arXiv subjects

Chufeng Nien

Publications and source records attributed to Chufeng Nien.

4 recordsLinked to original sources

arXiv2026

Associated Representations of finite pattern groups

In this paper, we consider the construction of irreducible representations of finite pattern groups in terms of Panov's associative polarization, which is a finite-field analogue of Kirillov's orbital method. Using this construction, first, we are able to classify the irreducible representations of the unipotent radical of the standard parabolic subgroups of $\mathrm{GL}_n$ with 4 parts; second, we can parameterize irreducible characters of degree $q$ in terms of coadjoint orbits of cardinality $q^2$, for any finite pattern groups $G$ over $\mathbb{F}_q,$ where $\mathbb{F}_q$ is a finite field with $q$ elements.

math.RT↗

arXiv2020

Representations of finite pattern groups

Let $G=1+A$ be a finite pattern group over the finite field ${\mathbb{F}}_q$. We give a natural bijection between coadjoint orbits of $G$ and its equivalent classes of irreducible representations. More precisely, given any $T\in A^t$, viewed as a representative of associated coadjoint orbit ${\mathfrak{O}}_T$ of $G$, we can explicitly construct a subgroup $H_T $ of $G$, such that ${\mathrm{Ind}}_{H_T}^G ψ_T$ is irreducible and ${\mathrm{Ind}}_{H_T}^G ψ_T \cong {\mathrm{Ind}}_{H_{T'}}^G ψ_{T'}$ if and only if $T$ and $ T'$ are in the same coadjoint orbit. Here $ψ_T(x)=ψ({\mathrm{tr}} Tx)\text{ for }x\in H_T,$ and $ψ$ is a fixed nontrivial additive character of ${\mathbb{F}}_q$.

math.RT↗

arXiv2018

Converse Theorem Meets Gauss Sums (with an appendix by Zhiwei Yun)

This paper verifies $n\times 1$ Local Converse Theorem for twisted gamma factors of irreducible cuspidal representations of ${\rm GL}_n({\mathbb F}_p)$, for $n\leq 5,$ and of irreducible generic representations, for $n<\frac{q-1}{2\sqrt{q}}+1$ in the appendix by Zhiwei Yun, where $p$ is a prime and q is a power of $p$. The counterpart of $n\times 1$ converse theorem for level zero cuspidal representations also follows the established relation between gamma factors of ${\rm GL}_n({\mathcal F})$ and that of ${\rm GL}_n({\mathbb F}_q)$, where ${\mathcal F}$ denotes a $p$-adic field whose residue field is isomorphic to ${\mathbb F}_q.$ For $n=6,$ examples failed $n\times 1$ Local Converse Theorem over finite fields are provided and the authors propose a set of primitive representations, for which $n\times 1$ gamma factors should be able to detect a unique element in it. For $m,\ n\in {\mathbb N},$ in the spirit of Langlands functorial lifting, we formulate a conjecture to relate $n\times m$ gamma factors of finite fields with Gauss sums over extended fields.

math.NT↗

arXiv2015

Towards the Jacquet Conjecture on the Local Converse Problem for $p$-adic $\mathrm{GL}_n$

The Local Converse Problem is to determine how the family of the local gamma factors $γ(s,π\timesτ,ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.

math.NT↗