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Anne-Marie Aubert

Publications and source records attributed to Anne-Marie Aubert.

At least 19 recordsLinked to original sources
arXiv2026

A nonabelian Fourier transform for tempered unipotent representations, II

We extend the construction of the nonabelian Fourier transform proposed in our previous paper from the space of unipotent elliptic characters to the space of unipotent compact characters of a reductive $p$-adic group, and we investigate its relation with the branching to maximal compact open subgroups. We prove that this Fourier transform is compatible with parahoric restriction for certain groups, including $\mathrm{SL}_n, \mathrm{GL}_n$, and $G_2$. We also relate the compact setting to the elliptic setting using parabolic induction.

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arXiv2026

The theta correspondence over finite fields

This set of lecture notes is an expanded version of a mini-course the author gave in March of 2025 for the program ``Representation Theory \& Noncommutative Geometry" at the Institut Henri Poincaré, Paris. The goal is to provide a survey of the main properties of the theta correspondence over finite fields of odd characteristic, including its compatibility with Harish-Chandra and Lusztig series, and with the Jordan decomposition of representations, as well as its full explicit description.

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arXiv2026

A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations

Let $F$ be a non-archimedean local field such that $4|q-1$, with $q$ the order of the residue field of $F$, and let $(M^0,σ^0)$ be the depth-zero cuspidal pair for the twisted Levi subgroup $G^0$ of $\mathrm{SL}_8$ arising from quadratic and quartic field extensions, as defined in the recent article by Adler-Fintzen-Ohara [AFO]. Then the corresponding Bernstein block is described by a twisted Hecke algebra $\mathcal{H}^0$. We describe $\mathcal{H}^0$ explicitly as a noncommutative $\mathbb{C}$-algebra with generators and relations. We describe explicitly the simple modules of $\mathcal{H}^0$. All the simple modules are $2$-dimensional. The primitive spectrum of $\mathcal{H}^0$ is then an explicit complex algebraic variety $\mathfrak{X}$. The maximal compact real form of $\mathfrak{X}$ is homeomorphic to a Klein bottle. This Klein bottle is a model of the unitary principal series of $G^0$ attached to the cuspidal pair $(M^0, σ^0)$. We make a full comparison with the classical situation in which $G = \mathrm{SL}_8$ and $(M,σ)$ is a cuspidal pair for $G$. The supercuspidal representation $σ$ is constructed from the same quadratic and quartic extensions of $F$. Let $\mathfrak{s}$ be the point in the Bernstein spectrum $\mathfrak{B} G$ determined by $(M,σ)$ and let $\mathfrak{s}^0$ be the point in the Bernstein spectrum $\mathfrak{B} G^0$ determined by $(M^0, σ^0)$. We compare the two points $\mathfrak{s}$ and $\mathfrak{s}^0$ and show explicitly that the corresponding Bernstein varieties are isomorphic. In that case, the Klein bottle re-appears, this time floating in the tempered dual of $\mathrm{SL}_8$.

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arXiv2025

Comparison of the Sally-Shalika character formulas with the endoscopic character identities for $\mathrm{SL}_2$

We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2(F)$ where $F$ is a non-archimedean local field of characteristic zero. We compare the explicit endoscopic character identities for $\mathrm{SL}_2(F)$ with the classical character formulas of Sally-Shalika. Our main result concerns the supercuspidal $L$-packet of size $4$. For this $L$-packet, we show how the norm $1$ groups $H_1, H_2, H_3$ in the three quadratic extensions of $F$ play a crucial role in the endoscopic character identities for $\mathrm{SL}_2(F)$.

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arXiv2025

On Kazhdan-Yom Din asymptotic Schur orthogonality for K-finite matrix coefficients

In a recent article, D. Kazhdan and A. Yom Din conjectured the validity of an asymptotic form of Schur's orthogonality for tempered irreducible unitary representations of semisimple groups defined over local fields. In the non-Archimedean case, they established such an orthogonality for $K$-finite matrix coefficients. Building on their work, and exploiting the admissibility of irreducible unitary representations, we prove the analogous result in the Archimedean case.

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arXiv2025

Affine Hecke algebras for classical p-adic groups

We consider four classes of classical groups over a non-archimedean local field F: symplectic, (special) orthogonal, general (s)pin and unitary. These groups need not be quasi-split over F. The main goal of the paper is to obtain a local Langlands correspondence for any group G of this kind, via Hecke algebras. To each Bernstein block Rep(G)^s in the category of smooth complex G-representations, an (extended) affine Hecke algebra H(s) can be associated with the method of Heiermann. On the other hand, to each Bernstein component $Φ_e (G)^{s^\vee}$ of the space Φ_e (G) of enhanced L-parameters for G, one can also associate an (extended) affine Hecke algebra, say $H (s^\vee)$. For the supercuspidal representations underlying Rep(G)^s, a local Langlands correspondence is available via endoscopy, due to Moeglin and Arthur. Using that we assign to each Rep(G)^s a unique $Φ_e (G)^{s^\vee}$. Our main new result is an algebra isomorphism $H(s)^{op} \to H (s^\vee)$, canonical up to inner automorphisms. In combination with earlier work, that provides an injective local Langlands correspondence Irr(G) -> Φ_e (G) which satisfies Borel's desiderata. This parametrization map is probably surjective as well, but we could not show that in all cases. Our framework is suitable to (re)prove many results about smooth G-representations (not necessarily reducible), and to relate them to the geometry of a space of L-parameters. In particular our Langlands parametrization yields an independent way to classify discrete series G-representations in terms of Jordan blocks and supercuspidal representations of Levi subgroups. We show that it coincides with the classification of the discrete series obtained twenty years ago by Moeglin and Tadić.

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arXiv2025

Generalizations of the Springer correspondence and cuspidal Langlands parameters

Let H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal H-representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein's theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for H, that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible H-representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for H-representations to the corresponding problem for supercuspidal representations of Levi subgroups of H. The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig's generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G. In 2025 an erratum was added, to repair Theorem 3.1.a.

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arXiv2025

On the Macdonald correspondence

In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of ${\mathrm{GL}}_n(k)$, where $k$ is a finite field, and inertia equivalence classes of admissible tamely ramified $n$-dimensional Weil-Deligne representations of $W_F$, where $F$ is a non-archimedean local field with residue field $k$ and $W_F$ the absolute Weil group of $F$. We describe a construction of the Macdonald correspondence based on the specialization to ${\mathrm{GL}}_n(k)$ of Lusztig's classification of irreducible representations of finite groups of Lie type, and review some properties of the correspondence. We define $ε$-factors for pairs of irreducible cuspidal representations of finite general linear groups, and show that they match with the expected Deligne $ε$-factors under the Macdonald correspondence. We use these $ε$-factors for pairs to obtain a characterization of the Macdonald correspondence for the irreducible cuspidal representations

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arXiv2024

Affine Hecke algebras for Langlands parameters

It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to create a similar role for affine Hecke algebras on the Galois side of the local Langlands correspondence. To every Bernstein component of enhanced Langlands parameters for G we canonically associate an affine Hecke algebra (possibly extended with a finite R-group). We prove that the irreducible representations of this algebra are naturally in bijection with the members of the Bernstein component, and that the set of central characters of the algebra is naturally in bijection with the collection of cuspidal supports of these enhanced Langlands parameters. These bijections send tempered or (essentially) square-integrable representations to the expected kind of Langlands parameters. Furthermore we check that for many reductive p-adic groups, if a Bernstein component B for G corresponds to a Bernstein component B^\vee of enhanced Langlands parameters via the local Langlands correspondence, then the affine Hecke algebra that we associate to B^\vee is Morita equivalent with the Hecke algebra associated to B. This constitutes a generalization of Lusztig's work on unipotent representations. It might be useful to establish a local Langlands correspondence for more classes of irreducible smooth representations.

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arXiv2024

Typical representations for $\mathrm{Sp}_4(F)$

Let $F$ be a non Archimedean local field with odd residual characteristic, and let $K$ be a hyperspecial maximal compact subgroup of the $p$-adic symplectic group $G=\mathrm{Sp}_4(F)$. Let $\mathfrak{s}$ be an inertial class for $G$ in the Bernstein decomposition of the category of smooth representations of $G$, which is attached to a proper Levi subgroup $L$ of $G$. We prove that the $\mathfrak{s}$-typical irreducible representations of $K$ are the irreducible components of $\mathrm{Ind}_{J_{\mathfrak{s}}}^{K}(λ_{\mathfrak{s}})$, where $(J_{\mathfrak{s}},λ_{\mathfrak{s}})$ is an $\mathfrak{s}$-type for $G$ such that $J_{\mathfrak{s}}\subset K$, and $(J_{\mathfrak{s}},λ_{\mathfrak{s}})$ is a $G$-cover of a Bushnell-Kutzko maximal simple type for $L$.

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arXiv2024

A nonabelian Fourier transform for tempered unipotent representations

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for $\mathsf{SL}_n$ and $\mathsf{PGL}_n$.

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arXiv2024

On congruent isomorphisms for tori

Let $F$ and $F'$ be two $l$-close nonarchimedean local fields, where $l$ is a positive integer, and let $\mathrm{T}$ and $\mathrm{T}'$ be two tori over $F$ and $F'$, respectively, such that their cocharacter lattices can be identified as modules over the ''at most $l$-ramified'' absolute Galois group $Γ_F/I_F^l \congΓ_{F'}/I_{F'}^l$. In the spirit of the work of Kazhdan and Ganapathy, for every positive integer $m$ relative to which $l$ is large, we construct a congruent isomorphism $\mathrm{T}(F)/\mathrm{T}(F)_m\cong\mathrm{T}'(F')/\mathrm{T}'(F')_m$, where $\mathrm{T}(F)_m$ and $\mathrm{T}(F')_m$ are the minimal congruent filtration subgroups of $\mathrm{T}(F)$ and $\mathrm{T}(F')$, respectively, defined by J.-K.~Yu. We prove that this isomorphism is functorial and compatible with both the isomorphism constructed by Chai and Yu and the Kottwitz homomorphism for tori. We show that, when $l$ is even larger relative to $m$, it moreover respects the local Langlands correspondence for tori.

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arXiv2023

Bruhat-Tits buildings, representations of $p$-adic groups and Langlands correspondence

The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of $p$-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the case of supercuspidal representations, in particular regarding the local Langlands correspondence and the internal structure of the $L$-packets. We prove that the enhanced $L$-parameters with semisimple cuspidal support are those which are obtained via the (ordinary) Springer correspondence. Let ${\mathbf G}$ be a connected reductive group over a non-archimedean field $F$ of residual characteristic $p$. In the case where ${\mathbf G}$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the Weyl group of ${\mathbf G}$, we show that the enhanced $L$-parameters with semisimple cuspidal support correspond to the irreducible smooth representations of ${\mathbf G}(F)$ with non-singular supercuspidal support via the local Langlands correspondence constructed by Kaletha, under the assumption that the latter satisfies certain expected properties. As a consequence, we obtain that every compound $L$-packet of ${\mathbf G}(F)$ contains at least one representation with non-singular supercuspidal support.

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arXiv2023

The Explicit Local Langlands Correspondence for $G_2$

We develop a general strategy for constructing the explicit Local Langlands Correspondences for $p$-adic reductive groups via reduction to LLC for supercuspidal representations of proper Levi subgroups, using Hecke algebra techniques. As an example of our general strategy, we construct the explicit Local Langlands Correspondence for the exceptional group $G_2$ over a nonarchimedean local field, with explicit $L$-packets and explicit matching between the group and Galois sides. We also give a list of characterizing properties for our LLC. In \cite{G2-stability}, we complete unique characterization using stability property of our $L$-packets. For intermediate series, we build on our previous results on Hecke algebras. For principal series, we improve previous works of Muic etc. and obtain more explicit descriptions on both group and Galois sides. Moreover, we show the existence of non-unipotent \textit{singular} supercuspidal representations of $G_2$, and exhibit them in \textit{mixed} $L$-packets mixing supercuspidal representations with non-supercuspidal ones. Furthermore, our LLC satisfies a list of expected properties, including the compatibility with cuspidal support.

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arXiv2022

Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

We study the endomorphism algebras attached to Bernstein components of reductive $p$-adic groups and construct a local Langlands correspondence with the appropriate set of enhanced $L$-parameters, using certain "desiderata" properties for the LLC for supercuspidal representations of proper Levi subgroups. We give several applications of our LLC to various reductive groups with Bernstein blocks cuspidally supported on general linear groups. In particular, for Levi subgroups of maximal parabolic of the split exceptional group $G_2$, we compute the explicit weight functions for the corresponding Hecke algebras, and show that they satisfy a conjecture of Lusztig's. Some results from $§4$ are used by the same authors to construct a full local Langlands correspondence in \cite{AX-LLC}. Moreover, we also prove a reduction to depth zero case result for the Bernstein components attached to regular supercuspidal representations of Levi subgroups.

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arXiv2021

$C^\ast$-blocks and crossed products for classical $p$-adic groups

Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of induced representations. For $p$-adic groups, each connected component of the tempered dual comes with a compact torus equipped with a finite group action, and we prove that a version of Wassermann's theorem holds true under a certain geometric assumption on the structure of stabilizers for that action. We then focus on the case where $G$ is a quasi-split symplectic, orthogonal or unitary group, and explicitly determine the connected components for which the geometric assumption is satisfied.

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arXiv2020

Groupoids, Geometric Induction and Gelfand Models

In this paper we introduce an intrinsic version of the classical induction of representations for a subgroup $H$ of a (finite) group $G$, called here {\em geometric induction}, which associates to any, not necessarily transitive, $G$-set $X$ and any representation of the action groupoid $A(G,X)$ associated to $G$ and $X$, a representation of the group $G$. We show that geometric induction, applied to one dimensional characters of the action groupoid of a suitable $G$-set $X$ affords a Gelfand Model for $G$ in the case where $G$ is either the symmetric group or the projective general linear group of rank $2$.

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