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Register a new userOn the Stable Transfer for $mathrm{Sym}^{n}$ Lifting of $mathrm{GL}_{2}$
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Following the paradigm of cite{MR3117742}, we are going to explore the stable transfer factors for $mathrm{Sym}^{n}$ lifting from $mathrm{GL}_{2}$ to $mathrm{GL}_{n+1}$ over any local fields $F$ of characteristic zero with residue characteristic not equal to $2$. When $F=mathbb{C}$ we construct an explicit stable transfer factor for any $n$. When $n$ is odd, we provide a reduction formula, reducing the question to the construction of the stable transfer factors when the $L$-morphism is the diagonal embedding from $mathrm{GL}_{2}(mathbb{C})$ to finitely many copies of $mathrm{GL}_{2}(mathbb{C})$ under mild assumptions on the residue characteristic of $F$. With the assumptions on the residue characteristic, the reduction formula works uniformly over any local fields of characteristic zero, except that for $p$-adic situation we need to exclude the twisted Steinberg representations.
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Let $mathsf k$ be a local field. Let $I_ u$ and $I_{ u}$ be smooth principal series representations of $mathrm{GL}_n(mathsf k)$ and $mathrm{GL}_{n-1}(mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $mathrm{GL}_n(mathsf k)times mathrm{GL}_n(mathsf k)$.
Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. In this paper we study the {em Whittaker coinvariants functor}, which is an exact functor from category $mathcal O$ for $mathfrak{gl}_{m|n}(mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergels functor $mathbb V$ in the setting of category $mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $mathcal O$ up to Morita/derived equivalence.
For a split reductive group $G$ over a number field $k$, let $rho$ be an $n$-dimensional complex representation of its complex dual group $G^vee(mathbb{C})$. For any irreducible cuspidal automorphic representation $sigma$ of $G(mathbb{A})$, where $mathbb{A}$ is the ring of adeles of $k$, in cite{JL21}, the authors introduce the $(sigma,rho)$-Schwartz space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho}$, and study the $(sigma,rho,psi)$-Poisson summation formula on $mathrm{GL}_1$, under the assumption that the local Langlands functoriality holds for the pair $(G,rho)$ at all local places of $k$, where $psi$ is a non-trivial additive character of $kbackslashmathbb{A}$. Such general formulae on $mathrm{GL}_1$, as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture (cite{L70}) on global functional equation for the automorphic $L$-functions $L(s,sigma,rho)$. In order to understand such Poisson summation formulae, we continue with cite{JL21} and develop a further local theory related to the $(sigma,rho)$-Schwartz space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho}$. More precisely, over any local field $k_ u$ of $k$, we define distribution kernel functions $k_{sigma_ u,rho,psi_ u }(x)$ on $mathrm{GL}_1$ that represent the $(sigma_ u,rho)$-Fourier operators $mathcal{F}_{sigma_ u,rho,psi_ u}$ as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands $gamma$-functions $gamma(s,sigma_ u,rho,psi_ u)$ as Mellin transform of the kernel function. As consequence, we show that any local Langlands $gamma$-functions are the gamma functions in the sense of Gelfand, Graev, and Piatetski-Shapiro in cite{GGPS}.
In this paper, we explore possibilities to utilize harmonic analysis on $mathrm{GL}_1$ to understand Langlands automorphic $L$-functions in general, as a vast generalization of the pioneering work of J. Tate (cite{Tt50}). For a split reductive group $G$ over a number field $k$, let $G^vee(mathbb{C})$ be its complex dual group and $rho$ be an $n$-dimensional complex representation of $G^vee(mathbb{C})$. For any irreducible cuspidal automorphic representation $sigma$ of $G(mathbb{A})$, where $mathbb{A}$ is the ring of adeles of $k$, we introduce the space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ of $(sigma,rho)$-Schwartz functions on $mathbb{A}^times$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho,psi}$ that takes $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ to $mathcal{S}_{widetilde{sigma},rho}(mathbb{A}^times)$, where $widetilde{sigma}$ is the contragredient of $sigma$. By assuming the local Langlands functoriality for the pair $(G,rho)$, we show that the $(sigma,rho)$-theta functions [ Theta_{sigma,rho}(x,phi):=sum_{alphain k^times}phi(alpha x) ] converges absolutely for all $phiinmathcal{S}_{sigma,rho}(mathbb{A}^times)$, and state conjectures on $(sigma,rho)$-Poisson summation formula on $mathrm{GL}_1$. One of the main results in this paper is to prove the conjectures when $G=mathrm{GL}_n$ and $rho$ is the standard representation of $mathrm{GL}_n(mathbb{C})$. The proof uses substantially the local theory of Godement-Jacquet (cite{GJ72}) for the standard $L$-functions of $mathrm{GL}_n$ and the Poisson summation formula for the classical Fourier transform on affine spaces.
Let $F$ be a quadratic extension of $mathbb{Q}_p$. We prove that smooth irreducible supersingular representations with central character of $mathrm{GL}_2(F)$ are not of finite presentation.
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