Generalizations of a Theorem of Hecke

Abstract

Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π⁺, π⁻ be the pair of cuspidal representations of SL₂(𝔽p). It is well known by Hecke that the difference mπ⁺ - mπ⁻ in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(-p) of the imaginary quadratic field ℚ(√(-p)). This thesis consists of two main parts. In the first part, we extend Hecke's result to all fundamental discriminants of imaginary quadratic fields, including the even case. The proof is geometric in nature and uses the holomorphic Lefschetz number. In the second part, we consider generalizations to groups with higher ℚ-rank. In particular, we focus on the rank 2 special unitary group SU(2, 2). On the representation theory side, we prove the regular unipotent classes have positive contribution to an alternating sum of multiplicities of certain irreducible cuspidal representations of SU(2, 2) over the finite field of p elements. We also show that the semisimple classes have zero contribution, which is again a direct generalization of the SL₂ case. To obtain these two results, we make use of the Deligne-Lusztig theory and the connection of the traces to the Gelfand-Graev representations.