arXiv subjects

Alan Xuelun Hou

Publications and source records attributed to Alan Xuelun Hou.

3 recordsLinked to original sources

arXiv2026

Twisted Jacquet modules and induction from Speh representations

Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field $F$ of characteristic zero. Let $q_F$ be the cardinality of its residue field. Let $J_s$ be the twisted Jacquet module of the representation induced from $Δ(τ,m+i)|\det|^s$. It carries an action of $G\times H$, where $G$ and $H$ are split symplectic or special orthogonal groups. Let $σ$ be an admissible representation of $G$ of finite length. We prove that $(J_s\otimesσ)_G$ is induced from $Δ(τ,i)|\det|^s$ and a fixed conjugate of $σ$, outside a finite set of values of $q_F^{-s}$ depending on $τ$ and $σ$. No genericity assumption on $σ$ is needed. For irreducible $σ$, this determines all irreducible quotients of $J_s$ of the form $σ^\vee\boxtimesπ$. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal $σ$ unless $G$ is the split group $\mathrm{SO}_2$. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when $τ$ is cuspidal on $\mathrm{GL}_n$ and $n$ is larger than the rank of $G$.

math.RT↗

arXiv2026

Distinguished standard modules for $\mathrm{GL}_{2m}(\mathbb{C})/\mathrm{GL}_m(\mathbb{H})$

We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $δ_1,\dots, δ_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = δ_1 \times \cdots \times δ_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${δ_i^*} (z) = δ_i(\overline{z})^{-1}$ for $ z \in \mathbb{C}^\times$. In particular, we conclude that a standard module for $\GL_{2m}(\mathbb{C})$ is distinguished by $\GL_{m}(\mathbb{H})$ if and only if there exists an involution $p\in S_{2m}$ without fixed points such that $δ_{p(i)}=δ_i^*$ for every $i$. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.

math.RT↗

arXiv2026

Fourier Coefficients of the Degenerate Eisenstein Series on Symplectic Groups

We study degenerate Eisenstein series on symplectic groups and construct a new automorphic descent. We show that a certain Fourier coefficient, after restriction to a smaller symplectic group, is itself an Eisenstein series with the same inducing parameter $s$. We describe the resulting holomorphic section explicitly in terms of the original section and local $L$-factors at unramified places. At every regular point of the local descent, we prove surjectivity at non-Archimedean places and dense image at Archimedean places. We finally indicate an application of the iterated descent to maximal Fourier coefficients of Eisenstein series.

math.NT↗