Lefschetz operator and local Langlands mod ℓ: the regular case

Jean-François Dat^{∗}

To the memory of Hiroshi Saito

Abstract Let p and ℓ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod ℓ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’ local Langlands correspondence modulo ℓ for unipotent representations.

Contents

1 Main theorem 1

2 Elliptic principal series 3 2.1 Parametrization and decomposition matrix . . . . . . . . . . . . . . . . . . . . . 3 2.2 Corresponding representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Computation of some Ext groups . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 The cohomology complex 14 3.1 The unipotent block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Proof of the main theorem 21 4.1 Computation of R_{π}^{∗} for π an elliptic principal series . . . . . . . . . . . . . . . . 21 4.2 The Lefschetz operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1 Main theorem

Let K be a local non-archimedean field with ring of integers O and residue field k ≃ 𝔽_{q}, q a power of a prime p. Let ℓ be another prime number and d an integer. As in [10], we consider the cohomology complex

RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) ∈ D^{b}(Rep_{ℤ_{ℓ}}^{∞,c}(G × D^{×} × W_{K}))

∗The author acknowledges support from the Institut Universitaire de France and the Agence Nationale de la Recherche under grant ANR-10-BLANC-0114 “ArShiFo”.

of the height d Lubin-Tate tower of K. Here G = GL_{d}(K), D is the division algebra which is central over K with invariant 1/d, and W_{K} is the Weil group of K. The category Rep_{ℤ_{ℓ}}^{∞,c} consists of ℤ_{ℓ}-representations of the triple product which are smooth for G and D^{×} and continuous for W_{K}. In [9], we defined a Lefschetz operator

L : RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) ⟶ RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ})[2](1)

as the cup-product by the Chern class of the tautological invertible sheaf on the associated Gross-Hopkins period domain.

To an irreducible \overline{𝔽}_{ℓ}-representation π of G, we associate its “derived π-coisotypical part”

R_{π} := RHom_{ℤ_{ℓ}G}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}), π) ∈ D^{b}(Rep_{\overline{𝔽}_{ℓ}}^{∞}(D^{×} × W_{K}))

which inherits a morphism L_{π} : R_{π} ⟶ R_{π}[2](1). We also denote by R_{π}^{∗} the total cohomology of R_{π}, a smooth graded \overline{𝔽}_{ℓ}-representation of W_{K} × D^{×}, and by L_{π}^{∗} : R_{π}^{∗} ⟶ R_{π}^{∗}[2](1) the corresponding morphism. Our aim here is to prove the following theorem, where we forget the grading.

Theorem.– Assume that the multiplicative order of q mod ℓ is d. Then for any unipotent irreducible representation π of GL_{d}(K), there is an isomorphism

(R_{π}^{∗}, L_{π}^{∗})^{ss} ≃ |LJ(π)| ⊗ (σ^{ss}(π), L(π)).

The congruence condition on q modulo ℓ will be called the Coxeter congruence relation, by analogy with the modular Deligne-Lusztig theory where this condition arises in the context of Broué’s conjecture, see [15] for example. The terminology unipotent was introduced by Vignéras to denote representations that belong to the same block as the trivial representation. Finite group theorists would rather call them principal block representations.

Let us explain the notation of the theorem. The symbol LJ(π) stands for the Langlands-Jacquet transfer of [11]. In general it is a virtual \overline{𝔽}_{ℓ}-representation of D^{×}, but under the congruence hypothesis, it is known to be effective up to sign, cf [11, (3.2.5)], so we can put LJ(π) = ±|LJ(π)| for some semi-simple \overline{𝔽}_{ℓ}-representation of D^{×}. The symbol (σ^{ss}(π), L(π)) denotes the (transposed) Weil-Deligne \overline{𝔽}_{ℓ}-representation associated to π by the Vignéras correspondence of [26, Thm 1.8.2]. This is the Zelevinski-like normalization of the local Langlands correspondence mod ℓ. Therefore, to put it in english words, the above theorem offers a geometric interpretation of the nilpotent part^{1} of this Vignéras correspondence, at least for those unipotent representations such that LJ(π) ≠ 0.

Let us say a few words about the proof of the theorem. Note first that, since LJ(π) is most often 0, we are soon reduced to the case when π is a subquotient of the smooth representation Ind_{B}^{G} (\overline{𝔽}_{ℓ}) induced from the trivial representation of some Borel subgroup. In section 2 we classify these subquotients in a suitable way, making thereby explicit the corresponding block of the decomposition matrix of G, and we compute the associated Weil-Deligne and D^{×} representations. In section 3, we study the unipotent summand of the cohomoloy complex. In particular, thanks to our congruence hypothesis, we may split it in a non-trivial way according to weights. Note that, in principle, all this study can be carried out in a purely local way, using Yoshida’s model of the tame Lubin-Tate space. However, for reference convenience, we invoke at some point Boyer’s description of the cohomology of the whole tower in [3], the proof of which uses global arguments. An alternative argument uses the Faltings-Fargues isomorphism of [16]. Then in section 4 we prove the theorem, by some fairly explicit computations.

1Note that, in contrast to the ℓ-adic setting, this nilpotent part has no obvious arithmetic interpretation, in the sense that it cannot be related to any infinitesimal action of the ℓ-inertia of W_{K}.

One crucial ingredient is that we easily, and without any computation, get a complete description of (R_{π}^{∗}, L_{π}^{∗}) for π the trivial representation, thanks to the properties of the Gross-Hopkins period map. The theorem above is expected to hold true for any smooth irreducible \overline{𝔽}_{ℓ}-representation π under the congruence hypothesis, but we are still missing some control on the pair (R_{π}^{∗}, L_{π}^{∗}) when π is a general Speh representation.

Remark.– The theorem is also true under the wider assumption that q has order ⩾ d. This wider assumption is what we have called the regular case in the title, a terminology which comes from the fact that in this case, the modulus character (mod ℓ) of the Borel subgroup is indeed regular w.r.t the action of the Weyl group. This regular case splits into the Coxeter congruence case explained above and the banal case. The latter is not treated in this paper because the overall strategy and all the necessary computations work exactly the same as in the ℓ-adic case. Note however that even in the banal case we are not able to go beyond unipotent representations regarding the computation of the Lefschetz operator.

2 Elliptic principal series

By definition, an irreducible smooth \overline{𝔽}_{ℓ}-representation is called elliptic if it is not a linear combination of proper parabolically induced representations. Note that by [11, Thm 3.1.4], this is equivalent to LJ(π) ≠ 0. According to [11, Cor 3.2.2]^{2}, any elliptic principal series is an unramified twist of a subquotient of the induced representation Ind_{B}^{G} (\overline{𝔽}_{ℓ}) for some Borel subgroup B. The converse is not true in general, but it is true under the Coxeter congruence relation, as we will see below.

2.1 Parametrization and decomposition matrix

(2.1.1) A reminder on the ℓ-adic case. We denote by B the subgroup of upper triangular matrices in G and by S the set of simple roots of the diagonal torus T in Lie(B). The power set P(S) of S is in 1-to-1 correspondence with the set of parabolic subgroups containing B. Namely, to each subset I ⊂ S is associated the unique parabolic subgroup P_{I} with Lie(P_{I}) = Lie(B) + ∑_{α∈ℤ⟨I⟩} Lie(G)_{α}. In particular we have P_{∅} = B and P_{S} = G.

Definition.– For any ring R, we put i_{I}(R) := Ind_{P_{I}}^{G} (R), and

v_{I}(R) := i_{I}(R)/ ∑_{J⊃I} i_{J}(R).

Let δ_{B} denote the R-valued modulus character of B. We assume that R contains a square root of q in R and we choose such a root in order to define δ := δ_{B}^{− \frac{1}{2}} as well as the normalized Jacquet functor r_{B} along B. Write X := X_{∗}(T) ⊗ ℝ, so that S is naturally a subset of the dual ℝ-vector space of X. Following [5, 2.2.3], we associate to each subset I ∈ P(S) a connected component

2where unfortunately the term elliptic has a slightly different meaning

X_{I} := {x ∈ X, ∀α ∈ S, ε_{I}(α)⟨x, α⟩ > 0}

of the complement of the union of simple root hyperplanes in X. Here ε_{I} is the sign function on S which takes α to −1 if and only if α ∈ I. In particular, X_{∅} is the Weyl chamber associated to B and X_{S} is that associated to the opposite Borel subgroup.

Fact. ([5], Lemme 2.3.3)– If R is a field of characteristic prime to ∏_{i=1}^{d}(q^{i} − 1), then the following hold :

i) for each I ⊆ S, the R-representation v_{I}(R) is irreducible and we have

r_{B}(v_{I}(R)) = ⨁_{w(X_{S})⊆X_{I}} w^{−1}(δ).

ii) the multiset JH(Ind_{B}^{G}(R)) of Jordan-Hölder factors of Ind_{B}^{G}(R) has multiplicity one, hence

is a set, and the map

I ∈ P(S) ↦ v_{I}(R) ∈ JH(Ind_{B}^{G} (R))

is a bijection.

Let us label t_{0}, t_{1}, ⋯, t_{d−1} the diagonal entries of an element t ∈ T (starting from the upper left corner). We get a labeling S = {α_{1}, ⋯, α_{d−1}}, where α_{i}(t) := t_{i−1}t_{i}^{−1}, and we get an identification of the Weyl group W of T with the symmetric group 𝔖_{d} of the set {0, ⋯, d − 1}. Then we see that the condition w(X_{S}) ⊆ X_{I} appearing in the summation of point i) above is equivalent to the condition

I = {α_{i} ∈ S, w(i − 1) < w(i)}.

(2.1.2) Classification under the Coxeter congruence relation. Here the coefficient field is 𝔽_{ℓ} or \overline{𝔽}_{ℓ} and we assume that the multiplicative order of q modulo ℓ is exactly d. Denote by ν_{G} the unramified character g ↦ q^{−val(det)(g)}. Observe that ν_{G} is trivial on the center of G and generates a cyclic subgroup ⟨ν_{G}⟩ of order d of the group of \overline{𝔽}_{ℓ}-valued characters of G.

We put S̃ := S ∪ {α_{0}} where α_{0} denotes the opposite of the longest root of T in Lie(B). Thus, if the diagonal entries of t ∈ T are t_{0}, t_{1}, ⋯, t_{d−1} as above, then α_{0}(t) = t_{d−1}t_{0}^{−1}. Note that S̃ is stable under the action of the Coxeter element c of W = 𝔖_{d} which takes i < d − 1 to i + 1 and d − 1 to 0. In fact S̃ is a principal homogeneous set under the cyclic subgroup ⟨c⟩ of order d generated by c. Therefore, it is convenient to identify {0, ⋯, d − 1} with ℤ/dℤ through the canonical bijection, so that we simply have

c(α_{i}) = α_{i+1}, ∀i ∈ ℤ/dℤ.

We denote by P′(S̃) the set of strict subsets of S̃. For any I ∈ P′(S̃) we can thus choose an i ∈ S̃ \ I. The translated subset c^{−i}(I) is then contained in S so we can consider the representation i_{c^{−i}I}(\overline{𝔽}_{ℓ}) ⊗ ν_{G}^{i}.

Lemma.– Up to semisimplification, the representation i_{c^{−i}I}(\overline{𝔽}_{ℓ}) ⊗ ν_{G}^{i} is independent of the choice of i in S̃ \ I. We denote by [i_{I}] its class in the Grothendieck group R(G, \overline{𝔽}_{ℓ}).

Proof. Up to translation by a power of c, we may assume that I ⊆ S, so that we have to compare i_{I}(\overline{𝔽}_{ℓ}) with i_{c^{−i}I}(\overline{𝔽}_{ℓ}) ⊗ ν_{G}^{i} (assuming that i ∉ I). Note that the parabolic subgroups P_{I} and P_{c^{−i}I} are associated. More precisely, any element of the normalizer of T in G which projects to c^{i} will conjugate the Levi component M_{c^{−i}I} to M_{I}. Therefore, [7, Lemme 4.13] shows that in the Grothendieck group R(G, \overline{𝔽}_{ℓ}) we have the equality [i_{c^{−i}I}(1)] = [i_{I}(γ)] with γ = (δ_{P_{I}}c^{i}(δ_{P_{c^{−i}I}}^{−1}))^{\frac{1}{2}} . Thus we have to prove that γ = ν_{G}^{−i}_{|M_{I}}. Since restriction to T is injective on characters of M_{I}, we may restrict both sides to T. Using that δ_{P_{I}}_{|T} = δ_{B}δ_{B∩M_{I}}^{−1}, we get that γ_{|T} = (δ_{B}.c^{i}(δ_{B}^{−1}))^{\frac{1}{2}} .

For k = 0, ⋯, d − 1, consider the smooth character of T defined by ε_{k}(t) = q^{−val(t_{k})} where t_{0}, ⋯, t_{d−1} are the diagonal entries of t ∈ T. Then we have

γ_{|T} = (∏_{k<l} ε_{k}ε_{l}^{−1} ∏_{k<l} ε_{k+i}^{−1}ε_{l+i})^{\frac{1}{2}} = ∏_{k<i⩽l} ε_{k}ε_{l}^{−1} = ∏_{k<i} ε_{k}^{d−i} ∏_{i⩽l} ε_{l}^{−i}.

Of course in the first equality, the indices k + i and l + i should be read modulo d. To get the second equality, we observe that for 0 ⩽ k < l < d, we have k + i > l + i(mod d) ⇔ k < d − i ⩽ l ⇔ l + i < i ⩽ k + i. Now using the fact that ε_{k}^{d} = 1, we get γ_{|T} = ∏_{k} ε_{k}^{−i} = (ν_{G}^{−i})_{|T} as desired.

Remark.– The particular case I = ∅, i = 1 of the above lemma tells us that [Ind_{B}^{G} (\overline{𝔽}_{ℓ})] = [Ind_{B}^{G}(\overline{𝔽}_{ℓ})⊗ν_{G}]. So the twisting action of the cyclic group ⟨ν_{G}⟩ on the set of classes of irreducible representations preserves the multiset JH(Ind_{B}^{G} (\overline{𝔽}_{ℓ})).

We now want to isolate a certain irreducible constituent of [i_{I}]. We follow the Zelevinski approach via degenerate Whittaker models, as in Vignéras’ work. First we associate a partition λ_{I} of d to I in the following way. As in the previous proof, I determines a conjugacy class of Levi subgroups, namely that of M_{c^{−i}I} for any i ∈ S̃ \ {i}. This conjugacy class corresponds to a partition μ_{I} of d, and we let λ_{I} be the transpose of μ_{I}. For example, λ_{∅} = (d), and λ_{I} = (1, ⋯, 1) whenever |I| = d − 1.

We refer to [27, III.1] for the basics on the theory of derivatives and to [25, V.5] for the notion of degenerate Whittaker models.

Fact.– For I ∈ P′(S̃), the representation [i_{I}] has a unique irreducible constituent π_{I} admitting a λ_{I}-degenerate Whittaker model. Moreover any other irreducible constituent has λdegenerate Whittaker models only for λ < λ_{I}.

Remark.– By the Lemma above, for any I ∈ P′(S̃) we have π_{c^{i}I} ≃ π_{I} ⊗ ν_{G}^{i}. In particular for any i ∈ S̃, we have π_{S̃\{i}} = ν_{G}^{i}. On the other hand π_{∅} is the only generic constituent of Ind_{B}^{G} (\overline{𝔽}_{ℓ}).

Before proceeding, we introduce some more notation. Similarly to the banal case, we consider the complement in X of the union of all hyperplanes attached to the roots in S̃. Its connected components are labeled by proper subsets of S̃, and given by

X̃_{I} := {x ∈ X, ∀α ∈ S̃, ε_{I}(α)⟨x, α⟩ > 0}

where ε_{I} is the sign function attached to I as before. Note that X̃S̃ = X̃_{∅} = ∅ and that X̃_{S} = X_{S} is again the opposite Weyl chamber to B. However for I a strict subset of S, we have X̃_{I} ≠ X_{I}.

Proposition.– i) The multiset JH(Ind_{B}^{G} (\overline{𝔽}_{ℓ})) is a set (multiplicity one).

ii) The map I ∈ P′(S̃) ↦ π_{I} ∈ JH(Ind_{B}^{G} (\overline{𝔽}_{ℓ})) is a bijection.

iii) For all I ∈ P′(S̃) the following equality holds in R(G, \overline{𝔽}_{ℓ}) :

[i_{I}] = ∑_{J⊇I}[π_{J}].

iv) π_{∅} is a cuspidal representation, and if I ≠ ∅ then

r_{B}(π_{I}) = ⨁_{w(X̃_{S})⊂X̃_{I}} w^{−1}(δ).

Proof. i) Suppose π is a non-cuspidal irreducible subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}). Let P = M_{P} U_{P} be a parabolic subgroup such that π_{U_{P}} ≠ 0. Since ℓ is banal for M_{P}, the Mackey formula (or “geometric lemma”) shows that in fact π_{U_{B}} ≠ 0. But the congruence relation and the Mackey formula imply that Ind_{B}^{G} (\overline{𝔽}_{ℓ})_{U_{B}} has the multiplicity one property, as a representation of T.

More precisely, we have r_{B}(Ind_{B}^{G} (\overline{𝔽}_{ℓ})) = ⨁_{w∈W} w(δ), and δ = νG^{\frac{1−d}{2}} ∏_{i=0}^{d−1} ε_{i}^{i} is a W -regular character since q has order d. Hence π occurs with multiplicity one in Ind_{B}^{G} (\overline{𝔽}_{ℓ}). Now any cuspidal representation is generic, so there is at most one cuspidal subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}).

ii) This follows from the proof of [11, Prop 3.2.4]. However the latter reference rests on Vignéras’ classification [25, V.12], so in particular on a difficult result of Ariki’s on the classification of simple modules of Hecke-Iwahori algebras at roots of unity. In fact, in our context the latter can be avoided and replaced by the more elementary partial classification of [23, 2.17]. Nevertheless, for the convenience of the reader, we sketch a complete and more direct proof.

Let us first show the injectivity of the map. Let π be some irreducible subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}), and let λ = λ_{π} be the partition of d obtained from π by taking successive higher derivatives. Hence λ_{1} is the order of the highest non-zero derivative of π, λ_{2} is that of the derivative π^{(λ_{1})}, etc. The partition λ is the greatest element in the set of all partitions λ′ such that r_{λ′}(π) admits a generic subquotient. Here r_{λ′} denotes the normalized Jacquet functor associated to the standard parabolic subgroup P_{λ′} = U_{λ′}M_{λ′} associated to λ′. Let τ denote a generic subquotient of r_{λ}(π). We can write τ = γ(A_{1}) ⊗ ⋯ ⊗ γ(A_{|λ|}) where A_{1} ⊔ ⋯ ⊔ A_{|λ|} = {0, ⋯, d−1} is a set-theoretical partition with |A_{i}| = λ_{i}, and for any subset A ⊂ {0, ⋯, d−1}, γ(A) denotes the unique generic subquotient of the normalized induction ×_{a∈A}ν^{a− \frac{d−1}{2}} (so this is a representation of GL_{|A|}(K)). If π = π_{I} for some I, then λ = λ_{I} and a computation shows that for k = 1, ⋯, |λ|,

A_{k} = {a ∈ {0, ⋯, d − 1}, {α_{a}, c^{−1}α_{a}, ⋯, c^{2−k}α_{a}} ⊂ I}
= {a ∈ ℤ/dℤ, {α_{a}, α_{a−1}, ⋯, α_{a−k+2}} ⊂ I}.

In particular, the following holds :

a) for all k = 1, ⋯, |λ| − 1, we have A_{k+1} ⊂ c(A_{k}) = A_{k} + 1

b) I = {α_{i} ∈ S̃, i ∉ A_{1}}.

Hence we see that π_{I} determines I, so that the map in point ii) is injective.

In order to prove the surjectivity, it is enough to prove that Ind_{B}^{G} (\overline{𝔽}_{ℓ}) has at most 2^{d} − 2 irreducible non cuspidal constituents. If π is such a constituent, there is a Borel subgroup B′ such that π is the unique irreducible quotient of the normalized induced representation i_{B′}^{G} (δ). However, the same argument as in [5, 2.5.4] shows that if both the chambers C(B′) and C(B′′) are contained in a component X̃_{I}, then the canonical interwining operator i_{B′}^{G} (δ) ⟶ i_{B′′}^{G} (δ) is an isomorphism. Indeed, we may assume as in loc.cit that B′ and B′′ are adjacent, with wall associate to some root r. Then the representation theory for GL_{2} (note that ℓ is banal w.r.t to GL_{2}(K)) tells us that the canonical intertwining operator is invertible unless q^{l(r)} = q^{±1} in \overline{𝔽}_{ℓ}, where l(r) is the height of the root. With our congruence hypothesis and the general inequality l(r) ⩽ n − 1, this implies that l(r) is either 1, −1 or n − 1, which is equivalent to ±r ∈ S̃. This gives the desired bound.

iii) By ii), we only have to show that if J is any other strict subset of S̃, then π_{J} occurs in [i_{I}] if and only if J ⊇ I. Start with J ⊇ I and choose i ∈ S̃ \ J. Then we have i_{c^{−i}J}(𝔽_{ℓ}) ⊗ ν_{G}^{i} ⊂ i_{c^{−i}I}(𝔽_{ℓ}) ⊗ ν_{G}^{i}, so π_{J} occurs in [i_{I}]. Conversely, suppose that π_{J} occurs in [i_{I}]. Assume first that J ∪ I ≠ S̃ and choose i ∈ S̃ \ (J ∪ I). Then we see that [π_{J} ⊗ ν_{G}^{−i}] occurs in i_{c^{−i}I}(𝔽_{ℓ}) ∩ i_{c^{−i}J}(𝔽_{ℓ}) = i_{c^{−i}(I∪J)}(𝔽_{ℓ}). Hence λ_{J} ⩽ λ_{I∪J}, so λ_{J} = λ_{I∪J} and finally I ∪ J = J, as desired. Assume now that J ∪ I = S̃, choose j ∈ J \ I and set J^{∗} := J \ {j}. We get that [π_{J} ⊗ ν_{G}^{−j}] occurs in i_{c^{−j}I}(𝔽_{ℓ}) ∩ i_{c^{−j}J^{∗}}(𝔽_{ℓ}) = i_{c^{−j}(I∪J^{∗})}(𝔽_{ℓ}) = i_{S̃\{0}} = π_{S̃\{0}}. Hence π_{J} = π_{S̃\{j}}, so J = S̃ \ {j} which is impossible by definition of j.

iv) We proved in point ii) that Ind_{B}^{G} (\overline{𝔽}_{ℓ}) has exactly 2^{d} − 2 non-cuspidal irreducible subquotients. But we constructed 2^{d} − 1 constituents, so Ind_{B}^{G} (\overline{𝔽}_{ℓ}) has exactly one cuspidal subquotient. We know it is generic, so it is, by definition, π_{∅}. Now, fix some proper subset I of S̃. Again by the proof of the surjectivity in point ii), there is a unique proper subset J such that

r_{B}(π_{I}) = ⨁_{w(X̃_{S})⊂X̃_{J}} w^{−1}(δ).

We still have to prove that I = J. Note that the condition w(X̃_{S}) ⊂ X̃_{J} is equivalent to

J = {α_{j} ∈ S̃, wc^{−1}(j) < w(j)}.

Now, items a) and b) in the proof of the injectivity in ii) show that I ⊆ J. Since the map I ↦ J is a bijection, it has to be the identity.

(2.1.3) The decomposition matrix for elliptic representations. Recall that an admissible smooth \overline{ℚ}_{ℓ}-representation π of G is called ℓ-integral if it contains a G-stable \overline{ℤ}_{ℓ}-lattice. Then it is known that the reduction to \overline{𝔽}_{ℓ} of such a lattice only depends on π up to semisimplification, see [27, II.5.1.b]. We denote by r_{ℓ}(π) the semisimple \overline{𝔽}_{ℓ}-representation thus obtained.

Proposition.– Let I ⊆ S. Then we have

r_{ℓ}(v_{I}(\overline{ℚ}_{ℓ})) = [v_{I}(\overline{𝔽}_{ℓ})] = \{ \begin{matrix}[π_{I}] + [π_{I∪\{0\}}] & if & I ≠ S \\ [π_{S}] & if & I = S \end{matrix}

Proof. Since parabolic induction commutes with inductive limits, we have i_{I}(R) ≃ i_{I}(ℤ) ⊗ R for any ring R. By its definition as a quotient v_{I}(R) = i_{I}(R)/ ∑_{J⊃I} i_{J}(R), we also have v_{I}(R) = v_{I}(ℤ) ⊗ R. Now, by [21, Coro 4. 5], we know that v_{I}(ℤ) is free over ℤ. The first equality follows.

If I = S, we have v_{S}(\overline{𝔽}_{ℓ}) = π_{S} = \overline{𝔽}_{ℓ} (trivial representation), so the second equality is clear in this case. Assume I ≠ S. By [21, Prop 6.13], the following simplical complex is exact :

(2.1.3.1)  0 ⟶ i_{S}(ℤ) ⟶ ⋯ ⟶ ⨁_{J⊃I,|J|=|I|+1} i_{J}(ℤ) ⟶ i_{I}(ℤ) ⟶ v_{I}(ℤ) ⟶ 0.

Since it consists of free ℤ-modules, it remains exact after base change to \overline{𝔽}_{ℓ}. Thus we get the equality

[v_{I}(\overline{𝔽}_{ℓ})] = ∑_{S⊇J⊇I}(−1)^{|J\I|}[i_{J}(\overline{𝔽}_{ℓ})]

in R(G, \overline{𝔽}_{ℓ}). On the other hand, Proposition (2.1.2) iii) provides us with the equality

(2.1.3.2)  [π_{I}] = ∑_{S̃⊃J⊇I}(−1)^{|J\I|}[i_{J}].

Thus we get

[π_{I}] − [v_{I}(\overline{𝔽}_{ℓ})] = ∑_{S̃⊃J⊇I∪\{0\}}(−1)^{|J\I|}[i_{J}] = −[π_{I∪\{0\}}].

Alternatively, one could have used point iv) in Proposition (2.1.2), and the easy fact that for any I ⊂ S, we have X_{I} = X̃_{I} ∪ X̃_{I∪{0}}.

2.2 Corresponding representations

(2.2.1) Langlands-Jacquet transfer. We refer to [11] for the definition of the Langlands-Jacquet transfer map LJ_{\overline{𝔽}_{ℓ}} : R(G, \overline{𝔽}_{ℓ}) ⟶ R(D^{×}, \overline{𝔽}_{ℓ}), which is induced by carrying Brauer characters through the usual bijection between regular elliptic conjugacy classes of G and D^{×}. We will need the \overline{𝔽}_{ℓ}-valued unramified character ν_{D} : d ↦ q^{−val∘Nrd(d)} of D^{×}.

Proposition.– For any strict subset I ⊂ S̃ we have

LJ_{\overline{𝔽}_{ℓ}}(π_{I}) = (−1)^{|I|} ∑_{j∈S̃\I}[ν_{D}^{j}].

Proof. Since the map LJ_{\overline{𝔽}_{ℓ}} kills all parabolically induced representations [11, Thm 3.1.4], equality (2.1.3.2) shows that

LJ_{\overline{𝔽}_{ℓ}}(π_{I}) = (−1)^{|S̃\I|+1} ∑_{j∈S̃\I} LJ_{\overline{𝔽}_{ℓ}}(π_{S̃\\{j\}}).

On the other hand π_{S̃\\{j\}} = ν_{G}^{j} = r_{ℓ}(v_{S}(\overline{ℚ}_{ℓ})) ⊗ ν_{G}^{j}. By compatibility of the LJ maps with reduction modulo ℓ [11, Thm 1.2.3] and with torsion by characters, we get that LJ_{\overline{𝔽}_{ℓ}}(π_{S̃\\{j\}}) = (−1)^{|S|}[ν_{D}^{j}].

(2.2.2) Different operations on Weil-Deligne representations. Before we proceed to a description of the Galois-type representations attached to the π_{I}’s, we need to make precise some formal properties of Weil-Deligne representations.

It is convenient to work in a fairly general setting. So let C be an essentially small, artinian, noetherian, abelian category and let C^{ss} be the full subcategory of semisimple objects. The Jordan-Hölder theorem yields a map

Ob(C)_{/∼} ⟶ K_{+}(C), V ↦ [V]

from the set of isomorphism classes of objects to the free monoid on simple objects. This map induces a bijection Ob(C^{ss})_{/∼} ⟶^{∼} K_{+}(C).

Assume further that C is endowed with an automorphism V ↦ V (1) and denote by V ↦ V (n) its n^{th} iteration. Consider the category N (C) with objects all pairs (V, N) with V ∈ Ob(C) and N : V ⟶ V (−1) a nilpotent morphism. With the obvious notion of morphisms, N (C) is an artinian, noetherian, abelian category. The formalism of Deligne’s filtration [12, (1.6)] yields a map

Ob(N (C))_{/∼} ⟶ K_{+}(C)^{(ℕ)}, (V, N) ↦ [V, N]

where the RHS is the set of almost zero sequences of elements in K_{+}(C). Namely, we put [V, N] := ([P_{−n}^{N}(V)])_{n∈ℕ} where P_{i}^{N} is the primitive part of the i-graduate of Deligne’s filtration attached to N. We leave the reader check the following fact.

Lemma.– The map (V, N) ↦ [V, N] induces a bijection Ob(N (C^{ss}))_{/∼} ⟶^{∼} K_{+}(C)^{(ℕ)}.

As a consequence, one gets :

– a “semi-simplification” process Ob(N (C))_{/∼} ⟶ Ob(N (C^{ss}))/ ∼.

– a “transposition” process Ob(N (C^{ss}))_{/∼} ⟶^{∼} Ob(L(C^{ss}))_{/∼}, where L(C) denotes the category of pairs (V, L) with L : V ⟶ V (1) nilpotent.

– a map Ob(N (C′^{ss}))_{/∼} ⟶ Ob(N (C^{ss}))_{/∼} for any map K_{+}(C′) ⟶ K_{+}(C).

As an example of application, let C = Rep_{\overline{𝔽}_{ℓ}}(W_{K}), resp. C′ = Rep_{\overline{ℚ}_{ℓ}}(W_{K}), be the category of finite dimensional representations of W_{K} with \overline{𝔽}_{ℓ}, resp. \overline{ℚ}_{ℓ} coefficients. In this paper, a Weil-Deligne \overline{𝔽}_{ℓ}-representation is an object of N (C^{ss}) (so our convention is that the Weil part of a WD representation is semisimple). Applying the last item to the decomposition map r_{ℓ} : K_{+}(Rep_{\overline{ℚ}_{ℓ}}(W_{K})) ⟶ K_{+}(Rep_{\overline{𝔽}_{ℓ}}(W_{K})) we get a reduction process

(σ^{ss}, N) ↦ r_{ℓ}(σ^{ss}, N) = (r_{ℓ}σ^{ss}, N)

for Weil-Deligne representations.

(2.2.3) The Zelevinski-Vignéras correspondence. According to [26, Thm 1.6], there is a unique map

π ↦ σ^{ss}(π), Irr_{\overline{𝔽}_{ℓ}} (G) ⟶ \{d-dimensional semi-simple \overline{𝔽}_{ℓ}-reps of W_{K}\}_{/∼}

which is compatible with the ℓ-adic semi-simple Langlands correspondence via reduction modulo ℓ in the following sense : if π is a constituent of r_{ℓ}(π̃) for π̃ ∈ Irr_{\overline{ℚ}_{ℓ}} (G), then σ^{ss}(π) = r_{ℓ}(σ^{ss}(π̃)). In [10], we gave a geometric realization of this map, as well as another proof of its existence.

Using her classification à la Zelevinski, Vignéras explained in [26, 1.8] that the above semisimple Langlands correspondence extends uniquely to a bijection :

Irr_{\overline{𝔽}_{ℓ}} (G) → \{d-dimensional Weil-Deligne \overline{𝔽}_{ℓ}-reps of W_{K}\}_{/∼}
π ↦ σ^{Z}(π) = (σ^{ss}(π), N^{Z}(π))

such that the following compatibility with the ℓ-adic Langlands correspondence via reduction modulo ℓ holds : if π is a constituent of r_{ℓ}(π̃) for π̃ ∈ Irr_{\overline{ℚ}_{ℓ}} (G), and if λ_{π} = λπ̃, then σ^{Z}(π) = r_{ℓ}(σ(Z(π̃))).

Here, Z denotes the Zelevinski involution for \overline{ℚ}_{ℓ}-representations, and the precise meaning of r_{ℓ} in the context of WD representations was explained in the last paragraph. Further, λ_{π} is the partition of d attached to π by taking successive higher non-zero derivatives, as in the proof of point ii) of Proposition (2.1.2). Note that the mere existence of a π̃ fulfilling the conditions above is highly non trivial in general, and rests on Ariki’s work on cyclotomic Hecke algebras.

Our aim in this paper is to provide a (partial) geometric interpretation of this enhanced correspondence, by means of a Lefschetz operator. Therefore we will focus on the “transposed” WD representation, as defined in the previous paragraph :

(σ^{ss}(π), L(π)) := ^{t}(σ^{ss}(π), N^{Z}(π)).

We now want to compute explicitly these transposed WD representations for the elliptic principal series. This will involve the \overline{𝔽}_{ℓ}-character ν_{W} : w ↦ q^{−val(Art_{K}^{−1}(w))} where Art_{K} is the local class field homomorphism with takes a uniformizer to a geometric Frobenius. For simplicity, we will use the so-called “Hecke normalization” of Langland’s correspondence.

Proposition.– For any strict subset I ⊂ S̃, we have σ^{ss}(π_{I}) ≃ ⨁_{i=0}^{d−1} ν_{W}^{i} and in a good eigenbasis, L(π_{I}) is given by the matrix ∑_{α_{i}∈I} E_{i−1,i}

Proof. The correspondence is compatible with twisting in the sense that σ^{Z}(π ⊗ ν_{G}) = σ^{Z}(π) ⊗ ν_{W}. Since our proposed solution is also compatible with twisting, we may assume that I ⊂ S. In this case we know that π_{I} appears in r_{ℓ}(v_{I}(\overline{ℚ}_{ℓ})). We also know that λ_{π_{I}} = λ_{v_{I}(\overline{ℚ}_{ℓ})} = λ_{I}. Therefore we have (σ^{ss}(π_{I}), L(π_{I})) = r_{ℓ} (σ^{ss}(v_{I}(\overline{ℚ}_{ℓ})), L(v_{I}(\overline{ℚ}_{ℓ}))). But the latter was computed in [9, 3.2.4].

2.3 Computation of some Ext groups

This section is rather technical in nature and should be skipped at first reading. We first check that some computations of Ext groups between the v_{J}’s and the i_{I}’s performed by Orlik in [20] remain valid in our present context, although Orlik’s hypotheses are not satisfied. Then we proceed to compute Ext groups between the π_{J}’s and i_{I}’s.

(2.3.1) Context and notation. We fix a uniformizer ϖ of K and we will consider Yoneda extensions in the category Rep_{\overline{𝔽}_{ℓ}}^{∞}(G/ϖ^{ℤ}) of smooth \overline{𝔽}_{ℓ}-representations of G/ϖ^{ℤ}. Recall that a subset I ⊆ S determines a standard parabolic subgroup P_{I}, the standard Levi component of which is denoted by M_{I}. We also denote by W_{I} the Weyl group of T in M_{I}, which is also the subgroup of W generated by reflections associated to roots in I. We define a \overline{𝔽}_{ℓ}-vector space

Y_{I} := X^{∗}(M_{I}/Z(G)) ⊗_{ℤ} \overline{𝔽}_{ℓ}

where X^{∗} denotes the group of K-rational characters and Z means “center”.

Symbols r_{P} and i_{P} will stand for normalized parabolic functors along the parabolic subgroup P and δ_{P} will denote the modulus character of P. With this notation we have e.g. i_{I}(\overline{𝔽}_{ℓ}) = i_{P_{I}}(δ_{P_{I}}^{− \frac{1}{2}} ). We will also put δ = δ_{B}^{− \frac{1}{2}} . Finally, the symbol Exp(T, σ) denotes the set of characters of T occuring as subquotients of the admissible \overline{𝔽}_{ℓ}T -representation σ.

Lemma.– Let I be a strict subset of S.

i) If π, π′ are two principal series of M_{I}, then

(W_{I} · Exp (T, r_{B∩M_{I}}(π)) ∩ W_{I} · Exp (T, r_{B∩M_{I}}(π′)) = ∅) ⇒ Ext_{M_{I}/ϖ^{ℤ}}^{∗} (π, π′) = 0.

ii) Ext_{M_{I}/ϖ^{ℤ}}^{∗} (\overline{𝔽}_{ℓ}, \overline{𝔽}_{ℓ}) = ⋀^{∗} Y_{I}

Proof. i) The assumption means that π and π′ have disjoint cuspidal supports. Since ℓ is banal for M_{I}, the vanishing of Ext follows from [24, 6.1].

ii) The argument in [20, Prop. 9] shows that Ext_{M_{I}/ϖ^{ℤ}}^{∗} (\overline{𝔽}_{ℓ}, \overline{𝔽}_{ℓ}) = Ext_{M_{I}/M_{I}^{0}ϖ^{ℤ}}^{∗} (\overline{𝔽}_{ℓ}, \overline{𝔽}_{ℓ}) where M_{I}^{0} is the subgroup of M_{I} generated by compact elements (note that ℓ is prime to the pro-index [M_{I}^{0} : [M_{I}, M_{I}]]). Since ℓ is also prime to the torsion in the abelian group M_{I}/M_{I}^{0}ϖ^{ℤ}, we know that Ext_{M_{I}/M_{I}^{0}ϖ^{ℤ}}^{∗} (\overline{𝔽}_{ℓ}, \overline{𝔽}_{ℓ}) = ⋀^{∗}(Hom_{gps}(M_{I}/M_{I}^{0}ϖ^{ℤ}, \overline{𝔽}_{ℓ})) = ⋀^{∗}(Hom_{gps}(M_{I}/M_{I}^{0}ϖ^{ℤ}, ℤ) ⊗_{ℤ} \overline{𝔽}_{ℓ}). Finally, the usual map χ ↦ val_{K}∘χ yields an isomorphism X^{∗}(M_{I}/Z(G)) ⟶ Hom_{gps}(M_{I}/M_{I}^{0}ϖ^{ℤ}, ℤ).

Remark.– A consequence of item ii) of the foregoing lemma and Frobenius reciprocity is that for any representation π of G/ϖ^{ℤ}, the graded space Ext_{G/ϖ^{ℤ}}^{∗}(π, i_{I}(\overline{𝔽}_{ℓ})) ≃ Ext_{M_{I}/ϖ^{ℤ}}^{∗}((π)_{U_{P_{I}}} , \overline{𝔽}_{ℓ}) is naturally a graded right module over the graded algebra ⋀^{∗} Y_{I}. In particular there is a canonical graded map

Hom_{G/ϖ^{ℤ}} (π, i_{I}(\overline{𝔽}_{ℓ})) ⊗_{\overline{𝔽}_{ℓ}} ⋀^{∗} Y_{I} ⟶ Ext_{G/ϖ^{ℤ}}^{∗} (π, i_{I}(\overline{𝔽}_{ℓ})) .

This map is clearly functorial in π. It is also functorial in I in the sense that if J ⊂ I we have a commutative diagram

\xymatrix{Hom_{G/ϖ^{ℤ}} (π, i_{I}(\overline{𝔽}_{ℓ})) ⊗_{\overline{𝔽}_{ℓ}} ⋀^{∗} Y_{I} \ar[r] \ar[d] & Ext_{G/ϖ^{ℤ}}^{∗} (π, i_{I}(\overline{𝔽}_{ℓ})) \ar[d] \\ Hom_{G/ϖ^{ℤ}} (π, i_{J}(\overline{𝔽}_{ℓ})) ⊗_{\overline{𝔽}_{ℓ}} ⋀^{∗} Y_{J} \ar[r] & _{Ext{∗ \atop G/ϖ^{ℤ}} (π, i_{J}(\overline{𝔽}_{ℓ}))}}

where the vertical maps are induced by the inclusion i_{I}(\overline{𝔽}_{ℓ}) ↪ i_{J}(\overline{𝔽}_{ℓ}) and the restriction map Y_{I} ⟶ Y_{J}.

(2.3.2) Proposition.– Let I, J be two subsets of S, with I a strict subset. Then the canonical map

Hom_{G/ϖ^{ℤ}} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ⊗_{\overline{𝔽}_{ℓ}} ⋀^{∗} Y_{I} ⟶ Ext_{G/ϖ^{ℤ}}^{∗} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ}))

is an isomorphism. In other words, we have

Ext_{G/ϖ^{ℤ}}^{∗} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ≃ \{ \begin{matrix}⋀^{∗} Y_{I} & if J ⊇ I \\ 0 & otherwise \end{matrix} .

Moreover, the natural map Ext_{G/ϖ^{ℤ}}^{∗} (i_{K}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ⟶ Ext_{G/ϖ^{ℤ}}^{∗} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) is an isomorphism for any J ⊇ K ⊇ I.

Proof. We follow [20, Prop. 15] but we avoid Lemma 16 of loc. cit. which might fail to be true in our context. By Frobenius reciprocity, we have

Ext_{G/ϖ^{ℤ}}^{∗} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) = Ext_{M_{I}/ϖ^{ℤ}}^{∗} (r_{P_{I}} ∘ i_{P_{J}} (δ_{P_{J}}^{− \frac{1}{2}} ), δ_{P_{I}}^{− \frac{1}{2}} )

and by the geometric Mackey formula, r_{P_{I}} ∘ i_{P_{J}} (δ_{P_{J}}^{− \frac{1}{2}} ) has a filtration with graded pieces of

the form Q_{w} := i_{M_{I}∩w(P_{J})} (w(δ_{P_{J∩w^{−1}(I)}}^{− \frac{1}{2}})), where w runs over all elements in W such that w(J) ⊂ Φ^{+} and w^{−1}(I) ⊂ Φ^{+} (this is a complete set of representatives of double cosets in W_{I}\W/W_{J}). Using again the geometric Mackey formula we get

W_{I} · Exp (T, r_{B∩M_{I}}(Q_{w})) = W_{I} · Exp (T, r_{B∩M_{I∩w(J)}}(w(δ_{P_{J∩w^{−1}(I)}}^{− \frac{1}{2}}))) = W_{I} · \{w(δ)\}.

On the other hand, we have

W_{I} · Exp (T, r_{B∩M_{I}}(δ_{P_{I}}^{− \frac{1}{2}} )) = W_{I} · \{δ\}.

Since δ is W -regular, item i) of the above Lemma tells us that Ext_{M_{I}/ϖ^{ℤ}}^{∗} (Q_{w}, δ_{P_{I}}^{− \frac{1}{2}} ) = 0 unless

w ∈ W_{I}. In this case, we must have w = 1 so that Q_{w} = Q_{1} = i_{M_{I}∩P_{J}} (δ_{P_{I∩J}}^{− \frac{1}{2}} ) is the top quotient of the geometric Mackey filtration and the canonical map

Ext_{M_{I}/ϖ^{ℤ}}^{∗} (i_{M_{I}∩P_{J}} (δ_{P_{I∩J}}^{− \frac{1}{2}} ), δ_{P_{I}}^{− \frac{1}{2}} ) ⟶ Ext_{G/ϖ^{ℤ}}^{∗} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ}))

is an isomorphism. Using Casselman’s reciprocity, the LHS identifies with

Ext_{M_{I∩J} /ϖ^{ℤ}}^{∗} (δ_{P_{I∩J}}^{− \frac{1}{2}} , r_{M_{I}∩\overline{P_{J}}} (δ_{P_{I}}^{− \frac{1}{2}} )) = Ext_{M_{I∩J} /ϖ^{ℤ}}^{∗} (δ_{P_{I∩J}}^{− \frac{1}{2}} , δ_{P′_{I∩J}}^{− \frac{1}{2}} )

where \overline{P_{J}} is the opposite parabolic subgroup to P_{J} w.r.t M_{J} and P′_{I∩J} is the semistandard parabolic subgroup with Levi component M_{I∩J} and unipotent radical U_{I}(\overline{U_{J}} ∩ M_{I}). Let B′ be the Borel subgroup with unipotent radical U_{I}(\overline{U_{J}} ∩ M_{I})(U_{∅} ∩ M_{I∩J}). Point i) of the previous Lemma tells us that the RHS of the last displayed formula vanishes unless there is w ∈ W_{I∩J} such that w(B) = B′. But then w(B) ∩ M_{I∩J} = B′ ∩ M_{I∩J} hence w = 1, thus P′_{I∩J} = P_{I∩J} which is possible only if J ⊇ I.

Eventually we have proved the desired vanishing when J does not contain I, and we have proved that if J ⊇ I, the canonical map

Ext_{M_{I}/ϖ^{ℤ}}^{∗} (δ_{P_{I}}^{− \frac{1}{2}} , δ_{P_{I}}^{− \frac{1}{2}} ) ⟶ Ext_{G/ϖ^{ℤ}}^{∗} (i_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ}))

is an isomorphism. We conclude the computation using item ii) of the previous Lemma. The last assertion follows from the functorial nature of the above map.

(2.3.3) The complex (2.1.3.1) yields a spectral sequence

E_{1}^{pq} = ⨁_{K⊇J,|K\J|=p} Ext_{G/ϖ^{ℤ}}^{q} (i_{K}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ⇒ Ext_{G/ϖ^{ℤ}}^{p+q} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ}))

whence in particular an edge map

(2.3.3.1)  Ext_{G/ϖ^{ℤ}}^{∗} (\overline{𝔽}_{ℓ}, i_{I}(\overline{𝔽}_{ℓ})) ⟶ Ext_{G/ϖ^{ℤ}}^{|S\J|+∗} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) .

Thanks to the last Proposition, the same argument as [20, Prop. 17] gives the following expression.

Corollary.– Let I, J be subsets of S with I a strict subset.

i) If I ∪ J ≠ S then Ext_{G/ϖ^{ℤ}}^{∗} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) = 0.

ii) If I ∪ J = S, then the map (2.3.3.1) is an isomorphism, so we get an isomorphism

Ext_{G/ϖ^{ℤ}}^{∗} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ≃ ⋀^{∗−|S\J|} Y_{I}.

Moreover, if I′ is another strict subset of S which contains I, then the natural map Ext_{G/ϖ^{ℤ}}^{∗} (v_{J}(\overline{𝔽}_{ℓ}), i_{I′}(\overline{𝔽}_{ℓ})) ⟶ Ext_{G/ϖ^{ℤ}}^{∗} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) is induced by the natural restriction map Y_{I′} ⟶ Y_{I}.

Remark.– We may recast the foregoing corollary by stating that the canonical map

Ext_{G/ϖ^{ℤ}}^{|S\J|} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ⊗_{\overline{𝔽}_{ℓ}} ⋀^{∗} Y_{I} ⟶ Ext_{G/ϖ^{ℤ}}^{∗+|S\J|} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ}))

is an isomorphism, and that Ext_{G/ϖ^{ℤ}}^{|S\J|} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ≃ \overline{𝔽}_{ℓ} if J ∪ I = S and is zero otherwise.

Next we turn to extensions between the π_{J}’s and the i_{I}’s.

(2.3.4) Proposition.– Let J be a strict subset of S̃, and I a strict subset of S.

i) If 0 ∈ J, then Ext_{G/ϖ^{ℤ}}^{∗} (π_{J}, i_{I}(\overline{𝔽}_{ℓ})) = 0.

ii) Otherwise, the natural map is an isomorphism

Ext_{G/ϖ^{ℤ}}^{∗} (v_{J}(\overline{𝔽}_{ℓ}), i_{I}(\overline{𝔽}_{ℓ})) ⟶^{∼} Ext_{G/ϖ^{ℤ}}^{∗} (π_{J}, i_{I}(\overline{𝔽}_{ℓ})) .

Proof. Note first that ii) follows from i) since [v_{J}(\overline{𝔽}_{ℓ})] = [π_{J}] + [π_{J∪\{0\}}]. Now, in order to prove i) we first use Frobenius reciprocity to get

Ext_{G/ϖ^{ℤ}}^{∗} (π_{J}, i_{I}(\overline{𝔽}_{ℓ})) = Ext_{M_{I}/ϖ^{ℤ}}^{∗} (r_{P_{I}}(π_{J}), δ_{P_{I}}^{− \frac{1}{2}} ) .

By Proposition (2.1.2) iv), we have Exp (T, r_{B∩M_{I}}(r_{P_{I}}(π_{J}))) = {w^{−1}(δ), w(X̃_{S}) ⊂ X̃_{J}} . Since

Exp (T, r_{B∩M_{I}}(δ_{P_{I}}^{− \frac{1}{2}} )) = \{δ\} and since δ is W -regular, Lemma (2.3.1) shows that we are left to

prove that {w ∈ W, w(X̃_{S}) ⊂ X̃_{J}} ∩ W_{I} = ∅. Now, identifying W with 𝔖_{d} as in paragraph

(2.1.1), the condition w(X̃_{S}) ⊂ X̃_{J} is equivalent to the condition J = {α_{j} ∈ S̃, wc^{−1}(j) < w(j)}, so in particular it implies the property w(n−1) < w(0). However, since I is proper, this property is never satisfied by some w ∈ W_{I}.

3 The cohomology complex

In this section, we focus on the useful part of the cohomology complex, namely on that which pertains to the unipotent block of the category of smooth \overline{ℤ}_{ℓ}-representations.

3.1 The unipotent block

According to Vignéras [25, IV.6.2], the category Rep_{\overline{𝔽}_{ℓ}}^{∞}(G) is a product of indecomposable Serre subcategories called “blocks”. This product of blocks corresponds to the partition of the set of irreducible \overline{𝔽}_{ℓ}-representations according to the “inertia class of supercuspidal support”. Among them, the unipotent block is by definition the one which contains the trivial representation. In representation theory of finite groups, this would be rather called the “principal block”. Here we want to lift this block to \overline{ℤ}_{ℓ}-representations. Note that the usual way of lifting idempotents via Hensel’s lemma is not adapted to the p-adic case, since Hecke algebras are not finitely generated modules over \overline{ℤ}_{ℓ}. Therefore, we will exhibit a progenerator of the desired block. In all this subsection, no congruence assumption on the pair (q, ℓ) is required.

(3.1.1) Unipotent blocks for a finite GL_{n}. For a finite group of Lie type Ḡ, we will denote by bḠ the central idempotent in the group algebra ℤ_{ℓ}[Ḡ] which cuts out the direct sum of all blocks which contain a unipotent \overline{ℚ}_{ℓ}-representation (in the sense of Deligne-Luzstig).

Lemma.– Let P̄ = M̄Ū be a parabolic subgroup of Ḡ, and let eŪ be the idempotent associated to the p-group Ū. Then we have eŪbḠ = eŪbM̄ = bM̄eŪ.

Proof. According to [4], an irreducible \overline{ℚ}_{ℓ}-representation π satisfies bḠπ ≠ \{0\} if and only if it belongs to the Deligne-Lusztig series associated to some semi-simple conjugacy class in the dual group Ḡ^{∗} which consists of ℓ-elements. We call such a representation ℓ-unipotent. In this case, all irreducible subquotients of π_{U} are ℓ-unipotent representations of M. Indeed, this follows by adjunction from the “dual” statement that, if σ is an ℓ-unipotent representation of M then all irreducible subquotients of Ind_{P}^{G} (σ) are ℓ-unipotent, see [17, Cor. 6]. This shows that, denoting by b′ ; Ḡ := 1 − bḠ the complementary idempotent, we have b′ ; M̄eŪbḠ = 0 and bM̄eŪb′ ; Ḡ = 0. Then we get eŪbḠ = (b′ ; M̄ + bM̄)eŪbḠ = bM̄eŪbḠ = bM̄eŪ(1 − b′ ; Ḡ) = bM̄eŪ.

Fact.– Assume Ḡ = GL_{n}(𝔽_{q}). Then an irreducible \overline{𝔽}_{ℓ}-representation π̄ of Ḡ satisfies bḠπ̄ ≠ 0 if and only if it is a subquotient of IndḠ ; B̄ (\overline{𝔽}_{ℓ}).

Proof. Any irreducible subquotient of IndḠ ; B̄ (\overline{𝔽}_{ℓ}) occurs in the reduction of a unipotent irreducible \overline{ℚ}_{ℓ}-representation, hence belongs to the category cut out by bḠ. Conversely, fix π̄ such that bḠπ̄ ≠ 0. We may assume that π̄ is cuspidal, since for P̄ = M̄Ū a parabolic subgroup such that π̄Ū ≠ 0 we also have bM̄(π̄Ū) ≠ 0 (as in the previous proof). But then in terms of the Dipper-James classification, π̄ is of the form D(s, 1) for some elliptic semi-simple ℓ-element of Ḡ^{∗} = Ḡ, see [13, Coro 5.23]. Thus, in terms of the James-Dipper classification, it is also of the form D(1, (n)), see [14, Thm 5.1], which means that π̄ is the only non-degenerate subquotient in Ind_{B}^{G} (\overline{𝔽}_{ℓ}).

(3.1.2) Construction of the block. Here we put Ḡ = GL_{d}(𝔽_{q}). We may view bḠ as a central idempotent of the ℤ_{ℓ}-algebra H_{ℤ_{ℓ}}(GL_{d}(O)) of locally constant distributions on GL_{d}(O). Then we put

P_{b} := ind_{GL_{d}(O)}^{G} (bḠH_{ℤ_{ℓ}}(GL_{d}(O)))

and we define Rep_{b}^{∞}(G) as the full subcategory of Rep_{ℤ_{ℓ}}^{∞}(G) consisting of all objects V that are generated by bḠV over ℤ_{ℓ}G.

We will use similar notation to denote somewhat more familiar objects ; letting eḠ be the idempotent attached to the pro-p-radical of GL_{d}(O), we also put

P_{e} := ind_{GL_{d}(O)}^{G} (eḠH_{ℤ_{ℓ}}(GL_{d}(O)))

and we define the category Rep_{e}^{∞}(G) as above. We recall the following result, which is a special case of “level decomposition”, see e.g. [7, App. A].

Fact.– The category Rep_{e}^{∞}(G) is a direct factor of Rep_{ℤ_{ℓ}}^{∞}(G) and is pro-generated by P_{e}. In particular, there is an idempotent e of the center of the category Rep_{ℤ_{ℓ}}^{∞}(G) such that for any object V we have eV = ∑_{g∈G/GL_{d}(O)} geḠV.

Now we can state the main result of this subsection.

Proposition.– The category Rep_{b}^{∞}(G) is a direct factor of Rep_{e}^{∞}(G) and is pro-generated by P_{b}. It consists of all objects V, all irreducible subquotients of which are not annihilated by bḠ.

Proof. From its definition, P_{b} clearly is a generator of the category Rep_{b}^{∞}(G), and is a finitely generated projective object of Rep_{\overline{ℤ}_{ℓ}}^{∞}(G).

Let us prove that Rep_{b}^{∞}(G) is a Serre subcategory. For this, we will apply the general result of [18, Thm 3.1]. For any vertex x of the semisimple building of G, we denote by G_{x} its stabilizer, G_{x}^{+} the pro-p-radical of its stabilizer and Ḡ_{x} := G_{x}/G_{x}^{+} the reductive quotient. We thus get an idempotent b_{x} ∈ H_{\overline{ℤ}_{ℓ}}(G_{x}) by inflation from bḠ_{x}. If g ∈ G, we clearly have b_{gx} = gb_{x}g^{−1}. Therefore, to apply [18, Thm 3.1] we are left to check the two following properties (cf [18, Def 2.1]):

i) b_{x}b_{y} = b_{y}b_{x} for any adjacent vertices x, y. ii) b_{x}b_{z}b_{y} = b_{x}b_{y} whenever z is adjacent to x and belongs to the convex simplicial hull of

{x, y}.

Note first that the definition of b_{x} extends to any facet F of the building. Further, let e_{F} := e_{G_{F}^{+}} denote the idempotent associated to the pro-p-group G_{F}^{+}. We know that properties i) and ii) are satisfied by the system (e_{x})_{x}, and more precisely we have e_{x}e_{y} = e_{[x,y]} whenever x and y are adjacent vertices. Therefore, the above lemma shows that b_{x}e_{y} = b_{x}e_{[x,y]} = b_{[x,y]} and thus b_{x}b_{y} = b_{[x,y]} = b_{y}b_{x}. As for property ii), starting from e_{x}e_{y} = e_{x}e_{z}e_{y}, we get b_{x}b_{y} = b_{x}e_{z}b_{y} = b_{x}b_{z}b_{y}, as desired.

We now know that Rep_{b}^{∞}(G) is a Serre subcategory of the Serre subcategory Rep_{e}^{∞}(G) cut out by the system (e_{x})_{x}. For a vertex x, define b′_{x} := e_{x} − b_{x}, which is lifted from the idempotent 1 − bḠ_{x} of \overline{ℤ}_{ℓ}[Ḡ_{x}]. The same argument as above shows that the system (b′_{x})_{x} satisfies properties i) and ii) and therefore cuts out a Serre subcategory of Rep_{e}^{∞}(G), which is easily seen to be a complement to Rep_{b}^{∞}(G). Therefore the latter is a direct factor in Rep_{e}^{∞}(G). The last statement of the proposition is clear.

Notation.– We will denote by b the idempotent of the center of the category Rep_{ℤ_{ℓ}}^{∞}(G) which projects a representation V to its largest subobject bV in Rep_{b}^{∞}(G). Concretely, we have bV = ∑_{g∈GL_{d}(K)/GL_{d}(O)} g.bḠV.

(3.1.3) Proposition.– A representation π ∈ Irr_{\overline{𝔽}_{ℓ}} (G) belongs to Rep_{b}^{∞}(G) if and only if it is an irreducible subquotient of some Ind_{B}^{G} (χ) with χ an unramified character of B. In particular, Rep_{b}^{∞}(G) ∩ Rep_{\overline{𝔽}_{ℓ}}^{∞}(G) is Vignéras’ unipotent block [25, IV.6.3].

Proof. Let eḠ ∈ H_{\overline{ℤ}_{ℓ}}(GL_{d}(O)) be the idempotent associated to the kernel of the reduction map GL_{d}(O) ⟶ GL_{d}(𝔽_{q}). By Mackey formula, the residual representation of Ḡ on eḠInd_{B}^{G} (χ) is isomorphic to IndḠ ; B̄ (\overline{𝔽}_{ℓ}) with obvious notation. Since Ind_{B}^{G} (χ) belongs to the level zero subcategory Rep_{e}^{∞}(G), so does each one of its irreducible subquotients. Hence for such a subquotient π, eḠπ is a non-zero subquotient of IndḠ ; B̄ (\overline{𝔽}_{ℓ}), so that bḠπ ≠ 0 and π belongs to Rep_{b}^{∞}(G).

Conversely, let π be an irreducible \overline{𝔽}_{ℓ}-representation such that bπ ≠ 0. Choose a parabolic subgroup P = MU and a supercuspidal \overline{𝔽}_{ℓ}-representation σ of M such that π occurs as a subquotient of Ind_{P}^{G} (σ). As above, Mackey formula tells us that eḠInd_{P}^{G} (σ) ≃ IndḠ ; P̄ (eM̄σ), with obvious notation. So by Lemma (3.1.1) we get bḠInd_{P}^{G} (σ) ≃ IndḠ ; P̄ (bM̄σ) and finally bM̄σ ≠ 0. By [27, 3.15], we know that σ is of the form ind_{M∩GL_{n}(O)}^{M} \binom{}{σ̄} for some supercuspidal \overline{𝔽}_{ℓ}-representation σ̄ of the Levi subgroup M̄ of Ḡ, image of M ∩ GL_{n}(O) by the projection to Ḡ. Here, supercuspidal is equivalent to the fact that the semisimple elliptic class s associated to σ̄ consists of ℓ′-elements. However, an easy computation [27, 3.14] shows that bM̄ σ̄ = bM̄σ. Therefore bM̄ σ̄ is non zero and s consists of ℓ-elements by definition. Hence s = 1, or equivalently, M is a torus and σ̄ the trivial representation of M̄.

Remark.– In terms of the Langlands correspondence, the irreducible \overline{𝔽}_{ℓ}-representations π of the principal/unipotent block are those such that σ(π)^{ss} is a sum of unramified characters. This formulation might extend to other p-adic groups, as suggested by the finite field picture.

3.2 The complex

In the first two paragraphs of this subsection, no congruence hypothesis on the pair (q, ℓ) is required. From paragraph (3.2.3) on, we will work under the Coxeter congruence relation.

(3.2.1) The tower and its cohomoly complexes. We refer to [22] or [6, 3.1] for the definition of the Lubin-Tate space M_{LT,n} of height d and level n, which we see as a K̆-analytic space, endowed with a continuous action of D^{×}, an action of GL_{d}(O/ϖ^{n}O) and a Weil descent datum to K. Although in this paper we will be mainly interested in the tame level M_{LT,1}, the formalism used to define the complex requires the whole “tower” (M_{LT,n})_{n∈ℕ} and in particular the action of G = GL_{d}(K) which can be defined on this tower. Maybe the most precise way to describe this action is to introduce the category ℕ(G) with set of objects ℕ and arrows given by Hom(n, m) := {g ∈ G, gM_{d}(O)g^{−1} ⊂ ϖ^{m−n}M_{d}(O)} and to note that the M_{LT,n}’s are the image of a functor from ℕ(G) to the category whose objects are K̆-analytic spaces with continuous action of D^{×} and Weil descent datum to K, and morphisms are finite étale equivariant morphisms. This allows one to define the complex

RΓ_{c} := RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) ∈ D^{b}(Rep_{ℤ_{ℓ}}^{∞,c}(G × D^{×} × W_{K}))

as in [6, 3.3.3]. Let us note that the diagonal subgroup K^{×} of G × D^{×} acts trivially on the tower hence also on the cohomology.

It is technically important to recall that the tower is “induced” from a “sub-tower” denoted (M_{LT,n}^{(0)})_{n∈ℕ} which is stable under the subgroup

(GDW)^{0} := {(g, δ, w) ∈ G × D^{×} × W_{K}, | det(g)|^{−1}|Nr(δ)||Art_{K}(w)| = 1} .

So we have a complex RΓ_{c}^{(0)} := RΓ_{c}(M_{LT}^{ca,(0)}, ℤ_{ℓ}) ∈ D^{b}(Rep_{ℤ_{ℓ}}^{∞,c}(GDW)^{0}) together with an isomorphism [6, (3.5.2)]

RΓ_{c} ≃ ind_{(GDW)^{0}}^{GDW}RΓ_{c}^{(0)}.

An important consequence of this is the following compatibility with twisting. For any smooth character χ of K^{×} and any representation π of G, we have

(3.2.1.1)  R_{(χ∘det)⊗π} ≃ (χ ∘ (Nr · Art_{K})) ⊗ R_{π} in D^{b}(Rep_{ℤ_{ℓ}}(D^{×} × W_{K})).

We need yet another variant. Let us fix a uniformizer ϖ of K and see it as a central element of G. Its action on the tower is free (it permutes the connected components), so we may consider the quotient tower (M_{LT,n}/ϖ^{ℤ})_{n∈ℕ} and its cohomology complex

RΓ_{c,ϖ} := RΓ_{c}(M_{LT}^{ca}/ϖ^{ℤ}, ℤ_{ℓ}) ∈ D^{b}(Rep_{ℤ_{ℓ}}^{∞,c}(G/ϖ^{ℤ} × D^{×} × W)).

We then have isomorphisms (cf [6, 3.5.3])

RΓ_{c,ϖ} ≃ RΓ_{c} ⊗_{ℤ_{ℓ}[ϖ^{ℤ}]}^{L} ℤ_{ℓ} ≃ ind_{(GDW)^{0}ϖ^{ℤ}}^{GDW}RΓ_{c}^{(0)}.

Because of the first isomorphism, if π is a representation on which ϖ acts trivially, then R_{π} ≃ RHom_{G/ϖ^{ℤ}}(RΓ_{c,ϖ}, π). Since any irreducible representation may be twisted to achieve this condition π(ϖ) = 1, we see that we don’t loose any generality in restricting attention to RΓ_{c,ϖ}.

(3.2.2) The tame part. We take up the notation e, eḠ of the previous subsection and denote by H_{e} the commuting algebra End_{ℤ_{ℓ}G} (P_{e}), which identifies with the Hecke algebra of compactly supported ℤ_{ℓ}-valued (1 + ϖM_{d}(O))-bi-invariant measures on G.

The complex eḠRΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) is naturally an object of D^{b}(Rep_{H_{e}}^{∞}(D^{×}×W_{K})) and we recover the direct summand eRΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) via the usual equivalence of categories. Namely we have, as in [6, Lemme 3.5.9],

eRΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) ≃ P_{e} ⊗_{H_{e}}^{L} eḠRΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}).

Now, if we restrict the action to GL_{d}(O), we have by construction an isomorphism in D^{b}(Rep_{ℤ_{ℓ}}^{∞,c}(Ḡ× D^{×} × W_{K})).

eḠRΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) ⟶^{∼} RΓ_{c}(M_{LT,1}^{ca}, ℤ_{ℓ}) ≃ ind_{Ḡ×(DW)^{0}}^{Ḡ×D^{×}×W_{K}}RΓ_{c}(M_{LT,1}^{ca,(0)}, ℤ_{ℓ}).

The tame Lubin-Tate space M_{LT,1}^{(0)} was studied by Yoshida in [28]. He exibited in particular a certain affinoid subset N of M_{LT,1}^{0} which acquires good reduction over K̆[ϖ^{1/(q^{d}−1)}], with special fiber equivariantly isomorphic to the Deligne-Lusztig covering Y (c) associated to the Coxeter element of Ḡ. In [8], we showed that the restriction map induces an isomorphism RΓ(M_{LT,1}^{ca,(0)}, ℤ_{ℓ}) ⟶^{∼} RΓ(N^{ca}, ℤ_{ℓ}). Taking duals, we thus get an isomorphism in D^{b}(Rep_{ℤ_{ℓ}}(Ḡ))

(3.2.2.1)  RΓ_{c}(Y (c), ℤ_{ℓ}) ⟶^{∼} RΓ_{c}(M_{LT,1}^{ca,(0)}, ℤ_{ℓ}).

In particular we get the following important property.

Proposition.– The cohomology spaces of both the complexes eRΓ_{c} and eRΓ_{c,ϖ} are torsionfree.

Indeed, the torsion-freeness for Y (c) follows from Lemma 3.9 and Corollary 4.3 of [1]. We emphasize the fact that no hypothesis on the pair (q, ℓ) is required here.

(3.2.3) The unipotent part : ℓ-adic cohomology. From this paragraph on, we assume that the order of q in 𝔽_{ℓ}^{×} is d. We take up the notation of the previous subsection, and we consider the direct summand bRΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}), or rather its variant bRΓ_{c,ϖ}. There is a fairly explicit description of the ℚ_{ℓ}-cohomology of this complex. We first recall a classical construction. Let θ : 𝔽_{q^{d}}^{×} ⟶ \overline{ℚ}_{ℓ}^{×} be a character which is Frob_{q}-regular. Define :

– a representation ρ(θ) := ind_{O_{D}^{×}ϖ^{ℤ}}^{D^{×}}(θ) of D^{×}, where O_{D}^{×}ϖ^{ℤ} acts via the reduction map O_{D}^{×} ⟶ 𝔽_{q^{d}}^{×}.

– a representation σ(θ) := ind_{I_{K}φ^{dℤ}}^{W_{K}} (θ) where I_{K}φ^{dℤ} acts via the tame inertia map I_{K} ⟶ μ_{q^{d}−1} ≃ 𝔽_{q^{d}−1}.

– a representation π(θ) := ind_{GL_{d}(O)ϖ^{ℤ}}^{G} (π_{θ}^{0}) where π_{θ}^{0} is the cuspidal representation of GL_{d}(𝔽_{q}) associated to θ by the Green (or Deligne-Lusztig) correspondence.

All these representations are irreducible and depend only on the Frob_{q}-conjugacy class of θ. Moreover, they are associate by the Langlands and Jacquet-Langlands correspondences.

Fact.– Let I_{ϖ} := ℤ_{ℓ}[G × D^{×} × W_{K}/(GDW)^{0}ϖ^{ℤ}].

i) For i = 1, ⋯, d − 1, there is an isomorphism

H^{d−1+i}(bRΓ_{c,ϖ}) ⊗ ℚ_{ℓ} ⟶^{∼} v_{{1,··· ,i}}(ℚ_{ℓ})(−i) ⊗ I_{ϖ}.

ii) For i = 0, there is a (split) exact sequence

0 ⟶ K ⟶ H^{d−1}(bRΓ_{c,ϖ}) ⊗ ℚ_{ℓ} ⟶ v_{∅}(ℚ_{ℓ}) ⊗ I_{ϖ} ⟶ 0

and an isomorphism K ⊗ \overline{ℚ}_{ℓ} ≃ ⨁_{θ} π(θ) ⊗ ρ(θ)^{∨} ⊗ σ(θ)^{∨}, where θ runs over Frob_{q}- conjugacy classes of Frob_{q}-regular characters 𝔽_{q^{d}}^{×} ⟶ \overline{ℤ}_{ℓ}^{×} which are ℓ-congruent to the trivial character.

Proof. The shortest argument here is to invoke Boyer’s description of the \overline{ℚ}_{ℓ}-cohomology of the whole Lubin-Tate tower in [3] (see [6, 4.1.2] for an account featuring a notation consistent with that of the present paper), together with the characterization of irreducible objects of the unipotent block in Proposition (3.1.3). We note that the maps H^{d−1+i}(bRΓ_{c,ϖ}) ⊗ ℚ_{ℓ} ⟶ v_{{1,··· ,i}}(ℚ_{ℓ})(−i) are induced by the canonical morphism

(3.2.3.1)  RΓ_{c}(M_{LT}^{ca,(0)}, ℤ_{ℓ}) ⟶ RΓ_{c}(M_{LT}^{ca,(0)}, ℤ_{ℓ}) ⊗_{ℤ_{ℓ}[O_{D}^{×}]}^{L} ℤ_{ℓ}.

Alternatively, if one wants to avoid Boyer’s machinery, it is possible to derive almost everything from Yoshida’s construction [28], via isomorphism (3.2.2.1). More precisely, put w_{i} := H^{d−1+i}(bRΓ_{c,ϖ} ⊗_{ℤ_{ℓ}[O_{D}^{×}/ϖ^{ℤ}]}^{L} ℤ_{ℓ}). Then, by using a similar feature of Deligne-Lusztig varieties, one can show that the above morphism of complexes induces isomorphisms

H^{d−1+i}(bRΓ_{c,ϖ}) ⊗ ℚ_{ℓ} ⟶^{∼} w_{i} ⊗ ℚ_{ℓ}

for i > 0, as well as an exact sequence

K ↪ H^{d−1}(bRΓ_{c,ϖ}) ⊗ ℚ_{ℓ} ↠ w_{0} ⊗ ℚ_{ℓ}.

Further, one finds an isomorphism eḠw_{i} ≃ eḠ(v_{{1,··· ,i}}(ℤ_{ℓ}) ⊗ I_{ϖ}). However, what is a priori missing is enough information on Hecke operators acting on eḠw_{i} in order to recognize w_{i} as isomorphic to v_{{1,··· ,i}}(ℤ_{ℓ}) ⊗ I_{ϖ}. One highly non trivial way to get around this problem is to invoke the Faltings-Fargues isomorphism of [16] (see [6, 3.4] for a brief description) to move to the so-called Drinfeld tower (see [6, 3.2] for an overview on this tower). Then the morphism of complexes (3.2.3.1) is carried to

(3.2.3.2)  RΓ_{c}(M_{Dr}^{ca,(0)}, ℤ_{ℓ}) ⟶ RΓ_{c}(M_{Dr,0}^{ca,(0)}, ℤ_{ℓ})

and the right hand side is the Drinfeld upper half space whose cohomology is computed by combinatorics, and shown by Schneider and Stuhler to be isomorphic to v_{{1,··· ,i}}(ℤ_{ℓ}).

We let Π be a uniformizer of D such that Π^{d} = ϖ, and we fix a “geometric” Frobenius element φ in W_{K}. We are going to decompose the complex bRΓ_{c,ϖ} in the category D^{b}(Rep_{ℤ_{ℓ}}(G/ϖ^{ℤ})) according to the action of Π and φ. Since K_{diag}^{×} acts trivially on the tower, the action of Π on RΓ_{c,ϖ} is obviously killed by the polynomial X^{d} − 1. Further, as a corollary to the description above and to the torsion-freeness result of Corollary (3.2.2), we get :

Corollary.– For any integer 0 ⩽ i ⩽ d − 1, the action of φ on H^{d−1+i}(bRΓ_{c,ϖ}) is killed by the polynomial X^{d} − q^{id}.

(3.2.4) The unipotent part : splitting. Put P_{φ}(X) := ∏_{i=0}^{d−1}(X^{d} − q^{id}). By the above corollary and [6, Lemme A.1.4 i)], P_{φ}(φ) acts by zero on the whole complex bRΓ_{c,ϖ}. The ring A_{φ} := ℤ_{ℓ}[X]/P_{φ}(X) is a semi-local ring, hence decomposes as a product A_{φ} = ∏_{𝔪} A_{φ}_{𝔪} of its localizations at maximal ideals. Since q is a primitive d-root of unity in 𝔽_{ℓ} by our congruence hypothesis, the maximal ideals of this ring are 𝔪_{i} := (ℓ, X − q^{i}), i = 0, ⋯, d − 1 and we denote A_{φ} = ∏_{i=0}^{d−1} A_{φ,i} the associated decomposition. Accordingly we get a decomposition [19, Prop 1.6.8]

bRΓ_{c,ϖ} ≃ ⨁_{i=0}^{d−1}(bRΓ_{c,ϖ})_{i} in D^{b}(Rep_{ℤ_{ℓ}}^{∞}(G/ϖ^{ℤ})).

Similarly, the ring A_{Π} := ℤ_{ℓ}[X]/(X^{d} − 1) is semi-local with maximal ideals (ℓ, X − q^{j}), j = 0, ⋯, d − 1 and we get a sharper decomposition

bRΓ_{c,ϖ} ≃ ⨁_{i,j=0}^{d−1}(bRΓ_{c,ϖ})_{i,j} in D^{b}(Rep_{ℤ_{ℓ}}^{∞}(G/ϖ^{ℤ})).

Note that each one of these direct summands is preserved by the action of φ and Π, but not necessarily by that of I_{K} and O_{D}^{×}. Let ζ denote the Teichmüller lift of q, i.e. the only primitive d-root of unity in ℤ_{ℓ} which is ℓ-congruent to q. By construction, the action of Π on (bRΓ_{c,ϖ})_{i,j} is by multiplication by ζ^{j}, while that of φ is killed by the polynomial ∏_{k}(X − q^{i−k}ζ^{k}).

Moreover these direct summands satisfy the following properties.

(3.2.4.1)

(bRΓ_{c,ϖ})_{i,j} ≃ ζ^{j val_{K} ∘ det^{−1}}(bRΓ_{c,ϖ})_{i−j,0} with action of φ and Π twisted by ζ^{j}.

This follows indeed from (3.2.1.1).

(3.2.4.2)

There is a distinguished triangle

c_{i}[0] ⟶ (bRΓ_{c,ϖ})_{i,0}[d − 1] ⟶ h_{i}(−i)[−i] ⟶^{+1}

with c_{i} a cuspidal ℓ-torsion free representation and h_{i} a G-invariant lattice in v_{{1,··· ,i}}(ℚ_{ℓ}).

This follows from Fact (3.2.3) and Proposition (3.2.2). Note that by convention we set {1, ⋯, i} = ∅ if i = 0.

Let us put h̄_{i} := h_{i} ⊗ \overline{𝔽}_{ℓ}. By Proposition (2.1.3), we have the following equality in the Grothendieck group :

(3.2.4.3)  [h̄_{i}] = [v_{\{1,··· ,i\}}(\overline{𝔽}_{ℓ})] = [π_{\{1,··· ,i\}}] + [π_{\{0,··· ,i\}}],

Remark.– For i ≠ d − 1, it can be shown that h̄_{i} is not isomorphic to v_{\{1,··· ,i\}}(\overline{𝔽}_{ℓ}). More precisely, v_{\{1,··· ,i\}}(\overline{𝔽}_{ℓ}) is a non-split extension of π_{\{0,··· ,i\}} by π_{\{1,··· ,i\}}, while h̄_{i} is a non-trivial extension going the other way. The same phenomenon appears for the Deligne-Lusztig variety, see in particular [15, Thm 4.1] which provides a description of the finite field analogue of h̄_{i}. In the present contex, let us simply mention without proof that the morphism (3.2.3.2) induces a map

(3.2.4.4)  H_{c}^{d−1+i}(M_{Dr,1}^{ca}, \overline{𝔽}_{ℓ}) = h̄_{i} ⟶ H_{c}^{d−1+i}(M_{Dr,0}^{ca}, \overline{𝔽}_{ℓ}) = v_{\{1,··· ,i\}}(\overline{𝔽}_{ℓ})

which is non-zero, with kernel and cokernel both isomorphic to π_{{0,··· ,i}}. Of course this map is also induced by the morphism (3.2.3.1).

4 Proof of the main theorem

Let π be a \overline{𝔽}_{ℓ}-representation of G. Recall the definition of the graded vector space R_{π}^{∗} from the introduction. For convenience, we will shift this definition by [1 − d], i.e. we consider now

R_{π}^{∗} := H^{∗}(RHom_{ℤ_{ℓ}G}(RΓ_{c}[d − 1], π)).

This is a graded smooth \overline{𝔽}_{ℓ}-representation of D^{×} × W_{K}, whose grading is supported in the range [1 − d, d − 1] by [10, Prop 2.1.3].

In all this section, we work under the Coxeter congruence hypothesis, i.e. we assume that the order of q in 𝔽_{ℓ}^{×} is d.

4.1 Computation of R_{π}^{∗} for π an elliptic principal series

(4.1.1) Preliminaries. Assume now that π belongs to the unipotent block and that its central character is trivial on ϖ. Then we have R_{π}^{∗} = H^{∗}(RHom_{ℤ_{ℓ}(G/ϖ^{ℤ})}(RΓ_{c,ϖ}[1 − d], π)), and according to (3.2.4), we may decompose it as

R_{π}^{∗} = ⨁_{i,j=0}^{d−1}(R_{π}^{∗})_{i,j}, where (R_{π}^{∗})_{i,j} := H^{∗}(RHom_{ℤ_{ℓ}(G/ϖ^{ℤ})}((bRΓ_{c,ϖ})_{i,j}, π)).

Concretely, (R_{π}^{∗})_{i,j} is the intersection of the generalized q^{−i}-eigenspace of φ with the generalized q^{−j}-eigenspace of Π. As already mentioned, these summands need not be stable under the action of I_{K} and O_{D}^{×}. However, the description of the ℓ-adic cohomology of bRΓ_{c,ϖ} in (3.2.3), together with the the ℓ-torsion freeness of its integral cohomology show that both I_{K} and O_{D}^{×} act trivially on the D^{×} × W_{K} semi-simplifications H^{k}(bRΓ_{c,ϖ} ⊗_{ℤ_{ℓ}}^{L} \overline{𝔽}_{ℓ})^{ss}, k ∈ ℕ. Therefore, the same is true for R_{π}^{∗,ss}. As a consequence, letting I_{K} and O_{D}^{×} act trivially on each (R_{π}^{∗})_{i,j}^{ss}, we get the following equality in R(D^{×} × W_{K}, \overline{𝔽}_{ℓ}) :

(4.1.1.1)  R_{π}^{∗,ss} ≃ ⨁_{i,j=0}^{d−1}(R_{π}^{∗})_{i,j}^{ss} = ⨁_{i,j=0}^{d−1}(νj_{D} ⊗ νi_{W})^{dim_{\overline{𝔽}ℓ}(R_{π}^{∗})_{i,j}}.

Recall also from property (3.2.1.1) that we have R_{ν_{G}π}^{∗} ≃ (ν_{D} ⊗ ν_{W}) ⊗ R_{π}^{∗}. Therefore we get isomorphisms

(4.1.1.2)  (R_{ν_{G}π}^{∗})_{i,j} ≃ (R_{π}^{∗})_{i−1,j−1}.

The aim of this subsection is to prove Theorem (4.1.3) below, which describes explicitly each (R_{π_{I}}^{∗})_{i,j}, i.e. which computes the dimension of each (R_{π_{I}}^{∗})_{i,j}. We first introduce some notation.

(4.1.2) For an integer k between 0 and d − 1 and a subset I of S, we put

∂_{I}(k) := k − δ(k, I) where δ(k, I) := |I ∪ {1, ⋯, k}| − |I ∩ {1, ⋯, k}|.

These functions already appear in [5], see in particular Lemma 4.4.1 of loc. cit. The following property is elementary.

Fact.– The map k ∈ {0, ⋯, d−1} ↦ ∂_{I}(k) ∈ ℤ is non-decreasing, with image {−|I|, −|I|+ 2, ⋯, |I| − 2, |I|}. More precisely, writing I = {i_{1}, ⋯, i_{|I|}} and putting i_{0} := 0 and i_{|I|+1} := d, we have ∂_{I}^{−1}(−|I| + 2j) = {i_{j}, ⋯, i_{j+1} − 1}.

In the next statement, we extend the function ∂_{I} to ℤ by making it d-periodic.

(4.1.3) Theorem.– Let I be a strict subset of S̃ and let i, j be integers between 0 and d−1. We have

(R_{π_{I}}^{∗})_{i,j} ≃ \{ \begin{matrix}\overline{𝔽}_{ℓ}[∂_{c^{−j}I}(i − j)] & if j ∉ I \\ 0 & if j ∈ I \end{matrix} .

Since π_{I} ≃ νj_{G}π_{c^{−j}I}, we see that the statement above is compatible with the twisting property (4.1.1.2). Therefore we only have to prove it when j = 0. We will treat separately the vanishing statement (when 0 ∈ I) and the non-zero cases (when 0 ∉ I), and we start with a special case.

(4.1.4) The case |I| = d − 1. Here we prove Theorem (4.1.3) for characters, i.e. for |I| = d − 1. By the above remark on the effect of twisting by ν_{G}, we may assume that I = S, so that π_{I} = \overline{𝔽}_{ℓ} is the trivial representation of G. In this case, we have

R_{\overline{𝔽}_{ℓ}}^{∗} = H^{∗} (RHom_{\overline{𝔽}_{ℓ}} (\overline{𝔽}_{ℓ} ⊗_{\overline{𝔽}_{ℓ}G}^{L} RΓ_{c}, \overline{𝔽}_{ℓ}) [1 − d]) .

By the second Lemma of paragraph (A.1.1) in [10], we have

H^{∗} (\overline{𝔽}_{ℓ} ⊗_{\overline{𝔽}_{ℓ}G}^{L} RΓ_{c}) ≃ H^{∗}(ℙ^{d−1,ca}, \overline{𝔽}_{ℓ}) = ⨁_{i=0}^{d−1} \overline{𝔽}_{ℓ}[−2i](−i),

where the action of D^{×} is trivial and that of W_{K} is described by the Tate twists. Forgetting about technicalities, this merely expresses the fact that G acts freely on the tower (M_{LT,n})_{n∈ℕ} and that the quotient is the so-called Gross-Hopkins period space, which is isomorphic to the projective space ℙ^{d−1} over K̂^{nr}. It follows that

R_{\overline{𝔽}_{ℓ}}^{∗} = ⨁_{i=0}^{d−1}(ν0_{D} ⊗ νi_{W})[1 − d + 2i].

Since ∂_{S}(i) = 1 − d + 2i, we have proved Theorem (4.1.3) for I = S, and thus for any I ⊂ S̃ of cardinality d − 1.

(4.1.5) Vanishing when j ∈ I. As already mentioned, we may assume that j = 0. Fix a strict subset I of S̃ which contains 0. We will prove in this paragraph that

(4.1.5.1)  for all i = 0, ⋯, d − 1, we have (R_{π_{I}}^{∗})_{i,0} = 0.

We argue by decreasing induction on |I|. The case |I| = d − 1 was treated in (4.1.4), so let us assume |I| < d − 1. Recall from Lemma (2.1.2) and Proposition (2.1.2) iii) that for any k ∈ S̃ \ I, we have c^{−k}I ⊂ S and

[νk_{G} ⊗ i_{c^{−k}I}(\overline{𝔽}_{ℓ})] = [i_{I}] = ∑_{J⊇I}[π_{J}].

Therefore, using the induction hypothesis, it is enough to find a k ∈ S̃ \ I such that

RHom_{ℤ_{ℓ}(G/ϖ^{ℤ})}((bRΓ_{c,ϖ})_{i,0}, νk_{G} ⊗ i_{c^{−k}I}(\overline{𝔽}_{ℓ})) = 0.

Let us start with a random k in S̃ \ I. By (3.2.4.2) and Frobenius reciprocity, we have an isomorphism

H^{∗}(RHom_{ℤ_{ℓ}(G/ϖ^{ℤ})}((bRΓ_{c,ϖ})_{i,0}, νk_{G} ⊗ i_{c^{−k}I}(\overline{𝔽}_{ℓ}))) ≃ Ext_{\overline{𝔽}_{ℓ}(G/ϖ^{ℤ})}^{∗+i} (ν−k_{G} ⊗ h̄_{i}, i_{c^{−k}I}(\overline{𝔽}_{ℓ})) .

Further, by (3.2.4.3) we have [ν−k_{G} ⊗ h̄_{i}] = [π_{c^{−k}{0,··· ,i}}] + [π_{c^{−k}{1,··· ,i}}]. Therefore, applying Proposition (2.3.4) i), we get

(4.1.5.2)  Ext_{G/ϖ^{ℤ}}^{∗} (ν−k_{G} ⊗ h̄_{i}, i_{c^{−k}I}(\overline{𝔽}_{ℓ})) = 0 whenever k ∈ \{1, ⋯, i\}.

In other words, if k ∈ {1, ⋯, i}, we are done. Let us thus assume that k ∉ {1, ⋯, i}. In this case, Proposition (2.3.4) ii) and Corollary (2.3.3) i) tell us that

Ext_{G/ϖ^{ℤ}}^{∗} (ν−k_{G} ⊗ h̄_{i}, i_{c^{−k}I}(\overline{𝔽}_{ℓ})) = 0 whenever c^{−k}\{1, ⋯, i\} ∪ c^{−k}I ≠ S.

This means that if I ∪ {1, ⋯, i} ≠ S̃ \ {k}, we are done. In particular, if i = 0 (in which case {1, ⋯, i} = ∅ by convention), we are done, because |I| < d − 1. Now let us assume the contrary, i.e. I ∪ {1, ⋯, i} = S̃ \ {k} (and therefore i ⩾ 1). Again because of |I| < d − 1, this means that {1, ⋯, i} contains an element k′ which does not belong to I. Applying (4.1.5.2) to this k′, we get

Ext_{G/ϖ^{ℤ}}^{∗} (ν−k′_{G} ⊗ h̄_{i}, i_{c^{−k′}I}(\overline{𝔽}_{ℓ})) = 0

and this finishes the proof of (4.1.5.1).

(4.1.6) Computation when j ∉ I. Again we may assume that j = 0, and hence that I ⊂ S. The vanishing property of (4.1.5) shows that the map π_{I} ↪ v_{I} := v_{I}(\overline{𝔽}_{ℓ}) induces isomorphisms

(4.1.6.1)  (R_{π_{I}}^{∗})_{i,0} ⟶^{∼} (R_{v_{I}}^{∗})_{i,0} for i = 0, ⋯, d − 1

because the cokernel v_{I}/π_{I} is isomorphic to π_{I∪{0}}.

Now, we will use the exact sequence (2.1.3.1) in order to compute (R_{v_{I}}^{∗})_{i,0}. It provides us with a spectral sequence

E_{1}^{pq} = ⨁_{\begin{matrix}S ⊇ J ⊇ I \\ |J \ I| = −p \end{matrix}} (R_{i_{J}}^{q})_{i,0} ⇒ (R_{v_{I}}^{p+q})_{i,0}

where we have abbreviated i_{J} := i_{J}(\overline{𝔽}_{ℓ}). A priori, this spectral sequence vanishes outside the range −|S \ I| ⩽ p ⩽ 0 and q ⩾ −i. Its differential d_{1} has degree (1, 0), and is given by the natural maps (R_{i_{J′}}^{∗})_{i,0} ⟶ (R_{i_{J}}^{∗})_{i,0} with signs associated to the simplicial set of subsets of S \ I.

The graded space R_{i_{J}}^{∗} is already known for J = S by (4.1.4) : we have R_{i_{S}}^{∗} = \overline{𝔽}_{ℓ}[1 − d + 2i]. Let us thus fix J ⊊ S. Using Frobenius reciprocity and (3.2.4.2) we then have isomorphisms

Ext_{G/ϖ^{ℤ}}^{∗+i} (h̄_{i}, i_{J}) ⟶^{∼} (R_{i_{J}}^{∗})_{i,0}

for i ∈ {0, ⋯, d − 1}. Further, using equality (3.2.4.3), Proposition (2.3.4), and Corollary (2.3.3) ii), we get isomorphisms

(R_{i_{J}}^{∗})_{i,0} ≃ Ext_{G/ϖ^{ℤ}}^{∗+i} (h̄_{i}, i_{J}) ≃ Ext_{G/ϖ^{ℤ}}^{∗+i} (π_{{1,··· ,i}}, i_{J}) ≃ Ext_{G/ϖ^{ℤ}}^{∗+i} (v_{{1,··· ,i}}, i_{J})
≃ \{ \begin{matrix}Ext_{G/ϖ^{ℤ}}^{∗+i−|S\\{1,··· ,i\}|} (\overline{𝔽}_{ℓ}, i_{J}) ≃ ⋀^{∗+2i+1−d} Y_{J} & if \{i + 1, ⋯, d − 1\} ⊆ J \\ 0 & otherwise \end{matrix} .

Observe that the smallest J which contributes is J(i, I) := I ∪ {i + 1, ⋯, d − 1}. In particular, the E_{1} page of the spectral sequence is supported in the vertical strip defined by

−|S \ I| ⩽ p ⩽ −|J(i, I) \ I|.

Moreover, since dim Y_{J} = d − 1 − |J|, we see that for each p in the above range, the column E_{1}^{p∗} is supported in the range

d − 1 − 2i ⩽ q ⩽ 2d − 2 − 2i − p − |I|.

In other words, the E_{1} page is supported in the half square with left corner

(−|S \ I|, d − 1 − 2i)

and right corners

(−|J(i, I) \ I|, d − 1 − 2i) and (−|J(i, I) \ I|, 2d − 2 − 2i − |J(i, I)|).

Now, we observe that the E_{1}^{∗•} of our spectral sequence is the same, up to some shifts, as that which occurs in [20, Proof of Thm 1] and [2, Ch. X, Prop 4.7]. We still have to compare the differential d_{1} with that of these two references. Using Corollary (2.3.4) ii) again, we see that for J′ ⊇ J with J′ a strict subset of S, the non-zero map (R_{i_{J′}}^{∗})_{i,0} ⟶ (R_{i_{J}}^{∗})_{i,0} is induced by the natural map Y_{J′} ⟶ Y_{J}. It follows that for p > −|S/I| (i.e. everywhere except maybe on the first non-zero column), the differential d_{1}^{pq} is the same as that of the two references cited above. In fact the only possible difference concerns d_{1}^{−|S/I|,d−1−2i} for which we have no control yet in our setting, except in the trivial case where i = 0, because in this case, the E_{1} is supported on one point. For i > 0, in order to ensure that d_{1}^{−|S/I|,d−1−2i} is the same as in the two references cited above, we have to prove that each map

(4.1.6.2)  (R_{i_{S}}^{d−1−2i})_{i,0} ≃ \overline{𝔽}_{ℓ} ⟶ (R_{i_{J}}^{d−1−2i})_{i,0} ≃ \overline{𝔽}_{ℓ}

is an isomorphism. However, since we know that all the other maps (R_{i_{J}}^{d−1−2i})_{i,0} ⟶ (R_{i_{J′}}^{d−1−2i})_{i,0} for J′ ⊂ J ⊊ S are isomorphisms, it is sufficient to prove that (4.1.6.2) is an isomorphism for a single J ! For this, we look at the special case I = ∅, so that the left corner is (1−d, d−1−2i). If all the maps (4.1.6.2) were zero, we would have E_{2}^{1−d,d−1−2i} ≃ \overline{𝔽}_{ℓ}, creating a non-zero R_{π_{∅}}^{−2i}, which is absurd since R_{π_{∅}}^{∗} has to vanish for ∗ < −i, and i was supposed to be positive.

Therefore, we have identified the first page of our spectral sequence with that of [20, Proof of Thm 1] and [2, Ch. X, Prop 4.7], up to schifts. Using their results, we get that E_{2}^{pq} is always 0 except in the upper right corner of the triangle, where it is 1-dimensional. Therefore we get

(R_{π_{I}}^{∗})_{i,0} ≃ \overline{𝔽}_{ℓ}[−(2d − 2 − 2i − 2|J(i, I)| + |I|)].

We still need to compute the schift a = −(2d − 2 − 2i − 2|J(i, I)| + |I|). Observe first that 2d − 2 − 2|J(i, I)| = 2|{1, ⋯, i} \ I|. Then, using i − |{1, ⋯, i} \ I| = |{1, ⋯, i} ∩ I|, we get a = 2|{1, ⋯, i}∩I|−|I|. Eventually, using the equality i+|I| = |{1, ⋯, i}∩I|+|{1, ⋯, i}∪I|, we get

a = |{1, ⋯, i} ∩ I| − |{1, ⋯, i} ∪ I| + i = ∂_{I}(i).

The proof of Theorem (4.1.3) is now complete. However, it will be important to keep some track of the isomorphism (R_{π_{I}}^{−∂_{I}(i)})_{i,0} ≃ \overline{𝔽}_{ℓ} we have just obtained, when we study the Lefschetz operator in next subsection. We may decompose this isomorphism in four steps :

i) The spectral sequence provides the isomorphism (R_{π_{I}}^{−∂_{I}(i)})_{i,0} ≃ (R_{i_{J(i,I)}}^{−∂_{I}(i)+|J(i,I)\I|})_{i,0}.

ii) Corollary (2.3.3) and Remark (2.3.3) exhibit an isomorphism

(R_{i_{J(i,I)}}^{d−1−2i})_{i,0} ⊗ ⋀^{max} Y_{J(i,I)} ⟶^{∼} (R_{i_{J(i,I)}}^{−∂_{I}(i)+|J(i,I)\I|})_{i,0}.

iii) The inclusion \overline{𝔽}_{ℓ} = i_{S} ↪ i_{J(i,I)} induces the isomorphism (R_{i_{S}}^{d−1−2i})_{i,0} ≃ (R_{i_{J(i,I)}}^{d−1−2i})_{i,0}, as was shown in the above proof.

iv) The geometric input from (4.1.4) provides the isomorphism (R_{\overline{𝔽}_{ℓ}}^{d−1−2i})_{i,0} ≃ \overline{𝔽}_{ℓ}.

4.2 The Lefschetz operator

We now study the Lefschetz operator recalled in the Introduction. We refer the reader to [9, 2.2.4] for the precise definition of this operator and will contend ourselves with recalling the relevant details when necessary in the proof of Theorem (4.2.2) below.

(4.2.1) Our aim is to describe the operator L_{π}^{∗} : R_{π}^{∗} ⟶ R_{π}^{∗}[2](1) for π a unipotent elliptic representation. Since this operator is D^{×} × W_{K}-equivariant, it decomposes as a sum L_{π}^{∗} = ∑_{i,j}(L_{π}^{∗})_{i,j} with

(L_{π}^{∗})_{i,j} : (R_{π}^{∗})_{i,j} ⟶ (R_{π}^{∗+2})_{i−1,j}.

It also satisfies the following compatibility with torsion :

(4.2.1.1)  (L_{ν_{G}π}^{∗})_{i,j} = (L_{π}^{∗})_{i−1,j−1},

where equality merely means that these morphisms are part of a commutative diagram involving isomorphisms (4.1.1.2). Thanks to (4.2.1.1), we may restrict our attention to the case j = 0.

Now, recall from Theorem (4.1.3) that each (R_{π_{I}}^{∗})_{i,0} is zero unless I ⊆ S. In the latter case, it is 1-dimensional and concentrated in degree −∂_{I}(i). Therefore (L_{π_{I}}^{∗})_{i,0} is necessarily 0 as soon as ∂_{I}(i) ≠ ∂_{I}(i − 1) + 2, which by Fact (4.1.2) is equivalent to i ∉ I. The following theorem asserts that (L_{π_{I}}^{∗})_{i,j} is non-zero in the remaining cases.

(4.2.2) Theorem.– Let I be a subset of S and let i ∈ I. Then the operator

(L_{π_{I}}^{∗})_{i,0} : (R_{π_{I}}^{∗})_{i,0} ≃ \overline{𝔽}_{ℓ}[∂_{I}(i)] ⟶ (R_{π_{I}}^{∗})_{i−1,0}[2] ≃ \overline{𝔽}_{ℓ}[∂_{I}(i − 1) + 2]

is non-zero, and thus is an isomorphism.

Proof. As in the proof of Theorem (4.1.3), the crucial input comes from geometry, which rules out the case of the trivial representation \overline{𝔽}_{ℓ} = π_{S}. Indeed, recall from (4.1.4) that the period map provides us with isomorphisms

R_{π_{S}}^{∗}[1 − d] ≃ (H^{∗}(ℙ^{d−1,ca}, \overline{𝔽}_{ℓ}))^{∨} = ⨁_{i=0}^{d−1} \overline{𝔽}_{ℓ}[2i](i).

But by its mere definition, the Lefschetz operator of [9, 2.2.4] induces the tautological Lefschetz operator on ℙ^{d−1}, namely that given by the Chern class of the tautological sheaf. It is well known to induce isomorphisms H^{i}(ℙ^{d−1,ca}, \overline{𝔽}_{ℓ}) ⟶^{∼} H^{i+2}(ℙ^{d−1,ca}, \overline{𝔽}_{ℓ})(1) for 0 ⩽ i < 2d − 2, thereby proving the theorem for I = S.

We now consider a general I ⊂ S. We will use the four steps gathered in the end of Paragraph (4.1.6), and which summarize the origin of the isomorphism (R_{π_{I}}^{−∂_{I}(i)})_{i,0} ≃ \overline{𝔽}_{ℓ}. Motivated by step iii) in that list, we consider for any J ⊂ S the following commutative diagram, which is functorially induced by the inclusion map i_{S} ↪ i_{J}

\xymatrix{(R_{i_{S}}^{d−1−2i})_{i,0} \ar[r] \ar[d]_{L_{i_{S}}^{d−1−2i}} & (R_{i_{J}}^{d−1−2i})_{i,0} \ar[d]^{L_{i_{J}}^{d−1−2i}} \\ (R_{i_{S}}^{d−1−2i+2})_{i−1,0} \ar[r] & (R_{i_{J}}^{d−1−2i+2})_{i−1,0}}

The two horizontal maps were shown to be isomorphisms in (4.1.6), and the left vertical map has just been shown to be so. We conclude that (L_{i_{J}}^{d−1−2i})_{i,0} is an isomorphism.

Further, let us consider the diagram for k ∈ ℕ

\xymatrix{(R_{i_{J}}^{d−1−2i})_{i,0} ⊗_{\overline{𝔽}_{ℓ}} ⋀^{k} Y_{J} \ar[r] \ar[d]_{L_{i_{J}}^{d−1−2i}⊗Id} & (R_{i_{J}}^{d−1−2i+k})_{i,0} \ar[d]^{L_{i_{J}}^{d−1−2i+k}} \\ (R_{i_{J}}^{d−1−2i+2})_{i−1,0} ⊗_{\overline{𝔽}_{ℓ}} ⋀^{k} Y_{J} \ar[r] & (R_{i_{J}}^{d−1−2i+2+k})_{i−1,0}}

The horizontal maps are explained in Remark (2.3.1) and the functoriality of these maps insures that the diagram is commutative. It follows from the identification (R_{i_{J}}^{∗})_{i,0} ≃ Ext_{G/ϖ^{ℤ}}^{∗+i}(v_{{1,··· ,i}}, i_{J}) explained in the course of (4.1.6), together with Remark (2.3.3) that these maps are isomorphisms. Since the left vertical map has just been shown to be an isomorphism, so is the right one (L_{i_{J}}^{d−1−2i+k})_{i,0}.

Recall now the notation J(i, I) = I∪{i+1, ⋯, d−1} of (4.1.6), and observe that J(i−1, I) = J(i, I) since we assume i ∈ I. Recall also that ∂_{I}(i) = ∂_{I}(i − 1) + 2 under this assumption, and consider the diagram

\xymatrix{(R_{π_{I}}^{−∂_{I}(i)})_{i,0} \ar[r] \ar[d]_{L_{π_{I}}^{−∂_{I} (i)}} & (R_{i_{J(i,I)}}^{−∂_{I}(i)+|J(i,I)\I|})_{i,0} \ar[d]^{L_{i_{J(i,I)}}^{−∂_{I} (i)+|J(i,I)\I|}} \\ (R_{π_{I}}^{−∂_{I}(i−1)})_{i−1,0} \ar[r] & (R_{i_{J(i,I)}}^{−∂_{I}(i−1)+|J(i,I)\I|})_{i−1,0}}

where the horizontal maps are provided by the spectral sequence considered in (4.1.6) (these are edge maps once we know enough on the support of the spectral sequence). These maps were shown to be isomorphisms in (4.1.6), and we have just proved that the vertical right hand map is also an isomorphism. We conclude that L_{π_{I}}^{−∂_{I}(i)} is an isomorphism, as desired.

(4.2.3) Recollection and proof of the Main Theorem. We now prove the theorem announced in the Introduction. In particular we forget all gradings. We first assume that π is a unipotent (or principal series) elliptic representation. Let I be the strict subset of S̃ such that π ≃ π_{I}. By (4.1.1.1) and (4.1.3), we have

R_{π}^{∗,ss} ≃ ⨁_{i,j=0}^{d−1}(R_{π_{I}}^{∗})_{i,j} ≃ ⨁_{j∉I} νj_{D} ⊗ (R_{π}^{∗,ss})_{j} with (R_{π}^{∗,ss})_{j} := ⨁_{i=0}^{d−1}(R_{π_{I}}^{∗})_{i,j} = ⨁_{i=0}^{d−1} νi_{W} .

According to Theorem (4.2.2) and the explicit description of Proposition (2.2.3) we have

((R_{π}^{∗,ss})_{0}, L_{π}^{∗}) ≃ (σ^{ss}(π), L(π)).

Applying again Theorem (4.2.2) to c^{−j}I and using compatibility with twisting (4.2.1.1), we get for any j ∉ I

((R_{π}^{∗,ss})_{j}, L_{π}^{∗}) ≃ ν_{W}^{j} ⊗ (σ^{ss}(π_{c^{−j}I}), L(π_{c^{−j}I})) ≃ (σ^{ss}(π), L(π)).

Recalling now Proposition (2.2.1), we eventually get

(R_{π}^{∗}, L_{π}^{∗})^{ss} ≃ |LJ(π)| ⊗ (σ^{ss}(π), L(π)),

as desired.

In order to finish the proof of the Main Theorem, we still have to deal with the case when π is not elliptic. In this case we must show that R_{π}^{∗} = 0. Here we use the full force of the Vignéras-Zelevinski classification in [25, V.12]^{3}. Following this classification, there is a proper parabolically induced representation ι which contains π as a subquotient with multiplicity one, and all other subquotients π′ of which satisfy the condition λ_{π′} < λ_{π}. Here, λ_{π} is the partition associated to π via the successive highest derivatives. Hence, arguing by induction on λ_{π}, we see that it suffices to prove that R_{ι}^{∗} = 0. Write ι = i_{P} (τ) for some proper standard parabolic subgroup P = MU and some irreducible representation τ. Then R_{ι}^{∗} = ⨁_{i=0}^{d−1} Ext_{M}^{∗} (r_{P} v_{{1,··· ,i}}, τ). But since π is not elliptic, the cuspidal support of τ is disjoint from W.δ. Therefore, item i) of Lemma (2.3.1) shows that each Ext group occuring in the above sum vanishes.

(4.2.4) Remark on non-unipotent representations. The Main Theorem may remain true for any irreducible \overline{𝔽}_{ℓ}-representation π of G, under the Coxeter congruence hypothesis. In fact, much is already known ; Boyer has described the cohomology of the whole tower and has announced recently that the integral cohomology is torsion-less. This allows to split the full complex according to weights. Then our arguments, which are somehow inductive on the “Whittaker level”, work fine for arbitrary elliptic representations, except that the induction has to be initialized at some point. For unipotent representations, the initialization was the computation of (R_{\overline{𝔽}_{ℓ}}^{∗}, L_{π}^{∗}) thanks to the period map.

All in all, our arguments show that the Main Theorem is true for any representation π, provided it holds true for any super-Speh representation, in the sense of [11, Def 2.2.3].

3A more detailed account of this classification will appear in a current work by Minguez and Secherre

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