Ihara’s Lemma for $\operatorname{GL}_{d}$: the limit case

In the Taylor-Wiles method Ihara’s lemma is the key ingredient to extend a $R=T$ property from the minimal case to a non minimal one. It is usually formulated by the injectivity of some map as follows.

Let $\Gamma=\Gamma_{0}(N)$ be the usual congruence subgroup of $SL_{2}({\mathbb{Z}})$ for some $N>1$, and for a prime $p$ not dividing $N$ let $\Gamma^{\prime}:=\Gamma\cap\Gamma_{0}(p)$. We then have two degeneracy maps

between the compactified modular curves of levels $\Gamma^{\prime}$ and $\Gamma$ respectively, induced by the inclusion

For $l\neq p$, we then have a map

Diamond and Taylor in [DT94] proved an analogue of Ihara’s lemma for Shimura curves over ${\mathbb{Q}}$. For a general totally real number field $F$ with ring of integers $\mathcal{O}_{F}$, Manning and Shotton in [MS21] succeeded to prove it under some large image hypothesis. Their strategy is entirely different from those of [DT94]but consists roughly

• •

  to carry Ihara’s lemma for a compact Shimura curve $Y_{\bar{K}}$ associated to a definite quaternion algebra $\overline{D}$ ramified at some auxiliary place $v$ of $F$, in level $\bar{K}=\bar{K}^{v}\bar{K}_{v}$ an open compact subgroup of $D\otimes{\mathbb{A}}_{F,f}$ unramified at $v$,

• •

  to the indefinite situation $X_{K}$ relatively to a quaternion division algebra $D$ ramified at all but one infinite place of $F$, and isomorphic to $\bar{D}$ at all finite places of $F$ different to $v$, and with level $K$ agreing with $\bar{K}^{v}$ away from $v$.

Indeed in the definite case Ihara’s statement is formulated by the injectivity of

where both $\overline{D}$ and $\bar{K}$ are unramified at the place $v$ and $\bar{K}_{0}(v)_{v}$ is the subgroup of $\operatorname{GL}_{2}(F_{v})$ of elements which are upper triangular modulo $p$.

The proof goes like this, cf. [MS21] theorem 6.8. Suppose $(f,g)\in\ker\pi^{*}$. Regarding $f$ and $g$ as $K^{v}$-invariant function on $\overline{G}(F)\backslash\overline{G}({\mathbb{A}}_{F,f})$, then $f(x)=-g(x\omega)$ where $\omega=\left(\begin{array}[]{cc}\varpi_{v}&0\\ 0&1\end{array}\right)$, $\varpi_{v}$ being an uniformizer for $F_{w}$ and $\overline{G}$ being the algebraic group over $\mathcal{O}_{F}$ associated to $\mathcal{O}_{\overline{D}}^{\times}$ the inversible group of the maximal order $\mathcal{O}_{\overline{D}}$ of $\overline{D}$: note that $\overline{G}(F_{v})\cong\operatorname{GL}_{2}(F_{v})$. Then $f$ is invariant under $K^{v}$ and $\omega^{-1}K^{v}\omega$ so that, using the strong approximation theorem for the subgroup of $\overline{G}$ of elements of reduced norm $1$, then $f$ factors through the reduced norm map, and so is supported on Eisenstein maximal ideals.

The link between $X_{K}$ and $Y_{K^{v}}$ is given by the geometry of the integral model of the Shimura curve $X_{K_{0}(v)}$ with $\Gamma_{0}(v)$-level structure. The main new ingredient of [MS21] to carry this geometric link to Ihara’s lemma goes through the patching technology which allows to obtain maximal Cohen-Macaulay modules over deformation rings. Using a flatness property and Nakayama’s lemma, there are then able to extend a surjective property, dual to the injectivity in the Ihara’s lemma, from the maximal unipotent locus on the deformation space to the whole space, and recover the Ihara’s statement reducing by the maximal ideal of the deformation ring.

Recently Caraiani and Tamiozzo following closely [MS21] also obtained Ihara’s lemma for Hilbert varieties essentially because Galois deformations rings are the same and so regular which is not the case beyond $\operatorname{GL}_{2}$.

To generalize the classical Ihara’s lemma for $\operatorname{GL}_{d}$, there are essentially two approaches.

The first natural one developed by Clozel, Harris and Taylor in their first proof of Sato-Tate theorem [CHT08], focuses on the $H^{0}$ with coefficients in ${\mathbb{F}}_{l}$ of a zero dimensional Shimura variety associated to higher dimensional definite division algebras. More precisely consider a totally real field $F^{+}$ and a imaginary quadratic extension $E/{\mathbb{Q}}$ and define $F=F^{+}E$. We then consider $\overline{G}/{\mathbb{Q}}$ an unitary group with $\overline{G}({\mathbb{Q}})$ compact so that $\overline{G}$ becomes an inner form of $\operatorname{GL}_{d}$ over $F$. This means, cf. §2.3, we have fixed a division algebra $\overline{B}$ with center $F$, of dimension $d^{2}$, provided with an involution of the second kind such that its restriction to $F$ is the complex conjugation. We moreover suppose that at every place $w$ of $F$, either $\overline{B}_{w}$ is split or a local division algebra.

Let $v$ be a place of $F$ above a prime number $p$ split in $E$ and such that $\overline{B}_{v}^{\times}\cong\operatorname{GL}_{d}(F_{v})$ where $F_{v}$ is the associated local field with ring of integers $\mathcal{O}_{v}$ and residue field $\kappa(v)$.

Consider then an open compact subgroup $\overline{K}^{v}$ infinite at $v$ in the following sense: $\overline{G}({\mathbb{Q}}_{p})\cong{\mathbb{Q}}_{p}^{\times}\times\prod_{v_{i}^{+}}\overline{B}_{v_{i}^{+}}^{op,\times}$ where $p=\prod_{i}v_{i}^{+}$ in $F^{+}$ and we identify places of $F^{+}$ over $p=uu^{c}\in E$ with places of $F$ over $u$. We then ask $\overline{K}^{v}_{p}={\mathbb{Z}}_{p}^{\times}\times\prod_{w|u}\overline{K}_{w}$ to be such that $\overline{K}_{v}$ is restricted to the identity element.

The associated Shimura variety with level $\overline{K}=\overline{K}^{v}\overline{K}_{v}$ for some finite level $\overline{K}_{v}$ at $v$, denoted by $\overline{\mathrm{Sh}}_{\overline{K}}$, is then such that its ${\mathbb{C}}$-points are $\overline{G}({\mathbb{Q}})\backslash\overline{G}({\mathbb{A}}_{{\mathbb{Q}}}^{\infty})/\overline{K}$ and for $l$ a prime not divisible by $v$, its $H^{0}$ with coefficients in $\overline{\mathbb{F}}_{l}$ is then identified with the space

Replacing $\overline{K}$ by $\overline{K}^{v}$, we then obtain an admissible smooth representation of $\operatorname{GL}_{d}(F_{v})$ equipped with an action of the Hecke algebra ${\mathbb{T}}(\overline{K}^{v})$ defined as the image of the abstract unramified Hecke algebra, cf. definition 3.2.1, inside $\operatorname{End}(S_{\overline{G}}(\overline{K}^{v},\overline{\mathbb{F}}_{l})\Bigr{)}$.

To a maximal ideal $\mathfrak{m}$ of ${\mathbb{T}}(\overline{K}^{v})$ is associated a Galois $\overline{\mathbb{F}}_{l}$-representation $\overline{\rho}_{\mathfrak{m}}$, cf. §4.2. We consider the case where this representation is irreducible. Note in particular that such an $\mathfrak{m}$ is then not pseudo-Eisenstein in the usual terminology.

For rank $2$ unitary groups, we recover the previous statement as the characters are exactly those representations which do not have a Whittaker model, i.e. are the non generic ones. For $d\geq 2$, over $\overline{\mathbb{Q}}_{l}$, the generic representations of $\operatorname{GL}_{d}(F_{v})$ are the irreducible parabolically induced representations $\mathrm{st}_{t_{1}}(\pi_{v,1})\times\cdots\times\mathrm{st}_{t_{r}}(\pi_{v,r})$ where for $i=1,\cdots,r$,

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  $\pi_{v,i}$ is an irreducible cuspidal representation of $\operatorname{GL}_{g_{i}}(F_{v})$,

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  $\mathrm{st}_{t_{i}}(\pi_{v,i})$ is a Steinberg representations, cf. definition 2.1.2,

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  $\sum_{i=1}^{r}t_{i}g_{i}=d$ where the Zelevinsky segments $[\pi_{v,i}\{\frac{1-t_{i}}{2}\},\pi_{v,i}\{\frac{t_{i}-1}{2}\}]$are not linked in the sense of [Zel80].

Over $\overline{\mathbb{F}}_{l}$ every irreducible generic representation is obtained as the unique generic subquotient of the modulo $l$ reduction of a generic representation. It can also be characterized intrinsically using representation of the mirabolic subgroup, cf. §2.1.

The second approach asks to find a map playing the same role as $\pi^{*}=\pi_{1}^{*}+\pi_{2}^{*}$. It is explained in section 5.1 of [CHT08] with the help of the element

constructed by Russ Mann, cf. proposition 5.1.7 of [CHT08], where $F_{v}$ is here a finite extension of ${\mathbb{Q}}_{p}$ with ring of integers $\mathcal{O}_{v}$.

Recall that $l$ is called quasi-banal for $\operatorname{GL}_{d}(F_{v})$ if either $l\nmid\sharp\operatorname{GL}_{d}(\kappa_{v})$ (the banal case) or $l>d$ and $q_{v}\equiv 1\mod l$ (the limit case).

In [Mos21], the author also proved that Ihara’s property in the quasi-banal case is equivalent to the following result.

With the previous notations, let $q_{v}$ be the order of the residue field of $F_{v}$. We fix some prime number $l$ unramified in $F^{+}$ and split in $E$ and we place ourself in the limit case where $q_{v}\equiv 1\mod l$ with $l>d$, which is, after by base change, the crucial case to consider.

• •

  The last two hypothesis come from theorem 5.1.5 of [Gee11] which is some level raising and lowering statement, cf. theorem 4.2.2. Any other similar statement, for example theorem 4.4.1 of [BLGGT14], with different hypothesis can then be used to modify the hypothesis of the theorem above.

• •

  Our techniques work also in the banal case as soon as you avoid cuspidal $\overline{\mathbb{F}}_{l}$-representations which are not supercuspidal which is for example the case if you suppose that, after semi-simplification, $\overline{\rho}_{\mathfrak{m},v}$ is a direct sum of characters. In particular the resulting statement is more general than those of [Boy22].

The basic idea¹¹1this explains the hypothesis on the existence of $p_{0}$ in the statement, cf. §4.1, as in [Boy22], is to introduce geometry and move from the Shimura variety associated to $\overline{G}$ which is of dimension zero, to another Shimura variety $\operatorname{Sh}_{K}$ associated to some reductive group $G$ and level $K$, of strictly positive dimension, so that $S_{\overline{G}}(\overline{K},\overline{\mathbb{F}}_{l})$ appears in a certain cohomology group of some sheaf over $\operatorname{Sh}_{K}$. The strategy is then to construct a filtration, coming from geometry, of this cohomology group so that the graded parts, which are expected to be more easy to handle with, also verify the genericity property of there irreducible sub-spaces.

More explicitly we study the middle degree cohomology group of the KHT Shimura variety $\operatorname{Sh}_{K^{v}(\infty)}$ associated to some similitude group $G/{\mathbb{Q}}$ such that $G({\mathbb{A}}_{\mathbb{Q}}^{\infty,p})\cong\overline{G}({\mathbb{A}}_{\mathbb{Q}}^{\infty,p})$, cf. §2.3 for more details, and with level $K^{v}(\infty):=\overline{K}^{v}$ meaning finite level outside $v$ and infinite level at $v$. The localization at $\mathfrak{m}$ of the cohomology groups of $\operatorname{Sh}_{K^{v}(\infty)}$ can be computed as the cohomology of the geometric special fiber $\operatorname{Sh}_{K^{v}(\infty),\bar{s}_{v}}$ of $\operatorname{Sh}_{K^{v}(\infty)}$, with coefficient in the complex of nearby cycles $\Psi_{K^{v}(\infty),v}$.

The Newton stratification of $\operatorname{Sh}_{K^{v}(\infty),\bar{s}_{v}}$ gives us a filtration of $\Psi_{K^{v}(\infty),v}$, cf. [Boy20], and so a filtration $\operatorname{Fil}^{\bullet}(K^{v}(\infty))$ of $H^{d-1}(\operatorname{Sh}_{K^{v}(\infty),\bar{\eta}_{v}},\overline{\mathbb{Z}}_{l})_{\mathfrak{m}}$ and the main point of [Boy22] is to prove that the modulo $l$ reduction of each graded part of this filtration verifies the Ihara property, i.e. each of their irreducible sub-space are generic. To realize this strategy we need first the cohomology groups of $\operatorname{Sh}_{K^{v}(\infty)}$ to be torsion free: this point is now essentially settled by the main result of [Boy23a]. More crucially the previous filtration $\operatorname{Fil}^{\bullet}(K^{v}(\infty))$ should be strict, i.e. its graded parts have to be torsion free, cf. theorem 3.2.3.

It appears that the graded parts of $\operatorname{Fil}^{\bullet}(K^{v}(\infty))$ are parabolically induced and in the limit case when the order $q_{v}$ of the residue field is such that $q_{v}\equiv 1\mod l$, the socle of the modulo $l$ reduction of these parabolic induced representations are no more irreducible and do not fulfill the Ihara property, i.e. some of their subspaces are not generic. It then appears that we have at least

• •

  to verify that the modulo $l$ reduction of the first non trivial graded part of $\operatorname{Fil}^{\bullet}(K^{v}(\infty))$ verifies the genericity property of its irreducible submodule. For this we need a level raising statement as theorem 5.1.5 in [Gee11], cf. theorem 4.2.2, or theorem 4.4.1 of [BLGGT14].

• •

  Then we have to understand that the extensions between the graded parts of $\operatorname{Fil}^{\bullet}(K^{v}(\infty))\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$ are non split.

One problem about this last point is that the $\overline{\mathbb{Q}}_{l}$-cohomology is split. For any irreducible automorphic representation $\Pi$ of $G({\mathbb{A}})$ cohomological for, say, the trivial coefficients, the $\overline{\mathbb{Z}}_{l}$-cohomology defines a lattice $\Gamma(\Pi)$ of $(\Pi^{\infty})^{K^{v}(\infty)}\otimes\sigma(\Pi)_{v}$ whose modulo $l$ reduction gives a subspace of the $\overline{\mathbb{F}}_{l}$-cohomology: Ihara’s lemma predicts that the socle of this subspace is still generic, i.e. it gives informations about the lattice $\Gamma(\Pi)$. We then see that non splitness of $\operatorname{Fil}^{\bullet}(K^{v}(\infty))\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$ should be understood in a very flexible point of view.

One possible strategy is, using the fact that the $\overline{\mathbb{Q}}_{l}$-cohomology is split, to start from the filtration $\operatorname{Fil}^{\bullet}(K^{v}(\infty))$ and modify it in order to arrive to another one where all the modulo $l$ reduction of the graded parts fulfill the Ihara property i.e. their irreducible subspaces are generic. The main ingredient to construct modifications of filtrations is to consider following situations:

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  a filtration $\operatorname{Fil}^{\bullet}$ of $H^{d-1}(\operatorname{Sh}_{K^{v}(\infty),\bar{\eta}_{v}},\overline{\mathbb{Z}}_{l})_{\mathfrak{m}}$ whose graded parts $\operatorname{gr}^{\bullet}$ are torsion free;

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  let $k$ and $X:=\operatorname{Fil}^{k}/\operatorname{Fil}^{k-2}$ such that $X\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{Q}}_{l}\cong(gr^{k-1}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{Q}}_{l})\oplus(\operatorname{gr}^{k}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{Q}}_{l})$.

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  We can then define $\widetilde{\operatorname{gr}^{k-1}}$ and $\widetilde{\operatorname{gr}^{k}}$ with

  obtained by taking $\widetilde{\operatorname{gr}^{k}}:=X\cap(\operatorname{gr}^{k}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{Q}}_{l})$, and where $T$ is torsion. Passing modulo $l$, we then obtained a priori two distinct filtrations.

Let us first explain why something interesting should happen during this process.

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  We can define a $\overline{\mathbb{F}}_{l}$-monodromy operator for the Galois action at the place $v$.²²2Note that over $\overline{\mathbb{F}}_{l}$ the usual arithmetic approach for defining the nilpotent monodromy operator, is hopeless because, up to consider a finite extension of $F_{v}$, such a $\overline{\mathbb{F}}_{l}$-representation has a trivial action of the inertia group. We are looking for a geometric monodromy operator $N^{geo}_{v}$ which then exists whatever are the coefficients, $\overline{\mathbb{Q}}_{l}$, $\overline{\mathbb{Z}}_{l}$ and $\overline{\mathbb{F}}_{l}$, compatible with tensor products. One classical construction is known in the semi-stable reduction case, cf. [Ill94] §3, which corresponds to the case where the level at $v$ of our Shimura variety is of Iwahori type.³³3This corresponds to automorphic representations $\Pi$ such that the cuspidal support of $\Pi_{v}$ is made of unramified characters, and so with the weak form of Ihara’s lemma of definition 1.2.3. Using our knowledge of the $\overline{\mathbb{Z}}_{l}$-nearby cycles described completely in [Boy23b], we can construct such a geometric nilpotent monodromy operator which generalizes the semi-stable case, cf. §3.3.

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  Taking this geometric monodromy operator, we then obtain a cohomological monodromy operator $N_{v,\mathfrak{m}}^{coho}$ acting on $H^{0}(\operatorname{Sh}_{K,\bar{s}_{v}},\Psi_{K^{v}(\infty),v})_{\mathfrak{m}}$ One of the main point, cf theorem 3.2.3, is that the graded parts of the filtration of $H^{0}(\operatorname{Sh}_{K,\bar{s}_{v}},\Psi_{K^{v}(\infty),v})_{\mathfrak{m}}$ induced by the Newton filtration on the nearby cycles spectral sequence, are all torsion free, so that in particular we are in position to understand quite enough the action of $\overline{N}^{coho}_{v,\mathfrak{m}}:=N^{coho}_{v,\mathfrak{m}}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$ on $H^{0}(\operatorname{Sh}_{K,\bar{s}_{v}},\Psi_{K^{v}(\infty),v})_{\mathfrak{m}}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$, and prove that its nilpotency order is as large as possible.

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  Note that as $\overline{\rho}_{\mathfrak{m}}$ is supposed to be irreducible, then the modulo $l$ reduction of the monodromy operator acting on $\rho_{\widetilde{\mathfrak{m}}}$ does not depend on the choice of the prime ideal $\widetilde{\mathfrak{m}}\subseteq\mathfrak{m}$ so that it is usually trivial.

Finally, as $N_{v}^{coho}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$ is far from being trivial, there should be non split extensions between the graded parts of $\operatorname{Fil}^{\bullet}(K^{v}(\infty))$.

However this strategy seems difficult to implement directly because of counting problems: to deal with finite number of representations you need to work with a finite level at the place $v$ and then pass to the limit. It seems first difficult to count liftings of a fixed representation and secondly when increasing the level, it should be not easy to glue back things together. Our approach in some sense consists to consider all the liftings together using typicness of the cohomology, cf. §4.3. The proof finally goes into three main steps:

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  we first prove, cf. theorem 3.2.3, that the filtration of the middle cohomology group constructed from the filtration of stratification of the nearby cycles perverse sheaf, has torsion free graded parts, otherwise the all cohomology would have non trivial torsion classes which is not the case by [Boy23a];

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  secondly, §3.3, using results from [Boy23b] about various filtrations of stratification of the nearby cycle perverse sheaf over $\overline{\mathbb{Z}}_{l}$, we define an integral monodromy operator on the nearby cycles perverse sheaf whose order of nilpotency modulo $l$ coincides with those over $\overline{\mathbb{Q}}_{l}$;

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  this integral geometric monodromy operator gives us a monodromy operator on $\rho_{\mathfrak{m}}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$, a representation with coefficients in an artinian local ring defined as the modulo $l$ reduction of the image of some Hecke algebra acting on the middle cohomology group with finite level. Working at various finite level and using Matlis duality for artinian ring, we prove, §4.3, the genericity of the socle of the middle cohomology group at infinite level at $v$, for the action of $\operatorname{GL}_{d}(F_{v})$.

To conclude this long introduction, note that Ihara’s lemma in Clozel-Harris-Taylor formulation, was stated in order to be able to do level raising. In our proof we use level raising statements, proved thanks to Taylor’s Ihara avoidance in [Tay08], in order to prove Ihara’s lemma. Then we can see our arguments as the proof that level raising implies Ihara’s lemma in the limit case.

Consider a finite extension $L/{\mathbb{Q}}_{p}$ with residue field ${\mathbb{F}}_{q}$. We denote by $|-|$ its absolute value. For a representation $\pi$ of $\operatorname{GL}_{d}(L)$ and $n\in\frac{1}{2}{\mathbb{Z}}$, set

Recall that a representation $\varrho$ of $\operatorname{GL}_{d}(L)$ is called cuspidal (resp. supercuspidal) if it is not a subspace (resp. subquotient) of a proper parabolic induced representation. When the field of coefficients is of characteristic zero, these two notions coincides, but this is no more true over $\overline{\mathbb{F}}_{l}$.

More generally the set of sub-quotients of the induced representation (1) is in bijection with the following set.

For any $\underline{s}\in\operatorname{Dec}(s)$, we the denote by $\mathrm{st}_{\underline{s}}(\pi)$ the associated irreducible sub-quotient of (1). Following Zelevinsky, we fix this bijection such that $\operatorname{Speh}_{s}(\pi)$ corresponds to $(s)$ and $\mathrm{st}_{s}(\pi)$ to $(1,\cdots,1)$. The Lubin-Tate representation $LT_{h,s}(\pi)$ will also appear in the following, it corresponds with $(\overbrace{1,\cdots,1}^{h},s-h)$.

We now suppose as explained in the introduction that

so the following facts are verified (cf. [Vig96] §III):

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  the modulo $l$ reduction of every irreducible cuspidal representation of $\operatorname{GL}_{g}(L)$ for $g\leq d$, is supercuspidal⁵⁵5In the banal case this is not always the case but it is when the cuspidal support contains only characters.: with the notation of [Boy11] proposition 1.3.5, $m(\varrho)=l>d$ for any irreducible $\overline{\mathbb{F}}_{l}$-supercuspidal representation $\varrho$.

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  For a $\overline{\mathbb{F}}_{l}$-irreducible supercuspidal representation $\varrho$ of $\operatorname{GL}_{g}(L)$, the parabolic induced representation $\varrho\times\cdots\times\varrho$, with $s$ copies of $\varrho$, is semi-simple with irreducible constituants the modulo $l$ reduction of the set of elements of $\{\mathrm{st}_{\underline{s}}(\pi)\hbox{ such that }\underline{s}\in\operatorname{Dec}(s)\}$, where $\pi$ is any cuspidal representation whose modulo $l$ reduction is isomorphic to $\varrho$.

Concerning the notion of genericity, consider the mirabolic subgroup $M_{d}(L)$ of $\operatorname{GL}_{d}(L)$ as the set of matrices with last row $(0,\cdots,0,1)$: we denote by

its unipotent radical. We fix a non trivial character $\psi$ of $L$ and let $\theta$ be the character of $V_{d}(L)$ defined by $\theta((m_{i,j}))=\psi(m_{d-1,d})$. For $G=\operatorname{GL}_{r}(L)$ or $M_{r}(L)$, we denote by $\operatorname{alg}(G)$ the abelian category of smooth representations of $G$ and, following [BZ77], we introduce

and

defined by $\Psi^{-}=r_{V_{d},1}$ (resp. $\Phi^{-}=r_{V_{d},\theta}$) the functor of $V_{d}$ coinvariants (resp. $(V_{d},\theta)$-coinvariants), cf. [BZ77] 1.8. For $\tau\in\operatorname{alg}(M_{d}(L))$, the representation

is called the $k$-th derivative of $\tau$. If $\tau^{(k)}\neq 0$ and $\tau^{(m)}=0$ for all $m>k$, then $\tau^{(k)}$ is called the highest derivative of $\tau$. In the particular case where $k=d$, there is an unique irreducible representation $\tau_{nd}$ of $M_{d}(L)$ with derivative of order $d$.

Let $\pi$ be an irreducible generic $\overline{\mathbb{Q}}_{l}$-representation of $\operatorname{GL}_{d}(L)$ and consider any stable lattice which gives us by modulo $l$ reduction a $\overline{\mathbb{F}}_{l}$- representation uniquely determined up to semi-simplification. Then this modulo $l$ reduction admits an unique generic irreducible constituant.

Recall that a Weil-Deligne representation of $W_{L}$ is a pair $(r,N)$ where

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  $r:W_{L}\longrightarrow\operatorname{GL}(V)$ is a smooth⁶⁶6i.e. continuous for the discrete topology on $V$ representation on a finite dimensional vector space $V$; and

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  $N\in\operatorname{End}(V)$ is nilpotent such that

  $$r(g)Nr(g)^{-1}=||g||N,$$

  where $||\bullet||:W_{L}\longrightarrow W_{L}/I_{L}\twoheadrightarrow q^{\mathbb{Z}}$ takes an arithmetic Frobenius element to $q$.

Main example: let $\rho:W_{L}\longrightarrow\operatorname{GL}(V)$ be a smooth irreducible representation on a finite dimensional vector space $V$. For $k\geq 1$ an integer, we then define a Weil-Deligne representation

where for $0\leq i\leq k-2$, the isomorphism $N:V(i)\cong V(i+1)$ is induced by some choice of a basis of $\overline{L}(1)$ and $N_{|V(k-1)}$ is zero. Then every Frobenius semi-simple Weil-Deligne representation of $W_{L}$ is isomorphic to some $\bigoplus_{i=1}^{r}\operatorname{Sp}(\rho_{i},k_{i})$, for smooth irreducible representations $\rho_{i}:W_{L}\longrightarrow\operatorname{GL}(V_{i})$ and integers $k_{i}\geq 1$. Up to obvious reorderings, such a writing is unique.

Let now $\rho$ be a continuous representation of $W_{L}$, or its Weil-Deligne representation $\operatorname{WD}(\rho)$, and consider its restriction to $I_{L}$, $\tau:=\rho_{|I_{L}}$. Such an isomophism class of a finite dimensional continuous representation of $I_{L}$ is then called an inertial type.

Let $\operatorname{Part}$ be the set of decreasing sequences of positive integers $\underline{d}=(\underline{d}(1)\geq\underline{d}(2)\geq\cdots)$ viewed as a partition of $\sum\underline{d}:=\sum_{i}\underline{d}(i)$.

The map from $\{f:\mathcal{I}_{0}\longrightarrow\operatorname{Part}\}$ to the set of inertial types given by $f\mapsto\tau_{f}$, is a bijection. The dominance order $\preceq$ on $\operatorname{Part}$ induces a partial order on the set of inertial types.

We let $\mathrm{rec}_{L}$ denote the local reciprocity map of [HT01, Theorem A]. Fix an isomorphism $\imath\overline{{\mathbb{Q}}}_{\ell}\buildrel\sim\over{\rightarrow}{\mathbb{C}}$. We normalize the local reciprocity map $\mathrm{rec}$ of [HT01, Theorem A], defined on isomorphism classes of irreducible smooth representations of $\operatorname{GL}_{n}(L)$ over ${\mathbb{C}}$ as follows: if $\pi$ is the isomorphism class of an irreducible smooth representation of $\operatorname{GL}_{n}(L)$ over $\overline{{\mathbb{Q}}}_{\ell}$, then

Then $\rho_{\ell}(\pi)$ is the isomorphism class of an $n$-dimensional, Frobenius semisimple Weil–Deligne representation of $W_{L}$ over $\overline{{\mathbb{Q}}}_{\ell}$, independent of the choice of $\imath$. Moreover, if $\rho$ is an isomorphism class of an $n$-dimensional, Frobenius semisimple Weil–Deligne representation of $W_{L}$ over $M$, then $\rho_{\ell}^{-1}(\rho)$ is defined over $M$ (cf. [CEG⁺16, §1.8]).

Recall the following compatibility of the Langlands correspondence.

Let $F=F^{+}E$ be a CM field where $E/{\mathbb{Q}}$ is a quadratic imaginary extension and $F^{+}/{\mathbb{Q}}$ is totally real. We fix a real embedding $\tau:F^{+}\hookrightarrow{\mathbb{R}}$. For a place $v$ of $F$, we will denote by $F_{v}$ the completion of $F$ at $v$, $\mathcal{O}_{v}$ its ring of integers with uniformizer $\varpi_{v}$ and residue field $\kappa(v)=\mathcal{O}_{v}/(\varpi_{v})$ of cardinal $q_{v}$.

Let $B$ be a division algebra with center $F$, of dimension $d^{2}$ such that at every place $v$ of $F$, either $B_{v}$ is split or a local division algebra and suppose $B$ provided with an involution of second kind $*$ such that $*_{|F}$ is the complex conjugation. For any $\beta\in B^{*=-1}$, denote by $\sharp_{\beta}$ the involution $v\mapsto v^{\sharp_{\beta}}=\beta v^{*}\beta^{-1}$ and let $G/{\mathbb{Q}}$ be the group of similitudes, denoted by $G_{\tau}$ in [HT01], defined for every ${\mathbb{Q}}$-algebra $R$ by

with $B^{op}=B\otimes_{F,c}F$. If $x$ is a place of ${\mathbb{Q}}$ split $x=yy^{c}$ in $E$ then

where $x=\prod_{i}v^{+}_{i}$ in $F^{+}$ and we identify places of $F^{+}$ over $x$ with places of $F$ over $y$.

In [HT01], the authors justify the existence of some $G$ like before such that

• •

  if $x$ is a place of ${\mathbb{Q}}$ non split in $M$ then $G({\mathbb{Q}}_{x})$ is quasi split;

• •

  the invariants of $G({\mathbb{R}})$ are $(1,d-1)$ for the embedding $\tau$ and $(0,d)$ for the others.

As in [HT01, page 90], a compact open subgroup $K$ of $G({\mathbb{A}}^{\infty}_{{\mathbb{Q}}})$ is said to be sufficiently small if there exists a place $x$ of ${\mathbb{Q}}$ such that the projection from $K^{x}$ to $G({\mathbb{Q}}_{x})$ does not contain any element of finite order except identity.