In this paper we characterize irreducible generic representations of $\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of $\SO_{2n+1}(k)$.
We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.
In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.
We prove a local converse theorem for $GL_n$ over the archimedean local fields which characterizes an infinitesimal equivalence class of irreducible admissible representations of $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ in terms of twisted local gamma factors.
In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic zero. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply new ideas of considering the summation of partial Bessel functions and overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups.
Let F be a non-archimedean local field of characteristic zero. In this paper we construct examples of supercuspidal representations showing that the bound $[N/2]$ for the local converse theorem of $GL_N(F)$ is sharp, N general, when the residual characteristic of $F$ is bigger than $N$.
Mod $\ell$ gamma factors and a converse theorem for finite general linear groups
Boston University（波士顿大学）；University of Toronto（多伦多大学）；Brooklyn College, City University of New York（纽约市立大学布鲁克林学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Maine（缅因大学）；Massachusetts Institute of Technology（麻省理工学院）
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 …
Heilbronn Institute for Mathematical Research, School of Mathematics University of Bristol（布里斯托大学数学学院海布隆数学研究所）；Dept of Mathematics and Statistics Williams College（威廉姆斯学院数学与统计系）
In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $G_2$ over finite fields $k$ of odd characteristic. Then we prove the first converse theorem for exceptional groups, namely, $GL_1$ and $GL_2$-twisted gamma factors will uniquely determine an irreducible generic cuspidal representation of $G_2(k)$.
In this paper we prove a local converse theorem for GL_n over the archimedean local fields, which characterizes an infinitesimal equivalence class of irreducible admissible representations of GL_n(R) (or GL_n(C)) in terms of twisted L-factors.
In this paper, we prove the local converse theorem for $\textrm{Sp}_{2r}(F)$ over a $p$-adic field $F$. More precisely, given two irreducible supercuspidal representations of $\textrm{Sp}_{2r}(F)$ with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of $\textrm{GL}_n(F)$ for all $1\le n\le r$, then these two representations must be isomorphic. Our proof is based on the local analysis of the local integrals which define local gamma factors. A key ingredient of the pr…
Let $E/F$ be a quadratic extension of $p$-adic fields and $\textrm{U}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for $\textrm{U}_{2r+1}$: given two irreducible generic supercuspidal representations $π,π_0$ of $\textrm{U}_{2r+1}$ with the same central character, if $γ(s,π\times τ,ψ)=γ(s,π_0\times τ,ψ)$ for all irreducible generic representation $τ$ of $\textrm{GL}_n(E)$ and for all $n$ with $1\le n\le r$, then $π\cong π_0$. The proof depends on analysis of the local integrals which define local gamma factors and uses certain properties of p…
