Nearly holomorphic Eisenstein liftings (Q1378247)

Let \(\mathcal H_n\) be the Siegel upper half space of degree \(n\), and let \(f: \mathcal H_n \to \mathbb C\) be a \(C^\infty\)-modular form for the Siegel modular group \(\Gamma_n = Sp (n, \mathbb Z)\). Given a nonnegative integer \(\nu\), assume that \(\det (y)^\nu f(z)\) is a polynomial in the entries of \(y\) whose coefficients are holomorphic functions in \(z = x +iy\) bounded on \(\{ z \in \mathcal H_n \mid y \geq \delta 1_n \}\) for all \(\delta >0\). Then \(f\) has a Fourier expansion of the form \[ f (z) = \det (y)^{-\nu} \sum_{h \geq 0} p(h,y) \exp (2\pi i \text{Tr} (hz)). \] It is also assumed that \(\det (y)^\nu p(h,y^{-1})\) is a polynomial in \(y\) for each \(h\) and that \(p(h,y)\) is a polynomial in \(y\) and in the entries in the upper left \(r \times r\) block of \(y\) if the only nonzero entries of \(h\) appear in the same block. The complex vector space consisting of such \(f\) is denoted by \(N_{k,\nu} (\Gamma_n)\). Then elements of \(N_{k,\nu} (\Gamma_n)\) are nearly holomorphic modular forms in the sense of \textit{G. Shimura} [Math. Ann., II. Ser. 123, 347-406 (1986; Zbl 0593.10022)]. Let \(S_k (\Gamma_r)\) be the complex vector space of holomorphic cusp forms of weight \(k\) for \(\Gamma_r\) for \(0 \leq r <n\). In this paper the author constructs a mapping from the direct sum of \((\nu+1)\) copies of \(S_k (\Gamma_r)\) to \(N_{k,\nu} (\Gamma_n)\) and proves that it is injective.