arXiv subjects

Chi-Heng Lo

Publications and source records attributed to Chi-Heng Lo.

13 recordsLinked to original sources

arXiv2026

Algorithms on the Pyasetskii involution on local Langlands parameters of classical groups

We give an algorithm to compute the Pyasetskii involution for $\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$ and $\mathrm{O}_{2n}$. The algorithm is a combination of Moeglin-Waldspurger's algorithm for the Pyasetskii involution for $\mathrm{GL}_n$ ([MW86]) and Lanard-M${\'i}$nguez's algorithm for the Aubert-Zelevinsky involution of bad parity representations for classical groups ([LM25]). In particular, we give a geometric interpretation of the bad parity case of Lanard-M${\'i}$nguez's algorithm.

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arXiv2026

On corank 4 unitary representations of classical groups

In this paper, we explicitly classify the corank 4 unitary representations of symplectic or split odd special orthogonal groups over non-Archimedean local fields of characteristic zero, by classifying Arthur representations of corank 4 and verifying the corresponding unitary dual conjecture recently proposed by Hazeltine-Jiang-Liu-Lo-Zhang in [HJLLZ24].

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arXiv2025

Covering Barbasch-Vogan duality and wavefront sets of genuine representations

In this paper, we start by defining a covering Barbasch-Vogan duality and prove some of its properties. Then, for genuine representations of $p$-adic covering groups we formulate an upper bound conjecture for their wavefront sets using this covering Barbasch-Vogan duality and reduce it to anti-discrete representations. The formulation generalizes that of Ciubotaru-Kim and Hazeltine-Liu-Lo-Shahidi for linear algebraic groups. We prove this upper bound conjecture for Kazhdan-Patterson coverings of general linear groups.

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arXiv2025

The FPP Conjecture for p-adic Groups

The FPP conjecture, proposed by J. Adams, S. Miller, and D. Vogan and proved by D. Davis and L. Mason-Brown in arXiv:2411.01372, imposes a strong upper bound on the infinitesimal character of a unitary representation of a real reductive group. In this paper, we formulate an analogous conjecture for $p$-adic groups. We prove our conjecture for pure rational forms assuming a version of the Local Langlands Correspondence.

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arXiv2025

On the complementary Arthur representations and unitary dual for p-adic classical groups

In [HJLLZ24], we proposed a new conjecture on the structure of the unitary dual of connected reductive groups over non-Archimedean local fields of characteristic zero based on their Arthur representations and verified it for all the known cases on the unitary dual problem. One step towards this conjecture involves the question whether certain complementary Arthur representations are unitary. In this paper, we give an explicit characterization of the complementary Arthur representations for symplectic and split odd special orthogonal groups. As applications, we obtain interesting constraints on local components of irreducible self-dual cuspidal automorphic representations of $\mathrm{GL}_N$, especially when $N=2,3$.

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arXiv2024

On Arthur representations and the unitary dual

In this paper, we propose a new conjecture describing the structure of the unitary dual in terms of Arthur representations for connected reductive algebraic groups defined over any non-Archimedean local field of characteristic zero. This conjecture provides a candidate set for the unitary dual, constructed from Arthur representations. For classical groups, we develop an explicit algorithm to generate this candidate set. Evidence for its exhaustiveness includes compatibility with the known generic unitary dual, unramified unitary dual, and low-corank representations. As further support, we verify the conjecture for the unitary dual of the exceptional group of type $G_2$.

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arXiv2024

On anti-tempered local Arthur packets and a lemma of Arthur

In this paper, following Arthur's ideas, we rework the process of constructing the anti-tempered local Arthur packets for quasi-split classical groups and their pure inner forms. In particular, we present explicit examples illustrating certain gap in a consequential lemma of Arthur and provide a uniform modification, based on the work of Moeglin, Waldspurger, and Xu.

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arXiv2024

On the enhanced Shahidi conjecture and global applications

In this paper, applying the intersection theory of local Arthur packets, for symplectic and split odd special orthogonal groups G_n, we give the first complete proof of the enhanced Shahidi conjecture on generic representations in local Arthur packets. We also classify unramified representations of Arthur type for G_n, and show that they lie in exactly one local Arthur packet, which is anti-generic. Then, we discuss the global applications of these results.

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arXiv2024

On the intersection of local Arthur packets for classical groups and applications

In this paper, for symplectic and split odd special orthogonal groups, we develop an account of theory on the intersection problem of local Arthur packets. Specifically, following Atobe's reformulation on Mœglin's construction of local Arthur packets, we give a complete set of operators on the construction data, based on which, we provide algorithms and Sage codes to determine whether a given representation is of Arthur type. Furthermore, for any representation $π$ of Arthur type, we give a precise formula for the set $$ Ψ(π)=\{ \text{local Arthur parameter }ψ\ | \ \text{the local Arthur packet } Π_ψ \text{ contains } π\}.$$ Our results have many applications, including the precise counting of tempered representations in any local Arthur packet, specifying and characterizing "the" local Arthur parameter in $Ψ(π)$ for $π$, especially when $π$ belongs to several local Arthur packets but does not belong to any local $L$-packet of Arthur type.

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arXiv2024

The closure ordering conjecture on local Arthur packets of classical groups

In this paper, we prove the closure ordering conjecture on the local $L$-parameters of representations in local Arthur packets of $\mathrm{G}_n=\mathrm{Sp}_{2n}, \mathrm{SO}_{2n+1}$ over a non-Archimedean local field of characteristic zero. Precisely, given any representation $π$ in a local Arthur packet $Π_ψ$, the closure of the local $L$-parameter of $π$ in the Vogan variety must contain the local $L$-parameter corresponding to $ψ$. This conjecture reveals a geometric nature of local Arthur packets and is inspired by the work of Adams, Barbasch, and Vogan, and the work of Cunningham, Fiori, Moussaoui, Mracek, and Xu, on ABV-packets. As an application, for general quasi-split connected reductive groups, we show that the closure ordering conjecture implies the enhanced Shahidi conjecture, under certain reasonable assumptions. This provides a framework towards the enhanced Shahidi conjecture in general. We verify these assumptions for $\mathrm{G}_n$, hence give a new proof of the enhanced Shahidi conjecture. At last, we show that local Arthur packets cannot be fully contained in other ones, which is in contrast to the situation over Archimedean local fields and has its own interests.

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arXiv2024

On the upper bound of wavefront sets of representations of p-adic groups

In this paper we study the upper bound of wavefront sets of irreducible admissible representations of connected reductive groups defined over non-Archimedean local fields of characteristic zero. We formulate a new conjecture on the upper bound and show that it can be reduced to that of anti-discrete series representations, namely, those whose Aubert-Zelevinsky duals are discrete series. Then, we show that this conjecture is equivalent to the Jiang conjecture on the upper bound of wavefront sets of representations in local Arthur packets and also equivalent to an analogous conjecture on the upper bound of wavefront sets of representations in local ABV packets.

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arXiv2023

Vogan's Conjecture on local Arthur packets of $p$-adic $\mathrm{GL}_n$ and a combinatorial Lemma

For $\mathrm{GL}_n$ over a $p$-adic field, Cunningham and Ray proved Vogan's conjecture, that is, local Arthur packets are the same as ABV packets. They used the endoscopic theory to reduce the general case to a combinatorial lemma for irreducible local Arthur parameters, and their proof implies that one can also prove Vogan's conjecture for $p$-adic $\mathrm{GL}_n$ by proving a generalized version of this combinatorial lemma. Riddlesden recently proved this generalized lemma. In this paper, we give a new proof of it, which has its own interest.

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arXiv2023

On the weak local Arthur packets conjecture for split classical groups

Recently, motivated by the theory of real local Arthur packets, making use of the wavefront sets of representations over non-Archimedean local fields $F$, Ciubotaru, Mason-Brown, and Okada defined the weak local Arthur packets consisting of certain unipotent representations and conjectured that they are unions of local Arthur packets. In this paper, we prove this conjecture for split classical groups with the assumption of the residue field characteristic of $F$ being large. In particular, this implies the unitarity of these unipotent representations. We also discuss the generalization of the weak local Arthur packets beyond unipotent representations which reveals the implications of a conjecture of Jiang on the structure of wavefront sets for representations in local Arthur packets.

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