Distinguished cusp forms are theta series
The author studies theta liftings defined by the oscillator representation of the dual reductive pair \((Sp_{2n}, O_ n)\) over a number field k. The main result of the paper states that if \(\pi\) is an irreducible cuspidal automorphic representation of \(Sp_{2n}({\mathbb{A}}_ k)\) which is distinguished and associated to a fixed non-singular symmetric matrix \(T_ 0\) of size n with entries in k, then \(\pi\) is a constituent of the ``space of theta functions'' attached to \(T_ 0\). Here \(\pi\) is defined to be distinguished (and associated to \(T_ 0)\) if all functions f in the space of \(\pi\) have vanishing Fourier coefficients \(f_ T\) for all non-singular T which are not \(GL_ n(k)\)-equivalent to \(T_ 0\). Also the ``space of theta functions'' attached to \(T_ 0\), by definition, is the closed subspace of automorphic forms on \(Sp_{2n}({\mathbb{A}}_ k)\) generated by all theta-lifts defined by the oscillator representation of the dual pair \((Sp_{2n}, O_{T_ 0})\) where \(O_{T_ 0}\) is the orthogonal group attached to \(T_ 0.\) The author's result can be viewed as a generalization of the work of \textit{S. Gelbart} and \textit{I. I. Piatetski-Shapiro} [Invent. Math. 59, 145-188 (1980; Zbl 0426.10027)] who have characterized distinguished forms on \(\tilde Sp_ 2\) as theta series. The main ingredients of the proof are i) the classification of the distinguished representations of \(Sp_{2n}(F)\) where F is a local field, in terms of the unitary dual of \(O_{T_ 0}(F)\) and ii) the characterization of theta functions in terms of L-series due to Piatetski-Shapiro and Rallis.
scientific article; zbMATH DE number 4128914
scientific article