Cycles with Local Coefficients for Orthogonal Groups and Vector-valued Siegel Modular Forms

The purpose of this paper is to generalize the relation [KM4] between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms. Let V be a non-degenerate quadratic space of dimension m and signature (p, q) over Q, for simplicity. The general case of a totally real number field is treated in the main body of the paper. We write V = V (R) for the real points of V and let G = SO0(V ). Let G ′ denote the nontrivial 2-fold covering group of the symplectic group Sp(n,R) (the metaplectic group) and K ′ be the 2-fold covering inherited by U(n). Let D = G/K resp. D = G/K ′ be the symmetric space of G resp. G. Note that D = Hn, the Siegel upper half space. In what follows we will choose appropriate (related) arithmetic subgroups Γ ⊂ G and Γ ⊂ G. We letM = Γ\D andM ′ = Γ\D be the associated locally symmetric spaces. If M is not compact, we let M denote the Borel-Serre compactification and ∂M denote the Borel-Serre boundary of M . We let En denote the holomorphic vector bundle over Hn associated to the standard representation of U(n), i.e., En = Sp(n,R) ×U(n) C . For each dominant weight λ of U(n), we have the corresponding irreducible representation space Sλ′(C ) of U(n) and the associated holomorphic vector bundle Sλ′En over M ′ (see §3, for the meaning of the Schur functor Sλ′(·)). For each half integer k/2 we have a character det k/2 of K . Let Lk/2 be the associated G -homogeneous line bundle over the Siegel space. For each dominant weight λ of G, we have the corresponding irreducible representation S[λ](V ) of G with highest weight λ and the flat vector bundle S[λ](V) over M with typical fiber S[λ](V ) (see §3, for the meaning of the harmonic Schur functor S[λ](·)). Let λ be a dominant weight for G. Let i(λ) be the number of nonzero entries in λ when λ is expressed in the coordinates relative to the standard basis {ǫi} of [Bou], Planche II and IV. Hence we have i(λ) ≤ [m/2]. We will assume (because of the choice of X below in the construction of our cycles CX , see Remark 4.7) that i(λ) ≤ p. Now we choose n as in the paragraph above to be any integer satisfying i(λ) ≤ n ≤ p and choose for our highest weight of U(n) corresponding to λ the unique dominant weight