On a construction of certain classes of cuspidal automorphic forms via Poincaré series
Let \(G\) be a semisimple algebraic group defined over a number field \(k\). Let \(\mathbb{A}\) denote the adelic ring of \(k\). Let \(\mathbb{A}_f\) denote its finite part. This paper discusses the construction of cusp forms on \(G(k)\setminus G(\mathbb{A})\) by the method of Poincaré series. More precisely, let \(L\) denote an open compact subgroup of \(G(\mathbb{A}_f)\) and let \(\text{char}_L\) denote the characteristic function on \(L\). Suppose \(\varphi_\infty\in C^\infty(G_\infty)\cap L^1(G_\infty)\) is a nonzero function which is right \(K_\infty\)-finite and \(Z({\mathfrak g}_\infty)\)-finite. Then it is necessary that \(G_\infty\) and \(K_\infty\) have the same rank, and \(\varphi_\infty\) is a matrix coefficient of a discrete series representation of \(G_\infty\). Let \(\varphi\) be a function on \(C^\infty(G_\infty)\otimes C^\infty_c(G(\mathbb{A}_f)\) such that \(\varphi\) is a finite sum of functions of the form \(\varphi_\infty\otimes\text{char}_L\) described above. The author denotes the set of such functions by \(C^\infty(G(\mathbb{A}))^{\text{fact,int}}_{K_\infty,Z({\mathfrak g}_\infty)\text{-finite}}\). We will write \({\mathcal C}\) for short. For \(\varphi\in{\mathcal C}\), we form the Poincaré series \(P(\varphi)(g)= \sum_{\gamma\in G(k)}\varphi(\gamma g)\) for \(g\in G(\mathbb{A})\). If the sum converges, then \(P(\varphi)\) is a function on \(G(k)\setminus G(\mathbb{A})\). Let \({\mathcal A}_{\text{cusp}}={\mathcal A}_{\text{susp}}(G(k)\setminus G(\mathbb{A}))\) denote the space of cuspidal automorphic forms on \(G(k)\setminus G(\mathbb{A})\). The main result of this paper states that for \(\varphi\in{\mathcal C}\), the sum in the Poincaré series converges absolutely and uniformly on compact sets, and \(P(\varphi)\) is a cuspidal automorphic form on \(G(k)\setminus G(\mathbb{A})\). The space \({\mathcal C}\) is a \(({\mathfrak g}_\infty, K_\infty)\times G(\mathbb{A}_f)\)-module and the map \({\mathcal C}\to{\mathcal A}_{\text{cusp}}\) given by \(\varphi\mapsto P(\varphi)\) is a \(({\mathfrak g}_\infty, K_\infty)\times G(\mathbb{A}_f)\)-module homomorphism. In the second half of the paper, the author studies and obtains a series of interesting results on the non-vanishing of the Poincaré series \(P(\varphi)\). One of the results is as follows: Pick a nonzero \(\varphi\in C^\infty(G_\infty)\cap L^1(G_\infty)\) which is right \(K_\infty\)-finite and \(Z({\mathfrak g}_\infty)\)-finite. Then for a sufficiently small open compact subgroup \(L\) in \(G(\mathbb{A}_f)\), \(P(\varphi_\infty\otimes \text{char}_L)\) is nonzero. In particular the map \({\mathcal C}\to{\mathcal A}_{\text{cusp}}\) given by \(\varphi\mapsto P(\varphi)\) above is nonzero. The paper is concise and well organized.
scientific article; zbMATH DE number 5376005
scientific article