JANTZEN FILTRATIONS

I. Grojnowski

14 Feb 1996

The following technical note is lifted from [Gr0], where it is applied to finite dimensional representations of quantum affine algebras and Nakajima varieties. In it we prove a result comparing various filtrations on algebras arising from geometry. This translates into a representation theoretic property of the filtrations which were first introduced and studied by Jantzen, in the context of Verma modules for Lie algebras.

When applied to the cotangent variety of the flag variety, this translates into the following theorem, which was first conjectured by Rogawski for GL_{n} (in 1985), and later (1987) by Ginzburg for arbitrary affine Hecke algebras.

Theorem (Jantzen conjecture). Let G be a reductive group with simply connected derived group, and H_{G}^{aff}(q) be the associated affine Hecke algebra. This is an algebra over C, where q ∈ C^{⋆}. We suppose q ≠ 1. Let H^{∗}(B_{n}^{s}) be a standard module for the affine Hecke algebra, as defined in [KL], and h ∈ 𝔤 part of an adapted 𝔰𝔩_{2} triple for (s, n). This defines a Jantzen filtration on the standard module. Then the subquotients of this filtration are semisimple, and their multiplicites are given by intersection cohomology Poincare polynomials. In particular, they are independant of the choice of h.

No proof has previously appeared, though the 𝔤-module story is in the wonderful paper [BB]. One should remark that for type A, the IC Poincare polynomials of the theorem are a subset of the affine Kazhdan-Lusztig polynomials of type A.

The deduction of the theorem from this note uses the intepretation of the affine Hecke algebra in equivariant K-homology due to Kazhdan-Lusztig, combined with some observations of Ginzburg. This is now textbook material [CG].

Once can articulate a similar theorem for finite dimensional representations of quantum affine algebras or Yangians. This uses Nakajima’s quiver varieties, and can be found in the monograph [Gr0].

There are two more filtrations one can place on standard modules, the socle and cosocle filtrations. Unlike the theorem of this note, which is a very general result that follows immediately from Deligne’s Hard Lefschetz theorem (as in [BB]), coincidence of the Jantzen filtrations with the socle and cosocle filtrations is a more specific result about the Hodge theory of the variety in questions.

These theorems comparing socle and Jantzen filtrations, and better exposition of the above theorem, will appear in [G0,G1]. This note has been circulating privately since February 1996.

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In the body of this article we define standard modules ^{!}M, ^{∗}M and their Jantzen and grading filtrations on modules arising from geometry, and show that these agree and have dimensions given by coefficients of Kazhdan-Lusztig polynomials.

2.1 We need some linear algebra in the style of [D,1.6].

Throughout this section, let C denote the category of graded C[t]-modules, where deg t = 2. Unless otherwise specified, all morphisms will be in C.

We suppose given an exact sequence in C

0 ⟶ ^{!}M ⟶ ^{∗}M ⟶ Ψ ⟶ 0

with the following properties i) ^{!}M, ^{∗}M are free C[t]-modules ii) If we set ^{!}M̄ = ^{!}M/t^{!}M and ^{∗}M̄ = ^{∗}M/t^{∗}M, then

!M̄^{i} = 0 for i < 0, ^{∗}M̄^{i} = 0 for i > 0.

iii) Multiplication by t^{i} induces an isomorphism Ψ^{−i−1} ⟶^{∼} Ψ^{i−1} for i ≥ 0.

2.2 Define Ξ = ^{∗}M/t^{!}M, so we have an exact sequence

0 ⟶ ^{!}M̄ ⟶ Ξ ⟶ Ψ ⟶ 0.

Then condition (i) implies that ^{!}M̄ = ker(t : Ξ → Ξ) and ^{∗}M̄ = coker(t : Ξ → Ξ).

Lemma. Multiplication by t^{i} induces an isomorphism Ξ^{−i} ⟶^{∼} Ξ^{i} for i > 0.

Proof. Factor this map as

M̄^{i−2}^{!}  M̄^{i}^{!}
↓  ↓
Ξ^{−i} ⟶^{α} Ξ^{i−2} ⟶^{β} Ξ^{i}
π_{−i}↓  ^{π_{i−2}}↓  ^{π_{i}}↓
Ψ^{−i} ⟶^{∼} Ψ^{i−2} ⟶ Ψ^{i}

where α is induced by multiplication by t^{i−1}, β by multiplication by t. By property (iii), the map Ψ^{i−2} → Ψ^{i} is a surjection for i + 1 ≥ 0, and the map Ψ^{−i} → Ψ^{i−2} an isomorphism. As ^{!}M^{i} = 0 for i < 0, π_{−i} is an isomorphism, so α is an injection. As ker β = ker π_{i−2}, βα is an injection. Multiplication by t induces the exact sequence

0 ⟶ Ψ^{i−2} ⟶^{t} Ξ^{i} ⟶ ^{∗}M̄^{i} ⟶ 0

For i > 0, we have ^{∗}M̄^{i} = 0, and so βα is surjective.

Corollary. ^{!}M̄^{i} ⟶^{∼} (t^{i}Ξ ∩ ker t)/(t^{i+1}Ξ ∩ ker t).

Define a decreasing filtration J_{.} on ^{!}M, the Jantzen filtration, by

J_{i} = ^{!}M ∩ t^{i} ^{∗}M

and denote the induced filtration on ^{!}M̄ by J̄_{.}. We can rephrase the corollary as Corollary. Under the conditions (i)-(iii) above, grJ̄ ; i^{!}M̄ ⟶^{∼} ^{!}M̄^{i}.

2.3 Let V be a vector space, λ : C^{∗} → GL(V) an action of C^{∗} with strictly positive weights (i.e. lim_{t→0} λ(t)v = 0 for all v ∈ V). We will consider complexes of sheaves on V equivariant with respect to this C^{∗}-action. Write P_{λ}V = (V \ 0)/C^{∗} for the weighted projective space. This is rationally smooth. Write p : V → 0, i : 0 → V for the obvious maps.

By applying p_{!} (respectively p_{∗}) to the adjunction map i_{!}i^{!} → Id (respectively Id → i_{∗}i^{∗}) we get arrows i^{!} → p_{!} → p_{∗} → i^{∗}. The following lemma is well known.

Lemma. If A is C^{∗}-equivariant on V, then the natural maps i^{!}A ⟶^{∼} p_{!}A, p_{∗}A ⟶^{∼} i^{∗}A are isomorphisms.

Now suppose A is a C^{∗}-equivariant pure perverse sheaf on V. As i^{!}, p_{∗} increase weights and i^{∗}, p_{!} decrease weights [D], the lemma implies p_{!}A and i^{∗}A are pure.

Write j : V \ 0 ↪ V for the inclusion. We have an exact triangle

j_{!}j^{∗}A ⟶ A ⟶ i_{!}i^{∗}A ⟶^{+1}

in D_{C^{∗}}(V). Applying p_{!} we get an exact triangle p_{!}j_{!}j^{∗}A ⟶ p_{!}A ⟶ i^{∗}A ⟶^{+1} in D_{C^{∗}}(pt). Applying H_{C^{∗}}^{∗} to this we get a long exact sequence.

Lemma. i) H_{C^{∗}}^{·}(p_{!}j_{!}j^{∗}A) ⟶^{∼} H^{·−1}(P_{λ}V, Ȧ)

ii) The long exact sequence splits into the short exact sequences

0 ⟶ H_{C^{∗}}^{k}(p_{!}A) ⟶ H_{C^{∗}}^{k}(i^{∗}A) ⟶ H^{k}(P_{λ}V, Ȧ) ⟶ 0.

iii) Write H_{C^{∗}}^{·}(pt) = C[t]. Then the action of t on H^{·}(P_{λ}V, Ȧ) induced from the exact sequence coincides with cup product with a generator of H^{2}(P_{λ}V, C); i.e. with the class of a hyperplane in P_{λ}V.

Proof. First, lets define Ȧ. Write PV = P_{λ}V to lighten the notation. As A is C^{∗}- equivariant, there is a unique sheaf on PV whose ∗-pullback to V \ 0 is isomorphic to A. Denote this sheaf Ȧ. Ȧ is pure, and Ȧ[−1] is perverse.

Choose N ≫ 0. Write

P^{N} ⟵^{α} (V \ 0) ×_{C^{∗}} (C^{N+1} \ 0) ⟶_{∼}^{β} PV × (C^{N+1} \ 0).

Then, writing Ȧ ⊠ C for the obvious sheaf on PV × (C^{N+1} \ 0), we have for k ≪ N

H_{C^{∗}}^{k}(p_{!}j_{!}j^{∗}A) = H^{k}(P^{N}, α_{!}β^{∗}(Ȧ ⊠ C))

by the definition of equivariant cohomology. But this last group is clearly isomorphic to H^{k−1}(PV, A) ⊗ H^{1}(C^{N+1} \ 0, C) = H^{k−1}(PV, Ȧ). Whence (i).

For (ii), recall that if A is pure of weight w, then i^{∗}A and p_{!}A are pure of weight w, as is Ȧ. But then in the long exact sequence

⟶ H^{k−1}(PV, Ȧ) ⟶^{δ_{k}} H_{C^{∗}}^{k}(p_{!}A) ⟶ H_{C^{∗}}^{k}(i^{∗}A) ⟶ H^{k}(PV, Ȧ) ⟶

δ_{k} must be zero, as H^{k−1}(PV, Ȧ) has weight k − 1, and H_{C^{∗}}^{k}(p_{!}A) has weight k.

Finally, take A = C, the constant sheaf on V. Then this exact sequence for k = 2 gives (iii).

Write ^{!}M = H_{C^{∗}}(p_{!}A), ^{∗}M = H_{C^{∗}}(i^{∗}A).

Proposition. Properties (i)–(iii) of §2 hold for ^{!}M, ^{∗}M. We have ^{!}M̄ ⟶^{∼} H^{·}(p_{!}A) = H_{c}^{·}(V, A) and ^{∗}M̄ ⟶^{∼} H^{·}(i^{∗}A) = H^{·}(V, A). If J̄ denotes the Jantzen filtration on

!M̄, then grJ̄ ; i^{!}M̄ = H^{i}(p_{!}A).

Proof. By purity of p_{!}A, p_{!}A ⟶^{∼} ⊕_{j}H^{j}(p_{!}A)[−j] in D_{C^{∗}}(pt). Hence

!M = ⊕_{j}H^{j}(p_{!}A)[−j] ⊗ C[t],

and ^{!}M is free and ^{!}M̄ = H^{·}(p_{!}A). ^{∗}M is similar. As A is perverse, H^{j}(i^{!}A) = 0 if j < 0, and H^{j}(i^{∗}A) =) if j > 0. So property (ii) holds. Finally, if A is perverse and pure, Ȧ[−1] is perverse and pure, and property (iii) is the Hard Lefschetz theorem [D,6.2.13], which is true for pure perverse sheaves on rationally smooth projective varieties.

The last statement of the proposition then follows from the corollary of §2.2.

2.4 We present a few complements to the ideas of Ginzburg [CG].

Let π : X → Y be a proper map, and F be a pure perverse sheaf on X of geometric origin. Set A = ⊕_{i}RHom_{DY}^{i} (π_{!}F, π_{!}F). By the decomposition theorem, there exists an isomorphism

(**)  π_{!}F ⟶^{∼} ⊕_{ϕ,i}L_{ϕ,i} ⊗ L_{ϕ}^{#}[i]

where L_{ϕ,i} are vector spaces, and L_{ϕ}^{#} are irreducible perverse sheaves on Y (so of the form j_{!∗}L_{ϕ}[dim Z], where j : Z ↪ Y is a locally closed smooth subvariety, and L_{ϕ} is a local system on Z).

Define \overline{π_{!}F} = ⊕_{i}^{p}H^{i}(π_{!}F) and Ā = ⊕_{k}RHom^{k}(\overline{π_{!}F}, \overline{π_{!}F}) = ⊕Ā^{k}. Then an isomorphism (**) defines an isomorphism between A and Ā:

A = ⊕_{α}RHom^{α}(π_{!}F, π_{!}F) ⟶^{∼} ⊕_{ϕ,ψ,i,j,α}RHom^{α+j−i}(L_{ϕ}^{#}, L_{ψ}^{#}) ⊗ Hom(L_{ϕ,i}, L_{ψ,j})

⟶^{∼} ⊕_{ϕ,ψ,k}RHom^{k}(L_{ϕ}^{#}, L_{ψ}^{#}) ⊗ Hom(L_{ϕ}, L_{ψ})

⟶^{∼} ⊕_{k}RHom^{k}(\overline{π_{!}F}, \overline{π_{!}F}) = Ā

where, for a perverse sheaf L_{ϕ}^{#} on Y, L_{ϕ} = RHom^{0}(L_{ϕ}^{#}, \overline{π_{!}F}) ⟶^{∼} ⊕_{i}L_{ϕ,i} is naturally an Ā^{0}-module. As observed by Ginzburg, ⊕_{k>0}Ā^{k} is the radical of Ā, and so the set of non-zero L_{ϕ} are the simple Ā-modules.

Let x ∈ Y, i_{x} : \{x\} ↪ Y the inclusion. Let us write ^{∗}M̄^{i} = ^{p}H^{i}(i_{x}^{∗}\overline{π_{!}F}), ∗M̄ = ⊕_{i} ^{∗}M̄^{i}. Then if α ∈ RHom^{k}(\overline{π_{!}F}, \overline{π_{!}F}), applying the functor ^{p}H^{i}i_{x}^{∗} to α we get a map

pH^{i}(i_{x}^{∗}\overline{π_{!}F}) ⟶ ^{p}H^{i}(i_{x}^{∗}\overline{π_{!}F}[k]) = ^{p}H^{i+k}(i_{x}^{∗}\overline{π_{!}F})

and so a graded action of Ā on ^{∗}M̄; i.e. Ā^{i} ^{∗}M̄^{j} ⊆ ^{∗}M̄^{i+j}. Define a filtration on ∗M̄, the grading filtration by ^{∗}M̄^{≥k} = ⊕_{j≥k} ^{∗}M̄^{j}. We clearly have

Lemma. gr_{k} ^{∗}M̄ ⟶^{∼} ^{∗}M̄^{k}, and gr_{.} ^{∗}M̄ is a semisimple Ā-module.

Now, \overline{π_{!}F} = ⊕L_{ϕ} ⊗ L_{ϕ}^{#}, so i_{x}^{∗}\overline{π_{!}F} = ⊕L_{ϕ} ⊗ i_{x}^{∗}L_{ϕ}^{#} and so

∗M̄^{i} = ⊕_{ϕ}L_{ϕ} ⊗ H^{i}(i_{x}^{∗}L_{ϕ}^{#}).

Similarly, defining ^{!}M̄^{i} = ^{p}H^{i}(i_{x}^{!}\overline{π_{!}F}), we have ^{!}M̄ is a graded Ā-module and

!M̄^{i} = ⊕_{ϕ}L_{ϕ} ⊗ H^{i}(i_{x}^{!}L_{ϕ}^{#}).

Finally, observe that choosing an isomorphism (**), we get H_{c}^{∗}(i_{x}^{∗}π_{!}F) is isomorphic to ^{∗}M̄, as

⊕_{k}H^{k}(i_{x}^{∗}π_{!}F) ⟶^{∼} ⊕_{ϕ,i,k}L_{ϕ,i} ⊗ H^{k+i}i_{x}^{∗}L_{ϕ}^{#} = ^{∗}M̄.

2.5 Suppose x ∈ Y has the property that there exists a subvariety V, x ∈ V ⊆ Y such that

i) V is in generic position with respect to π_{!}F; i.e. the map i_{V}^{!} π_{!}F[2d](d) → i_{V}^{∗} π_{!}F is an isomorphism, where d is the codimension of V in Y, and

ii) There exists an isomorphism C^{r} ⟶^{∼} V sending 0 to x and such that i_{V}^{!} π_{!}F is C^{∗}-equivariant for some given C^{∗}-action on C^{r} with strictly positive weights.

We call such a V a transverse slice. In this case, applying §2.3,2.4 to A = i_{V}^{!} π_{!}F[d] we obtain

Theorem. The Jantzen filtration on ^{!}M̄ coincides with the grading filtration, up to shift, and its (i − d)’th graded piece is ^{!}M̄^{i} = ⊕_{ϕ}L_{ϕ} ⊗ H^{i}(i_{x}^{!}L_{ϕ}^{#}). In particular, the subquotients of the Jantzen filtration are semisimple Ā-modules, and the Jantzen filtration is independant of the choice of the transverse slice V and C^{∗}-action. Similar statements hold for the Jantzen filtration on ^{∗}M̄.

2.6 The above story now applies to affine Hecke algebras (for any q), as well as Lusztig’s graded affine Hecke algebras (for any cuspidal), quantum affine algebras, …, as there are well known transverse slices in each of these cases.

References

[BB] A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, Adv. in Soviet Math. 16 (1993), 1–50.

[CG] N. Chriss and V. Ginzburg, Representation theory and Complex Geometry, Birkhauser, 1997.

[D] P. Deligne, La conjecture de Weil II, Publ. IHES 52 (1980), 137–252.

[Gr0] I. Grojnowski, Representations of quantum affine algebras, book in preparation.

[Gr1] I. Grojnowski, Affine and cyclotomic Hecke algebras at roots of unity, unfinished preprint.

[KL] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math 87 (1987), 153–215.