Multiplicity formulas for discrete series representations in L2(Γ\Sp(2,R))L2(Γ\Sp(2,R))

Let Sp(2)Sp(2) denote the split symplectic group of rank 2 over QQ. Fix a prime p  . Let KpKp be a parahoric subgroup of Sp(2,Qp)Sp(2,Qp). An arithmetic subgroup Γ   is defined by Γ=Sp(2,Q)∩(Sp(2,R)K0)Γ=Sp(2,Q)∩(Sp(2,R)K0), where K0=Kp∏v<∞,v≠pSp(2,Zv). In this paper, we calculate Arthurʼs L2L2-Lefschetz trace formula for Sp(2)Sp(2) in order to obtain an explicit formula for multiplicities of discrete series and some non-tempered unitary representations in the discrete spectrum of L2(Γ\Sp(2,R))L2(Γ\Sp(2,R)) for each such Γ  . From them we derive explicit multiplicity formulas for large discrete series, which are our main results. The multiplicity formulas are applied to a study on numbers of cuspidal automorphic representations of PGSp(2)PGSp(2).