An Application of the Hard Lefschetz Theorem

Abstract

In the later parts of this chapter, we will use the hard Lefschetz theorem to show that an irreducible cuspidal automorphic representation π = π∞πfin of the group GSp(4,A), whose Archimedean component π∞ belongs to the discrete series, gives a contribution to a cohomology group of some canonical associated locally constant sheaves on Siegel modular threefolds only if the cohomology degree is 3, provided π is not a cuspidal representation associated with parabolic subgroup (CAP representation). In other words, under the action of the adele group GSp(4,A_fin), all irreducible constituents occur in the middle degree, if one discards so-called CAP representations. Since CAP representations are well understood for the group GSp(4), this result is important for the analysis of the supertrace of Hecke operators acting on the cohomology of Siegel modular threefolds, and hence for the proof of the generalized Ramanujan conjecture for holomorphic Siegel modular forms of genus 2 and weight 3 or more.