We prove a conjecture of the first-named author ([J14]) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of split classical groups over any number field.
We complete the proof of Proposition 5.3 of [GJR04].
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $pi$ of $G$, determine the occurrence of an irreducible representation $sigma$ of $H$ in the restriction of $pi$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $sigma$ of $H$, find an irreducible representation $pi$ of $G$ such that $sigma$ occurs in the restriction of $pi$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.
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 $mathrm{GL}_n(mathbb{A})$ of form $$ Delta(tau_1, b_1) boxplus Delta(tau_2, b_2) boxplus cdots boxplus Delta(tau_r, b_r),, $$ where $Delta(tau_i,b_i)$s are Speh representations in the discrete spectrum of $mathrm{GL}_{a_ib_i}(mathbb{A})$ with $tau_i$s being unitary cuspidal representations of $mathrm{GL}_{a_i}(mathbb{A})$, and $n = sum_{i=1}^r a_ib_i$. Endoscopic lifting images of the discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.
In this paper, we reprove a global converse theorem of Cogdell and Piatetski-Shapiro using purely global methods.
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 study model transition for representations occurring in the local theta correspondence between split even special orthogonal groups and symplectic groups, over a non-archimedean local field of characteristic zero.
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].
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.
