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Register a new userRamanujan duals and automorphic spectrum
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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 arithmetic vanishing theorems of ``Ramanujan type as well as to give a new construction of automorphic forms.
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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.
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.
In this paper we first review the setting for the geometric Langlands functoriality and establish a result for the `backward functoriality functor. We illustrate this by known examples of the geometric theta-lifting. We then apply the above result to obtain new Hecke eigen-sheaves. The most important application is a construction of the automorphic sheaf for G=GSp_4 attached to a G^L-local system on a curve X such that its standard representation is an irreducible local system of rank 4 on X.
Let $(V,omega)$ be an orthosympectic $mathbb Z_2$-graded vector space and let $mathfrak g:=mathfrak{gosp}(V,omega)$ denote the Lie superalgebra of similitudes of $(V,omega)$. When the space $mathscr P(V)$ of superpolynomials on $V$ is emph{not} a completely reducible $mathfrak g$-module, we construct a natural basis $D_lambda$ of Capelli operators for the algebra of $mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_lambda$ on maximal indecomposable components of $mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $mathscr P(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_lambda$ are emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators ${mathbf D_{t,lambda}}_{lambdainmathcal P}^{}$ that induce morphisms of indecomposable components of symmetric powers of $mathsf V_t$, where $mathsf V_t$ is the generating object of $mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the $left{mathbf D_{t,lambda}right}_{lambdainmathcal P}$ that are analogous to our results for the operators ${D_lambda}_{lambdainmathcal P}^{}$.
We prove the Ramanujan-Petersson conjecture for Maass forms of the group $SL(2,Z)$, with the help of automorphic distribution theory: this is an alternative to classical automorphic function theory, in which the plane takes the place usually ascribed to the hyperbolic half-plane.
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