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Masao Oi

Publications and source records attributed to Masao Oi.

18 recordsLinked to original sources
arXiv2026

Twisted endoscopic character relation for toral supercuspidal L-packets of classical groups

We prove that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation in some cases, including the case of general linear groups equipped with an involution. Consequently, we verify that Kaletha's construction of the local Langlands correspondence for toral supercuspidal representations of quasi-split symplectic or special orthogonal groups coincides with Arthur's. The strategy is to emulate Kaletha's proof of the standard endoscopic character relation in the twisted setting by appealing to Waldspurger's framework ``l'endoscopie tordue n'est pas si tordue''.

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arXiv2025

Twisted character formula for toral supercuspidal representations

We establish an explicit formula for twisted Harish-Chandra characters of toral supercuspidal representations of p-adic reductive groups under several technical assumptions. Our setup especially includes the case of a quasi-split group equipped with an involution.

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arXiv2025

The B(G)-parametrization of the local Langlands correspondence

This article is on the parametrization of the local Langlands correspondence over local fields for non-quasi-split groups according to the philosophy of Vogan. We show that a parametrization indexed by the basic part of the Kottwitz set (which is an extension of the set of pure inner twists) implies a parametrization indexed by the full Kottwitz set. On the Galois side, we consider irreducible algebraic representations of the full centralizer group of the $L$-parameter (i.e not a component group). When $F$ is a $p$-adic field, we discuss a generalization of the endoscopic character identity.

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arXiv2025

Simple supercuspidal L-packets of split special orthogonal groups over dyadic fields

We consider the split special orthogonal group $\mathrm{SO}_{N}$ defined over a $p$-adic field. We determine the structure of any $L$-packet of $\mathrm{SO}_{N}$ containing a simple supercuspidal representation (in the sense of Gross--Reeder). We also determine its endoscopic lift to a general linear group. Combined with the explicit local Langlands correspondence for simple supercuspidal representations of general linear groups, this leads us to get an explicit description of the $L$-parameter as a representation of the Weil group of $F$. Our result is new when $p=2$ and our method provides a new proof even when $p\neq2$.

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arXiv2025

Green functions for positive-depth Deligne--Lusztig induction

Under a largeness assumption on the size of the residue field, we give an explicit description of the positive-depth Deligne--Lusztig induction of unramified elliptic pairs $(T,θ)$. When $θ$ is regular, we show that positive-depth Deligne--Lusztig induction gives a geometric realization of Kaletha's Howe-unramified regular $L$-packets. This is obtained as an immediate corollary of a very simple "litmus test" characterization theorem which we foresee will have interesting future applications to small-$p$ constructions. We next define and analyze Green functions of two different origins: Yu's construction (algebra) and positive-depth Deligne--Lusztig induction (geometry). Using this, we deduce a comparison result for arbitrary $θ$ from the regular setting. As a further application of our comparison isomorphism, we prove the positive-depth Springer hypothesis in the $0$-toral setting and use it to give a geometric explanation for the appearance of orbital integrals in supercuspidal character formulae.

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arXiv2023

On Swan exponents of symmetric and exterior square Galois representations

Let $F$ be a local non-Archimedean field and $E$ a finite Galois extension of $F$, with Galois group $G$. If $ρ$ is a representation of $G$ on a complex vector space $V$, we may compose it with any tensor operation $R$ on $V$, and get another representation $R\circρ$. We study the relation between the Swan exponents $\mathrm{Sw}(ρ)$ and $\mathrm{Sw}(R\circρ)$, with a particular attention to the cases where $R$ is symmetric square or exterior square. Indeed those cases intervene in the local Langlands correspondence for split classical groups over $F$, via the formal degree conjecture, and we present some applications of our work to the explicit description of the Langlands parameter of simple cuspidal representations. For irreducible $ρ$ our main results determine $\mathrm{Sw}(\mathrm{Sym}^{2}ρ)$ and $\mathrm{Sw}(\wedge^{2}ρ)$ from $\mathrm{Sw}(ρ)$ when the residue characteristic $p$ of $F$ is odd, and bound them in terms of $\mathrm{Sw}(ρ)$ when $p$ is $2$. In that case where $p$ is $2$ we conjecture stronger bounds, for which we provide evidence.

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arXiv2023

Characterization of supercuspidal representations and very regular elements

We prove that regular supercuspidal representations of $p$-adic groups are uniquely determined by their character values on very regular elements -- a special class of regular semisimple elements on which character formulae are very simple -- provided that this locus is sufficiently large. As a consequence, we resolve a question of Kaletha by giving a description of Kaletha's $L$-packets of regular supercuspidal representations which mirrors Langlands' construction for real groups following Harish-Chandra's characterization theorem for discrete series representations. Our techniques additionally characterize supercuspidal representations in general, giving $p$-adic analogues of results of Lusztig on reductive groups over finite fields. In particular, we establish an easy, non-cohomological characterization of unipotent supercuspidal representations when the residue field of the base field is sufficiently large.

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arXiv2022

Simple supercuspidal L-packets of symplectic groups over dyadic fields

We consider the symplectic group $\mathrm{Sp}_{2n}$ defined over a $p$-adic field $F$, where $p=2$. We prove that every simple supercuspidal representation (in the sense of Gross--Reeder) of $\mathrm{Sp}_{2n}(F)$ corresponds to an irreducible $L$-parameter under the local Langlands correspondence for $\mathrm{Sp}_{2n}$ established by Arthur.

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arXiv2022

Iwahori-Hecke algebra and unramified local L-functions

In this paper, we compute the Hecke action of a certain test function on the space of an unramified principal series of a connected reductive group over a non-archimedean local field by using the theory of Iwahori--Hecke algebra. As an application, we obtain a new expression of the local L-functions of unramified representations.

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arXiv2021

Simple supercuspidal L-packets of quasi-split classical groups

In this paper, for quasi-split classical groups over p-adic fields, we determine the L-packets consisting of simple supercuspidal representations and their corresponding L-parameters, under the assumption that p is not equal to 2. The key is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation, which is a characterization of the local Langlands correspondence for quasi-split classical groups.

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arXiv2021

Geometric L-packets of Howe-unramified toral supercuspidal representations

We show that $L$-packets of toral supercuspidal representations arising from unramified maximal tori of $p$-adic groups are realized by Deligne--Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series representations by their character on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction of $L$-packets of these representations. In parallel to the real case, we characterize the members of Kaletha's toral $L$-packets by their character on sufficiently regular elements of elliptic maximal tori.

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arXiv2020

Local Langlands correspondence for regular supercuspidal representations of GL(n)

In this paper, we prove the coincidence of Kaletha's recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris--Taylor's one in the case of general linear groups. The keys are Bushnell--Henniart's essentially tame local Langlands correspondence and Tam's result on Bushnell--Henniart's rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha's and Tam's $χ$-data.

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arXiv2018

Endoscopic lifting of simple supercuspidal representations of unramified U(N) to GL(N)

Let U(N) be the quasi-split unitary group in N variables for a quadratic unramified extension of p-adic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.

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arXiv2018

Depth preserving property of the local Langlands correspondence for non-quasi-split unitary groups

In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups (arXiv:1804.10901) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence established by Gan--Ichino. To combine them, we compare splittings for metaplectic covers of unitary groups constructed by Kudla with those constructed by Pan.

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arXiv2018

Depth preserving property of the local Langlands correspondence for quasi-split classical groups in a large residual characteristic

For a quasi-split classical group over a p-adic field with sufficiently large residual characteristic, we prove that the maximum of depth of representations in each L-packet equals the depth of the corresponding L-parameter. Furthermore, for quasi-split unitary groups, we show that the depth is constant in each L-packet. The key is an analysis of the endoscopic character relation via harmonic analysis based on the Bruhat--Tits theory. These results are slight generalizations of a result of Ganapathy and Varma.

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arXiv2015

On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of positive characteristic III

For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin-Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant r_x defined by Kato [2]. The main theme of this paper is to give a simple formula to compute r_x' defined in [4], which is equal to r_x for good Artin-Schreier extension. We also prove Kato's conjecture for upper bound of r_x.

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