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Jiandi Zou

Publications and source records attributed to Jiandi Zou.

9 recordsLinked to original sources
arXiv2026

Wavefront sets for genuine representations of $\rm GL$-covers of Kazhdan--Patterson or Savin types

First, we consider general Brylinski--Deligne covers of the $p$-adic general linear groups, and discuss the theory of Bernstein--Zelevinsky derivatives. We also recall the Zelevinsky-type classification of the irreducible genuine spectrum for the Kazhdan--Patterson and Savin covers. Following this, for these two special families of covers, we determine the wavefront sets of their irreducible genuine representations, expressed in terms of the iterated degrees of the highest Bernstein--Zelevinsky derivatives. Finally, for Kazhdan--Patterson covers, we reinterpret this result on the wavefront set using a version of the local Langlands correspondence and the covering Barbasch--Vogan duality.

math.RT↗
arXiv2026

Ramanujan Complexes from Unitary Groups over Number Fields

In this article, we construct new families of Ramanujan complexes with local structure distinct from all previously known examples. Our approach is based on unitary groups over number fields, more specifically on what we call super-definite unitary groups, that is definite unitary groups that are anisotropic modulo their center at a finite place. These arise naturally as groups of units in central division algebras with involution of the second kind. Our first main result gives a general construction of infinite families of Ramanujan complexes associated with a super-definite unitary group $G$ over a totally real number field and a finite place $v_0$. The structure of the resulting complex is governed by the type of the Bruhat-Tits building at $v_0$. It includes new examples of type $A_n$ when $v_0$ is split, and novel families of type ${}^2\!A'_n$, ${}^2 \! A''_n$ (with $n$ even), $B$-$C_n$, ${}^2 \! B$-$C_n$ and $C$-$BC_n$ in the non-split case. This construction works uniformly across all ranks. Since much of the motivation for constructing expander complexes comes from computer science, we investigate the algorithmic explicitness of our construction in the latter part of the paper, and provide an example in rank 5 where it becomes fully explicit. In particular, this example yields golden gates for the real Lie group $PU(5)$.

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arXiv2025

Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$

For an $n$-fold Kazhdan--Patterson cover or Savin's cover of a general linear group over a non-archimedean local field of residual characteristic $p$ with $\mathrm{gcd}(n,p)=1$, we realize the Gelfand--Graev representation as a Hecke algebra module of a simple type and study its explicit expression. As a main corollary, we calculate the Whittaker dimension of every discrete series representation of such a cover. Using Zelevinsky's classification, this theoretically gives the Whittaker dimension of every irreducible representation.

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arXiv2024

Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution

Let $F/F_{0}$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\neq 2$. Let $R$ be an algebraically closed field of characteristic different from $p$. For $π$ a supercuspidal representation of $G=\mathrm{GL}_{n}(F)$ over $R$ and $G^τ$ a unitary group in $n$ variables contained in $G$, we prove that $π$ is distinguished by $G^τ$ if and only if $π$ is Galois invariant. When $R=\mathbb{C}$ and $F$ is a $p$-adic field, this result first as a conjecture proposed by Jacquet was proved in 2010's by Feigon-Lapid-Offen by using global method. Our proof is local which works for both complex case and $l$-modular case with $l\neq p$. We further study the dimension of $\mathrm{Hom}_{G^τ}(π,1)$ and show that it is at most one.

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arXiv2024

Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution

Let $F$ be a non-archimedean locally compact field of residue characteristic $p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of $G$. For $π$ a complex smooth supercuspidal representation of $G$, we give a full characterization for the distinguished space $\mathrm{Hom}_{H}(π,1)$ being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of $π$ in the space of smooth functions on the set of symmetric matrices in $G$, as a complex vector space, is non-zero and of dimension four, if and only if the central character of $π$ evaluating at $-1$ is $1$.

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arXiv2024

Local metaplectic correspondence and applications

We revisit the local metaplectic correspondence previously constructed and studied by Flicker and Kazhdan. After restoring and generalizing some of their results, we get several interesting applications to the representation theory of a Kazhdan-Patterson covering group over a $p$-adic field, including a full criterion of the irreducibility of the Bernstein-Zelevinsky product of two cuspidal representations, the classification of essentially square integrable and tempered representations, and more interestingly, the calculation of the Whittaker dimension of an irreducible representation.

math.RT↗
arXiv2024

Simple type theory for metaplectic covers of $\mathrm{GL}(r)$ over a non-archimedean local field

Let $F$ be a non-archimedean locally compact field of residual characteristic $p$, let $G=\mathrm{GL}_{r}(F)$ and let $\widetilde{G}$ be an $n$-fold metaplectic cover of $G$ with $\mathrm{gcd}(n,p)=1$. We study the category $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$ of complex smooth representations of $\widetilde{G}$ having inertial equivalence class $\mathfrak{s}=(\widetilde{M},\mathcal{O})$, which is a block of the category $\mathrm{Rep}(\widetilde{G})$, following the "type theoretical" strategy of Bushnell-Kutzko. Precisely, first we construct a "maximal simple type" $(\widetilde{J_{M}},\widetildeλ_{M})$ of $\widetilde{M}$ as an $\mathfrak{s}_{M}$-type, where $\mathfrak{s}_{M}=(\widetilde{M},\mathcal{O})$ is the related cuspidal inertial equivalence class of $\widetilde{M}$. Along the way, we prove the forklore conjecture that every cuspidal representation of $\widetilde{M}$ could be constructed explicitly by a compact induction. Secondly, we construct "simple types" $(\widetilde{J},\widetildeλ)$ of $\widetilde{G}$, and prove that each of them is an $\mathfrak{s}$-type of a certain block $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$. When $\widetilde{G}$ is either a Kazhdan-Patterson cover or Savin's cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde{G}$. Finally, for a simple type $(\widetilde{J},\widetildeλ)$ of $\widetilde{G}$ we describe the related Hecke algebra $\mathcal{H}(\widetilde{G},\widetildeλ)$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde{G}$ is one of the two special covers mentioned above. We leave the construction of a "semi-simple type" related to a general block $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$ to a future phase of the work.

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arXiv2024

Distinction of the Steinberg representation with respect to a symmetric pair

Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $θ$ be an involution of $G$ over $K$, and let $H$ be the connected component of $θ$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Courtès, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincaré series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result.

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