arXiv subjects

Yongqi Feng

Publications and source records attributed to Yongqi Feng.

5 recordsLinked to original sources

arXiv2025

Class numbers and invariant characters of $\mathfrak{sl}_2(\mathbb{F}_p)$

Let $p$ be a prime and let $S_2(Γ(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $Γ(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(Γ(p))$ in a natural way. Around 1928, Hecke proved that if $p>3$ and $p\equiv 3\mod 4$, then the class number of $\mathbb{Q}(\sqrt{-p})$ is equal to the difference between the multiplicities of two particular irreducible representations of $\mathrm{SL}_2(\mathbb{F}_p)$ in $S_2(Γ(p))$. In this paper we prove a Lie algebra analogue of this result. As an application we extend Hecke's result to $\mathrm{SL}_2(\mathbb{Z}/p^r)$ (acting on $S_2(Γ(p^r))$) for any $r\geq 2$.

math.RT↗

arXiv2021

Supercuspidal unipotent representations: L-packets and formal degrees

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $γ$-factors of L-parameters. This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensueing L-packets, in terms of data from the affine Dynkin diagram of G. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.

math.RT↗

arXiv2020

On a uniqueness property of cuspidal unipotent representations

The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible factors of this rational function are $q$ and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of Mark Reeder, and for the remaining exceptional cases this is verified by the first name author in arXiv:1708.09547. In the present paper we treat the classical families. The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in arXiv:1310.7193) between unipotent affine Hecke algebras (up to twisting by unramified characters). In arXiv:1310.7790 the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.

math.RT↗

arXiv2019

On formal degrees of unipotent representations

Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation $π$ can be expressed in terms of the adjoint $γ$-factor of the enhanced L-parameter of $π$. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $γ$-factors.

math.RT↗

arXiv2018

A Note on the Spectral Transfer Morphisms for Affine Hecke Algebras

E. Opdam introduced the tool of spectral transfer morphism (STM) of affine Hecke algebras to study the formal degrees of unipotent discrete series representations. He established a uniqueness property of STM for the affine Hecke algebras associated of unipotent discrete series representations. Based on this result, Opdam gave an explanation for Lusztig's arithmetic/geometric correspondence (in Lusztig's classification of unipotent representations of $p$-adic adjoint simple groups) in terms of harmonic analysis, and partitioned the unipotent discrete series representations into $L$-packets based on the Lusztig-Langlands parameters. The present paper provides some omitted details for the argument of the uniqueness property of STM. In the last section, we prove that three finite morphisms of algebraic tori are spectral transfer morphisms, and hence complete the proof of the uniqueness property.

math.RT↗