An Exceptional Representation of Sp(4, Fq)

Let p be an odd prime, and let q be a power of p. Let G = Sp(4). B.Srinivasan (in [4]) discovered an irreducible representation (denoted by θ10) of Sp(4,Fq) with the following remarkable combination of properties, namely it is cuspidal(Defn. 4.1), unipotent(Defn. 5.8) as well as degenerate, i.e. it does not admit a Whittaker model(Defn. 4.2). The groups SLn(Fq) and GLn(Fq) do not have any unipotent cuspidal representations and neither do they have any degenerate cuspidal representations. Hence the existence of such a representation for Sp(4,Fq) is somewhat surprising. We will describe a folklore construction of θ10, which is different from [4]. It is based on the Weil representation of Sp(8,Fq) and Howe duality. This article was a part of my master’s thesis during my graduate studies at the University of Chicago. My advisor, V.Drinfeld, suggested that I publish this article in the e-print archive, since there were apparently no references for this construction of θ10.