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Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={mathbb G}_mtimes{mathrm Sp}_{2n}$. It is associated to the Langlands $gamma$-functions attached to any irreducible admissible representations $chiotimespi$ of $G(F)$ and the standard representation $rho$ of the dual group $G^vee({mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${rm GL}_1(F)$, which is associated to a $gamma$-function $beta_psi(chi_s)$ (a product of $n+1$ certain abelian $gamma$-functions). Our work on ${rm GL}_1(F)$ plays an indispensable role in the development of our work on $G(F)$. These two types of harmonic analyses both specialize to the well-known local theory developed in Tates thesis when $n=0$. The approach is to use the compactification of ${rm Sp}_{2n}$ in the Grassmannian variety of ${rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.
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We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.
In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [Sar05].
We study in detail relevant spectral properties of the adjacency matrix of inhomogeneous amenable networks, and in particular those arising by negligible additive perturbations of periodic lattices. The obtained results are deeply connected to the systematic investigation of the Bose--Einstein condensation for the so called Pure Hopping model describing the thermodynamics of Bardeen--Cooper pairs of Bosons in arrays of Josephson junctions.
The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{rho}$ for each irreducible unitary $G$-representation $rho$, a relation $g_{rho}=g_{rho}g_{rho}$ whenever $rho$ is weakly contained in $rhootimes rho$, and $g_{rho^*}=g_{rho}^{-1}$ for the representation $rho^*$ contragredient to $rho$. $G$ satisfies chain-center duality if assigning to each $g_{rho}$ the central character of $rho$ is an isomorphism of $C(G)$ onto the dual $widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M{u}gers result compact groups satisfy chain-center duality.
In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $sigma$ of symplectic groups $mathrm{Sp}_{2n}(mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,sigmatimeschi)$ for some character $chi$ of $F^timesbackslashmathbb{A}^times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(chi,b)$ in the global Arthur parameter $psi$ attached to $sigma$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.
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