Related papers: Theta Correspondence of Automorphic Characters

Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an…

We prove that for any Hecke eigenform f of level 1 and arbitrary weight there is a self-dual cuspidal automorphic form $\pi$ of $GL_6(\Q)$ corresponding to $\Symm^5 (f)$, i.e., such that the system of Galois representations attached to…

We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some…

We consider the action of an irreducible outer automorphism $\phi$ on the closure of Culler--Vogtmann Outer space. This action has north-south dynamics and so, under iteration, points converge exponentially to $[T^\phi_+]$. For each $N \geq…

We systematically study how the integrality of the conformal characters shapes the space of fermionic rational conformal field theories in two dimensions. The integrality suggests that conformal characters on torus with a given choice of…

We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $\pi$ of a connected reductive $p$-adic group, subject to the assumption that $\pi$ is obtained via induction from a representation of…

A systematic way to organise the interesting periods of automorphic forms on a reductive group $G$ is via the theory of nilpotent orbits of $G$. On the other hand, it is known that the theta correspondence can be used effectively to relate…

In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of…

Given two distinct number fields $K$ and $M$, and finite order Hecke characters $\chi$ of $K$ and $\eta$ of $M$ respectively, we say that the pairs $(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated L-functions…

Let $F$ be a field which is, either local non archimedean, or finite, of residual charcateristic $p$ but of characteristic different from $2$. Let $W$ be a symplectic space of finite dimension over $F$. Suppose $R$ is a field of…

Let $A$ be a finite-dimensional algebra over a field $F$ with char$(F)\ne 2$. We show that a linear map $D:A\to A$ satisfying $xD(x)x\in [A,A]$ for every $x\in A$ is the sum of an inner derivation and a linear map whose image lies in the…

In 2014, Yang showed that for $F \in \mathcal{A}_{r, s, 1, 1_N}$, we have $\textup{Sh}_{r}(F \mid V_{24}) = G \otimes \chi_{12}$ where $G\in S^{new}_{r+2s - 1}(\Gamma_{0}(6), - \left( \frac{8}{r} \right), - \left( \frac{12}{r} \right))$,…

A general framework of constructions of endoscopy correspondences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms.…

Let $G$ be a connected graph, and let $\lambda_1$ and $\rho$ denote the spectral radius of $G$ and the universal cover of $G$, respectively. In \cite{Fri03}, Friedman has shown that almost every $n$-lift of $G$ has all of its new…

The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for…

Let $\phi$ be an automorphism of a free group $F_n$ of rank $n$, and let $M_{\phi}=F_n \rtimes_{\phi} \mathbb{Z}$ be the corresponding mapping torus of $\phi$. We study the group $Out(M_{\phi})$ under certain technical conditions on $\phi$.…

We prove that the global geometric theta-lifting functor for the pair (H, G) is compatible with the Whittaker normalization, where (H,G) is one of the pairs (SO_{2n}, Sp_{2n}), (Sp_{2n}, SO_{2n+2}) or (GL_{n},GL_{n+1}). That is, the…

Let $(X_{A},\sigma_{A})$ be a shift of finite type and $\text{Aut}(\sigma_{A})$ its corresponding automorphism group. Associated to $\phi \in \text{Aut}(\sigma_{A})$ are certain Lyapunov exponents $\alpha^{-}(\phi), \alpha^{+}(\phi)$ which…

We explore the relation between the singleton and adjoint modules of higher-spin algebras via so(2,d) characters. In order to relate the tensor product of the singleton and its dual to the adjoint module, we consider a heuristic formula…

In a previous paper (arxiv:1409.7353), we introduced a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of…