Multiplicities of representations in spaces of modular forms
نویسنده
چکیده
This paper shows that for a given irreducible representation ρ of Γ/Γ1, the two functions dim(Mk(Γ1, ρ)) and dim(Sk(Γ1, ρ)) of k are almost linear functions. Let Γ and Γ1 be two Fuchsian subgroups of SL2(Q) of the first kind, with Γ1 a normal subgroup of Γ of index μ. Let H denote the extended upper half-plane [4] and put Y = Γ\H and X = Γ1\H . Let Ak(∗) denotes the space of meromorphic modular forms for ∗ of weight k. We define a representation πk of Γ on Ak(Γ1) by the formula πk(γΓ1)(f) = f |[γ ]k (γ ∈ Γ, f ∈ Ak(Γ1)), where the notation [∗]k is as in Shimura [4]. The representation πk factors through Γ/Γ1 to give a representation – also denoted πk – of the latter group. The space Mk(Γ1) of holomorphic modular forms of weight k for Γ1 and the subspace Sk(Γ1) of cusp forms of weight k for Γ1 are both stable under πk, and the resulting representations of Γ/Γ1 on Mk(Γ1) and Sk(Γ1) will be denoted ρk and σk respectively. Henceforth ρ denotes an irreducible complex representation of Γ/Γ1. If −I ∈ Γ then the value of ρ on the coset of −I in Γ/Γ1 is a scalar by Schur’s lemma, and we say that ρ is even or odd according as the scalar is 1 or −1. If −I ∈ Γ1 then ρ is automatically even. If π is any finite-dimensional complex representation of Γ/Γ1 then we write 〈ρ, π〉 for the multiplicity of ρ in π. For example 〈ρ, ρreg〉 = dim ρ, where ρreg is the regular representation of Γ/Γ1. Theorem Fix an irreducible complex representation ρ of Γ/Γ1, and put
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تاریخ انتشار 2005