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Guy Henniart

Publications and source records attributed to Guy Henniart.

29 records · Page 2Linked to original sources
arXiv2016

A classification of irreducible admissible mod p representations of p-adic reductive groups

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

math.NT↗
arXiv2016

Strong exponent bounds for the local Rankin-Selberg convolution

Let $F$ be a non-Archimedean locally compact field. Let $σ$ and $τ$ be finite-dimensional semisimple representations of the Weil-Deligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of $σ\otimesτ$ in terms of those of $σ$ and $τ$. We give a different lower bound in terms of $σ\otimes\checkσ$ and $τ\otimes\checkτ$. Using the Langlands correspondence, we obtain the bounds for Rankin-Selberg exponents.

math.NT↗
arXiv2013

To an effective local Langlands Corrspondence

Let $F$ be a non-Archimedean local field. Let $\Cal W_F$ be the Weil group of $F$ and $\Cal P_F$ the wild inertia subgroup of $\scr W_F$. Let $\hat{\Cal W}_F$ be the set of equivalence classes of irreducible smooth representations of $\Cal W_F$. Let $\Cal A^0_n(F)$ denote the set of equivalence classes of irreducible cuspidal representations of $\roman{GL}_n(F)$ and set $\hat{\roman{GL}}_F = \bigcup_{n\ge1} \Cal A^0_n(F)$. If $σ\in \hat{\Cal W}_F$, let $\upr Lσ\in \hat{\roman{GL}}_F$ be the cuspidal representation matched with $σ$ by the Langlands Correspondence. If $σ$ is totally wildly ramified, in that its restriction to $\Cal P_F$ is irreducible, we treat $\upr Lσ$ as known. From that starting point, we construct an explicit bijection $\Bbb N:\hat{\Cal W}_F \to \hat{\roman{GL}}_F$, sending $σ$ to $\upr Nσ$. We compare this "naïve correspondence" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of "internal twisting" of a suitable representation $π$ (of $\Cal W_F$ or $\roman{GL}_n(F)$) by tame characters of a tamely ramified field extension of $F$, canonically associated to $π$. We show this operation is preserved by the Langlands correspondence.

math.RT↗
arXiv2013

La transformée de Fourier pour les espaces tordus sur un groupe réductif $\mathfrak{p}$-adique I. Le théorème de Paley-Wiener

Let ${\boldsymbol{G}}$ be a connected reductive group defined over a non--Archimedean local field $F$. Put $G={\boldsymbol{G}}(F)$. Let $θ$ be an $F$--automorphism of ${\boldsymbol{G}}$, and let $ω$ be a smooth character of $G$. This paper is concerned with the smooth complex representations $π$ of $G$ such that $π^θ=π\circθ$ is isomorphic to $ωπ=ω\otimesπ$. If $π$ is admissible, in particular irreducible, the choice of an isomorphism $A$ from $ωπ$ to $π^θ$ (and of a Haar measure on $G$) defines a distribution $Θ_π^A={\rm tr}(π\circ A)$ on $G$. The twisted Fourier transform associates to a compactly supported locally constant function $f$ on $G$, the function $(π,A)\mapsto Θ_π^A(f)$ on a suitable Grothendieck group. Here we describe its image (Paley--Wiener theorem), and we reduce the description of its kernel (spectral density theorem) to a result on the discrete part of the theory.

math.RT↗
arXiv2013

Uniqueness of Rankin-Selberg products

In the present paper, we show the equality of the $γ$-factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands-Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local-global principle for ${\rm GL}_n$ which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of $γ$-factors is made via a uniqueness result for Rankin-Selberg $γ$-factors.

math.NT↗
arXiv2013

Weil representations over finite fields and Shintani lift

Let Sp_V(F) be the group of isometries of a symplectic vector space V over a finite field F of odd cardinality. The group Sp_V(F) possesses distinguished representations--- the Weil representations. We know that they are compatible with base change in the sense of Shintani for a finite extension F'/F. The result is also true for the group of similitudes of V.

math.RT↗
arXiv2012

Characterization of γ-factors: the Asai case

Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai γ-factors are defined for smooth irreducible representations πof ${\rm GL}_n(E)$. If σis the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to πunder the local Langlands correspondence, we show that the Asai γ-factor is the same as the Deligne-Langlands γ-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from σunder tensor induction. This is achieved by proving that Asai γ-factors are characterized by their local properties together with their role in global functional equations for $L$-functions. As an immediate application, we establish the stability property of γ-factors under twists by highly ramified characters.

math.NT↗
arXiv2011

A congruence property of the local Langlands correspondence

Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.

math.NT↗
arXiv2011

Intertwining of simple characters in GL(n)

Let $F$ be a non-Archimedean local field and let $G$ be the general linear group $G = \text{\rm GL}_n(F)$. Let $θ_1$, $θ_2$ be simple characters in $G$. We show that $θ_1$ intertwines with $θ_2$ if and only if $θ_1$ is endo-equivalent to $θ_2$. We also show that any simple character in $G$ is a $G$-type.

math.RT↗
arXiv2004

Functorial products for $GL_2\times GL_3$ and the symmetric cube for $GL_2$

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $GL_2$. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Section 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for $GL_2$. Many other applications are obtained.

math.NT↗