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We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li, Dvorsky--Sahi, and Kobayashi--Savin. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian unipotent radicals, our proof works uniformly for all of the (classical as well as exceptional) groups under consideration. Our result is an extension of the statement known for several semisimple groups that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal.
Let $G$ be a connected reductive group over the non-archime-dean local field $F$ and let $pi$ be a supercuspidal representation of $G(F)$. The local Langlands conjecture posits that to such a $pi$ can be attached a parameter $L(pi)$, which is an equi
It is conjectured by Adams-Vogan and Prasad that under the local Langlands correspondence, the L-parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the L-group. We verif
We introduce the notion of the automorphic dual of a matrix algebraic group defined over $Q$. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to deduce ar
We propose a new method to construct rigid $G$-automorphic representations and rigid $widehat{G}$-local systems for reductive groups $G$. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.
We study the dimension of the space of Whittaker functionals for depth zero representations of covering groups. In particular, we determine such dimensions for arbitrary Brylinski-Deligne coverings of the general linear group. The results in the pape