t and the global vector fields givenby elements of g. To this end, let us recall some of our previous notation: we havethe cocharacter ν : Gm → G, and we write h for the semisimple element in g given bydν(t∂t ). First, under the map ã the global vector field (t∂t , 0) is mapped to the global
vector field (t∂t , h). Hence,(4.4)

t∂t · (f (t)m) = (t∂t f (t))m + f (t)(hm).

Second, under the map ã the global vector field (0, v) given by an element v ∈ g ismapped to the global vector field (0, Adν −1 (t) v). Hence,(4.5)

v · (f (t)m) = f (t)((Adν −1 (t) v)m).

In particular, if v is in the β root space gβ , then
(4.6)

v · m = t−<ν,β> vm.

(The inverse of the adjoint action arises for the following reason. For a vector v ∈ gthought of as a vector field on X, the bracket [h, v] may be calculated by pullingback v along the integral curves given by the action of ν(t). Thus the derivative of thepushforward of v under the action of ν(t) is the negative of [h, v].)
To write down the V -filtration on M̃ , let us recall the discussion of Section 2. There
we constructed an increasing C-filtration F• (M ) on M . We now argue that the V filtration on M̃ may be written in terms of this filtration as follows
M
M
(4.7)
V α (M̃ ) =tk F−α+k (M ) ⊂tk M = M̃ .
k∈Z

k∈Z

A GEOMETRIC JACQUET FUNCTOR

7

To show that this is the V -filtration we must verify conditions (3.1) and (3.2). The
fact that (3.2) holds is clear from the construction (taking into account equation 4.4).
To show that (3.1) holds, we show that
(4.8)

V α+m (M̃ ) = V m (Ũλ )V α (M̃ ),for all α and m,

and that(4.9)

each V α (M̃ ) is finitely generated over V 0 (Ũλ ).

The first statement (4.8) is clear as multiplication by t clearly induces an isomorphism
between V α (M̃ ) and V α+1 (M̃ ). Therefore it remains to argue (4.9). We will in fact
show that V α (M̃ ) is finitely generated over C[t] ⊗C U (n̄). By Lemmas 2.3 and 2.5, wehave(4.10)(4.11)

F−α+k (M ) = M , for sufficiently large k,
F−α−k (M )/(F−k (U (n̄))Fα (M )) is finite-dimensional, for all k ≥ 0,

and
(4.12)

Fα−k−l (M ) = F−l (U (n̄))Fα−k (M ), for sufficiently large k and all l ≥ 0.

These three statements together (taking into account equation 4.6) imply that V α (M̃ )is finitely generated over C[t] ⊗C U (n̄), and hence that it is finitely generated overV 0 (Ũλ ) = C[t, t∂t ] ⊗C Uλ .
Now by Section 3, we have
(4.13)

Ψ(M ) = direct sum of generalised h-eigenspaces in M̂ .

Part (i) of Theorem 1.1 thus follows from Proposition 2.4.
We turn to proving part (ii) of the theorem. Thus we suppose that M is an arbitrary(Dλ , K)-module. If we write M = Γ(M), then there is a canonical map Dλ ⊗Uλ M → M,
whose kernel and cokernel are both taken to zero by the exact functor Γ. Taking into
account the fact that the passage to nearby cycles is exact, as well as part (i) of thetheorem, we see that part (ii) follows from the following statement:(4.14)

if Γ(M) = 0, then Γ(Ψ(M)) = 0.

To prove (4.14), we will argue as follows. First, it suffices to prove it in the case when
M is irreducible. Then, we will appeal to the results of [K2], §8. The final remark of[K2], p. 103, shows that we may find a Weyl group element w so that w(λ) is antidominant, and satisfies condition (8.3.2). Applying the intertwining functor Iw induces
an equivalence of categories between the category of (Dλ , K)-modules and the categoryof (Dw(λ) , K)-modules, which is easily verified to be compatible with the functor Ψ
on each of these categories. Thus in verifying 4.14 we may replace λ by w(λ). The
remark preceding the statement of [K2], Prop. 8.2.1 shows that we may also replace G
by its simply-connected cover, and thus assume that the sheaf O(ρ) is a G-equivariant
line-bundle on X. Proposition 8.2.1 and Theorem 8.3.1 of [K2] now show there exists
a simple root α such that α̌(λ) = 0 and M = O(−ρ) ⊗ p∗ M0 , where p : X → Xα is the
projection from the full flag manifold X to the partial flag manifold Xα of parabolics ofthe type associated to the simple root α, and M0 is a Dλ+ρ -module on Xα . The functor

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M. EMERTON, D. NADLER, AND K. VILONEN

Ψ has its analogue Ψα defined on Dλ+ρ -modules on Xα , compatible with Ψ. Thus weconclude(4.15) Γ(Ψ(M)) = Γ(Ψ(O(−ρ) ⊗ p∗ M0 )) = Γ(O(−ρ) ⊗ Ψ(p∗ M0 ))= Γ(O(−ρ) ⊗ p∗ Ψα (M0 )) = Γ(p∗ (O(−ρ) ⊗ p∗ Ψα (M0 ))) = 0;
here the second equality follows from the evident compatibility of the functor Ψ with
twisting by G-equivariant line-bundles, the third equality follows from the compatibilityof p∗ with Ψ and Ψα , and the the last equality holds because, by [K2], Prop. 8.2.1, we
see that p∗ (O(−ρ) ⊗ p∗ (M0 )) is zero for any Dλ -module M0 on Xα , when α̌(λ) = 0.
5. A geometric interpretation
As mentioned in the introduction, the arguments which go into the proof of our main
theorem are primarily algebraic. In this section, we informally describe the geometry
those arguments formalise and also discuss explicitly the example GR = SL2 (R). The
discussion is in the language of λ-twisted perverse sheaves rather than holonomic Dλ modules.
A review of our notation: pR is a minimal parabolic subalgebra of gR with complexification p; n is the nilpotent radical of p; P is the parabolic subgroup with Lie algebrap; P̄ is the parabolic subgroup opposite to P ; L is the Levi subgroup P ∩ P̄ ; and X isthe flag manifold of g.
The basic ingredient in defining the geometric Jacquet functor is the action(5.1)

a : Gm × X → X

defined as the composition of the action of G on X with a cocharacter ν : Gm → G.
Recall that we choose ν such that when paired with the roots of G it is positive precisely
on the roots in n.
The action a defines two Morse stratifications of X, one by ascending manifolds,
one by descending manifolds, which have the same critical manifolds. The ascending
manifolds are the P -orbits, the descending manifolds are the P̄ -orbits, and the criticalmanifolds are the L-orbits. Indeed, each connected component X α of the fixed pointsof a is a single L-orbit, the P -orbit XPα through X α satisfies(5.2)

XPα = {x ∈ X| lim a(z) · x ∈ X α },z→0and the P̄ -orbit XP̄α through X α satisfies(5.3)

XP̄α = {x ∈ X| lim a(z) · x ∈ X α }.
z→∞

Given a λ-twisted K-equivariant perverse sheaf M on X we think of its geometric
Jacquet module Ψ(M) as the result of ”flowing” M along the trajectories of the actiona and ”passing to the limit”. The limit Ψ(M) is constructible with respect to thestratification by the ascending manifolds {XPα }, and an ascending manifold XPα is in
the support of Ψ(M) exactly when the corresponding descending manifold XP̄α intersects
the support of M. The restriction of Ψ(M) to a critical manifold X α may be calculated
as the integral of M over the fibers of the corresponding descending manifold XP̄α .

A GEOMETRIC JACQUET FUNCTOR

9

To see this in a concrete example, consider the real group SL2 (R), maximal compact
subgroup SO2 (R), and corresponding Harish-Chandra pair (sl2 (C), SO2 (C)). We may
identify the flag manifold of sl2 (C) with the complex projective line CP1 = C ∪ {∞}so that (i) the SO2 (C)-orbits are the points i and −i, and their complement CP1 \{i, −i}, and (ii) the point 0 represents a Borel subalgebra b ⊂ sl2 (C) which is thecomplexification of a minimal parabolic subalgebra of pR ⊂ sl2 (R).
Fix a character λ : b/[b, b] → C. Let δ0 be the irreducible λ-twisted perverse
sheaf on CP1 supported at the point 0, and let N be the middle-extension to CP1of the irreducible λ-twisted local system on CP1 \ {0}. For λ non-integral, there are
two irreducible λ-twisted SO2 (C)-equivariant perverse sheaves on CP1 : the middleextensions M1 and M−1 of the irreducible trivial and twisted SO2 (C)-equivariant local
systems on the complement CP1 \ {i, −i}. For λ integral, there are in addition twoother irreducible λ-twisted SO2 (C)-equivariant perverse sheaves on CP1 : the irreducible
SO2 (C)-equivariant perverse sheaves δi and δ−i supported at the points i and −i.
Choose the cocharacter ν : C× → SL2 (C) so that the induced action a : C× × CP1 →
CP1 is the standard multiplication a(z) · p = z −2 p. Note that in the limit z → ∞, thepoints i and −i flow to 0, and this completely describes what happens to the SO2 (C)orbits. With this picture in mind, we calculate the geometric Jacquet module of theirreducible λ-twisted SO2 (C)-equivariant perverse sheaves on CP1 . For λ integral, we
have
Ψ(δi ) = Ψ(δ−i ) = δ0 Ψ(M1 ) = N Ψ(M−1 ) = L
where L is the unique non-semisimple self-dual perverse sheaf with three-step filtration
L0 ⊂ L1 ⊂ L satisfying
L0 = δ0

L1 /L0 = N

L/L1 = δ0 .

For λ non-integral, we haveΨ(M1 ) = Ψ(M−1 ) = N ⊕ δ0 ⊕ δ0 .
To help explain why the result for M1 and λ integral has different simple constituentsfrom the other perverse sheaves supported on the open orbit, note that only for M1 andλ integral is the characteristic variety of such a perverse sheaf the zero section alone.
References
[BB1]

[BB2]
[CC]

[K1]

[K2]
[W]

A. Beilinson and J. Bernstein, A generalisation of Casselman’s submodule theorem, Representation theory of reductive groups (Peter C. Trombi, ed.), Progress in Mathematics 40,
Birkhäuser (1983), 35–52.
A. Beilinson and J. Bernstein, Localisations de g–modules, C. R. Acad. Sci. Paris, vol 292(1981), 15–18.
L. G. Casian and D. H. Collingwood, Complex geometry and the asymptotics of HarishChandra modules for real reductive Lie groups. I, Trans. Amer. Math. Soc., vol 300 (1987),
73–107
M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry, proceedings, Tokyo/Kyoto 1982 (M. Raynaud and T. Shioda, eds.), Lecture
Notes in Mathematics 1016, Springer-Verlag (1983), 134–142.
M. Kashiwara, Representation theory and D-modules on flag varieties, Astérisque, vol 173–174(1989), pp. 55–109.
N. Wallach, Real reductive groups. I, Pure and Applied Math, vol 132, Academic Press (1988).

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M. EMERTON, D. NADLER, AND K. VILONEN

Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA