arXiv subjects

Tasho Kaletha

Publications and source records attributed to Tasho Kaletha.

At least 19 recordsLinked to original sources

arXiv2026

The Aubert and Bernstein involutions for disconnected groups

We extend to arbitrary disconnected reductive $p$-adic groups the duality on the category of smooth finite-length complex representations defined by Aubert, as well as its cohomological analog defined by Bernstein, and prove various properties of these functors, such as uniqueness, preservation of irreducibility, compatibility with parabolic induction and restriction, and a character formula. As a sample application, we obtain a definition of the Steinberg representation for a disconnected reductive $p$-adic group, compute its character, and discuss its twisted endoscopic properties.

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arXiv2026

On the refined local Langlands conjecture for discrete $L$-parameters of inner forms of quasi-split disconnected real reductive groups

Given a quasi-split connected reductive $\mathbb{R}$-group $G$ and a finite group $A$ acting on $G$ by $\mathbb{R}$-automorphisms that preserve an $\mathbb{R}$-pinning, we construct for each discrete $L$-parameter for $G$ a corresponding $L$-packet of irreducible discrete series representations on each inner forms $\tilde G_z(\mathbb{R})$ of the disconnected group $\tilde G = G \rtimes A$. We prove that these $L$-packets satisfy the endoscopic character identities with respect to normalized transfer factors. This proves the conjectural refined local Langlands correspondence for inner forms of quasi-split disconnected real reductive groups, as recently formulated by the first author.

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arXiv2024

Local Intertwining Relations and Co-tempered $A$-packets of Classical Groups

The local intertwining relation is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. We prove several instances of the local intertwining relation for quasi-split classical groups and the twisted general linear group, as they are required in the inductive proof of the endoscopic classification for quasi-split classical groups due to Arthur and Mok. In addition, we construct the co-tempered local $A$-packets by Aubert duality and verify their key properties by purely local means, which provide the seed cases needed as an input to the inductive proof. Together with further technical results that we establish, this makes the endoscopic classification conditional only on the validity of the twisted weighted fundamental lemma.

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arXiv2024

Discrete series L-packets for real reductive groups

We give a modern exposition of the construction, parameterization, and character relations for discrete series L-packets of real reductive groups, which are fundamental results due to Langlands and Shelstad. This exposition incorporates recent developments not present in the original sources, such as normalized geometric transfer factors and the canonical double covers of tori and endoscopic groups, allowing for simpler statements and proofs. We also prove some new results, such as a simple criterion for detecting generic representations for a prescribed Whittaker datum, and an explicit formula for the factor $\Delta_I$ in terms of covers of tori.

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arXiv2023

Galois cohomology of reductive groups over global fields

We give closed formulas for the abelian Galois cohomology groups H^1_{ab}(F,G) and H^2_{ab}(F,G) of a connected reductive group G over a global field F in terms of the algebraic fundamental group \pi_1(G) introduced earlier by one of us (M.B.). We further give closed formulas for the effects of restriction, corestriction, and localization, in terms of these formulas and the analogous known formulas in the case of local fields. Building on this, we give formulas, suitable for computer computations, for the first nonabelian Galois cohomology set H^1(F,G) of G and for the second Galois cohomology group H^2(F,T) of an F-torus T. As a preparation for the derivation of our formulas, we review the interpretation of Tate cohomology of a finite group in terms of the stable derived category of Z[\Gamma]-modules due to Buchweitz, and relate it to the explicit definition via cochains due to Kottwitz-Shelstad. We use this to construct the Tate-Nakayama isomorphisms for bounded complexes of tori over local and global fields, whose specialization to complexes of length 2 is then applied to obtain the desired formulas.

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arXiv2022

On the local Langlands conjectures for disconnected groups

We extend the local Langlands conjectures to a certain class of disconnected groups, allowing non-abelian component groups, and recast in this language some aspects of twisted endoscopy. We further introduce normalized twisted transfer factors and a normalized correspondence between an $L$-packet for a disconnected group and the set of representations of the centralizer groups of its Langlands parameter. We prove the first instance of this conjecture, in which the identity component of the (possibly non-abelian) disconnected group is a torus.

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arXiv2022

Covers of reductive groups and functoriality

For a quasi-split connected reductive group $G$ over a local field $F$ we define a compact abelian group $\tilde\pi_1(G)$ and an extension $1 \to \tilde\pi_1(G) \to G(F)_\infty \to G(F) \to 1$ of topological groups equipped with a splitting over $G_\textrm{sc}(F)$. Any character $x : \tilde\pi_1(G) \to \mu_n(\mathbb{C})$ leads to an $n$-fold cover $G(F)_x$ of $G(F)$ via pushout. We define an $L$-group $^LG_x$ for this cover that is generally a non-split extension of $\textrm{Gal}(F^s/F)$ by $\hat G$. We prove a refined local Langlands correspondence for $G(F)_x$, assuming it is known for connected reductive groups with the same adjoint group as $G$. Motivation for this construction comes from considerations of Langlands' functoriality conjecture, where subgroups $\mathcal{H} \subset {^LG}$ of the $L$-group of $G$ arise that need not be $L$-groups of other reductive groups. If such a subgroup is full and intersects $\hat G$ in a connected reductive subgroup of maximal rank, we construct a natural triple $(H,x,\xi)$ consisting of a quasi-split connected reductive group $H$, a double cover $H(F)_x$, and an $L$-embedding $\xi : {^LH}_x \to {^LG}$ that is an isomorphism onto $\mathcal{H}$. We expect that genuine representations of $H(F)_x$ transfer functorially to representations of $G(F)$. In the special case of endoscopy, we show that the construction of transfer factors simplifies when the natural double cover $H(F)_x$ of the endoscopic group is used. The transfer factor becomes the product of two natural invariants that do not depend on auxiliary choices. One of them is closely related to Kottwitz's work on transfer factors for Lie algebras. The other one is not specific to the case of endoscopy, and will likely play a role in general functoriality questions. Our work is motivated by work of Adams and Vogan over the real numbers.

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arXiv2022

Representations of reductive groups over local fields

We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties of the refined local Langlands correspondence.

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arXiv2021

A twisted Yu construction, Harish-Chandra characters, and endoscopy

We give a modification of Yu's construction of supercuspidal representations of a connected reductive group over a non-archimedean local field. This modification restores the validity of certain key intertwining property claims made by Yu, which were recently proven to be false for the original construction. This modification is also an essential ingredient in the explicit construction of supercuspidal L-packets. As further applications, we prove the stability and many instances of endoscopic character identities of these supercuspidal L-packets, subject to some conditions on the base field. In particular, for regular supercuspidal parameters we prove all instances of standard endoscopy. In addition, we prove that these supercuspidal L-packets satisfy a certain property, which, together with standard endoscopy, uniquely characterizes the local Langlands correspondence for supercuspidal L-packets (again subject to the above mentioned conditions on the base field). These results are based on a statement of the Harish-Chandra character formula for the supercuspidal representations arising from the twisted Yu construction.

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arXiv2019

Supercuspidal L-packets

Let F be a non-archimedean local field and let G be a connected reductive group defined over F. We assume that G splits over a tame extension of F and that the residual characteristic p does not divide the order of the Weyl group. To each discrete Langlands parameter of the Weil group of F into the complex L-group of G we associate explicitly a finite set of irreducible supercuspidal representations of G(F), and relate its internal structure to the centralizer of the parameter. We give evidence that this assignment is an explicit realization of the local Langlands correspondence.

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arXiv2019

On certain sign characters of tori and their extensions to Bruhat-Tits groups

We consider two sign characters defined on a tamely ramified maximal torus T of a twisted Levi subgroup M of a reductive p-adic group G. We show that their product extends to the stabilizer M(F)_x of any point x in the Bruhat-Tits building of T, and give a formula for this extension. This result is used in the passage between zero and positive depth in the explicit construction of supercuspidal L-packets, as well as in forthcoming work on the Harish-Chandra character formula for supercuspidal representations.

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arXiv2019

On L-embeddings and double covers of tori over local fields

To a torus T over a local field F and a subset of its character module subject to certain properties, we associate a canonical double cover of the topological group T(F). We further associate an L-group to this double cover and establish a natural bijection between L-parameters valued in this L-group and genuine characters of the double cover. When T is a maximal torus of a connected reductive group G, we show that there is a canonical L-embedding from the L-group of the double cover of T to the L-group of G. This leads to a canonical factorization of Langlands parameters. We associate to a genuine character of the double cover subject to certain conditions a Harish-Chandra character formula and use it to give a conjectural characterization of the supercuspidal local Langlands correspondence for G, subject to a certain condition on p. This generalizes previous work of Adams and Vogan F=R, and reinterprets computations of Langlands and Shelstad.

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arXiv2018

Global rigid inner forms vs isocrystals

We compare the cohomology of the global Galois gerbes constructed in [Kot] and [Kal18a], respectively, and give applications to the theory of endoscopy.

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arXiv2018

Commensurability growths of algebraic groups

Fixing a subgroup $\Gamma$ in a group $G$, the full commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] \leq n$. For pairs $\Gamma \leq G$, where $G$ is a Chevalley group scheme defined over $\mathbb{Z}$ and $\Gamma$ is an arithmetic lattice in $G$, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on $G$.

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arXiv2017

On the Kottwitz conjecture for local shtuka spaces

Kottwitz's conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze's more general spaces of local shtukas. Using a new Lefschetz-Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that for an irreducible smooth representation of an inner form of $\mathrm{GL}_n$, the $L$-parameter constructed by Fargues-Scholze agrees with the usual semisimplified parameter arising from local Langlands.

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arXiv2016

Regular supercuspidal representations

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.

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arXiv2015

Rigid inner forms of real and p-adic groups

We define a new cohomology set for an affine algebraic group G and a multiplicative finite central subgroup Z, both defined over a local field of characteristic zero, which is an enlargement of the usual first Galois cohomology set of G. We show how this set can be used to normalize the Langlands-Shelstad endoscopic transfer factors and to give a conjectural description of the internal structure and endoscopic transfer of L-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the real case, we show that this description is correct using Shelstad's work.

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arXiv2015

Rigid inner forms vs isocrystals

We compare two statements of the refined local Langlands correspondence for connected reductive groups defined over a p-adic field -- one involving Kottwitz's set B(G) of isocrystals with additional structure, and one involving the cohomology set H^1(u -> W,Z -> G) introduced in arXiv:1304.3292. We show that if either statement is valid for all connected reductive groups, then so is the other. We also discuss how the second statement depends on the choice of element of H^1(u -> W,Z -> G).

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