On counting cuspidal automorphic representations for GSp(4)

Abstract

Abstract We find the number sk\u2062(p,Ω)s_{k}(p,\\Omega) of cuspidal automorphic representations of GSp\u2062(4,AQ)\\mathrm{GSp}(4,\\mathbb{A}_{\\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k≥3k\\geq 3, and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for sk\u2062(p,Ω)s_{k}(p,\\Omega) generalizes to the vector-valued case and a finite number of ramified places.