Mixed automorphic vector bundles on Shimura varieties (Q1919923)

Let \(S^0 (G, X)\), \(S^0 (G', X')\) be connected Shimura varieties associated to semisimple algebraic groups \(G\), \(G'\) defined over \(\mathbb{Q}\) and Hermitian symmetric domains \(X\), \(X'\). Let \(\rho: G\to G'\) be a homomorphism of algebraic groups over \(\mathbb{Q}\) that induces a holomorphic map \(\omega: X\to X'\) mapping special points of \(X\) to special points of \(X'\). Given equivariant vector bundles \({\mathcal J}\), \({\mathcal J}'\) on the compact duals \(\check X\), \(\check X'\) of the symmetric domains \(X\), \(X'\), we can construct a mixed automorphic vector bundle \({\mathcal M} ({\mathcal J}, {\mathcal J}', \rho)\), on \(S^0 (G, X)\) whose sections can be interpreted as mixed automorphic forms. We prove that the space of sections of a certain mixed automorphic vector bundle is isomorphic to the space of holomorphic forms of the highest degree on the fiber product of a finite number of Kuga fiber varieties. We also prove that for each automorphism \(\tau\) of \(\mathbb{C}\) the conjugate \(\tau{\mathcal M} ({\mathcal J}, {\mathcal J}', \rho)\) of a mixed automorphic vector bundle \({\mathcal M} ({\mathcal J}, {\mathcal J}', \rho)\) on a connected Shimura variety \(S^0 (G, X)\) can be canonically realized as a mixed automorphic vector bundle \({\mathcal M} ({\mathcal J}_1, {\mathcal J}_1', \rho_1)\) on another connected Shimura variety \(S^0 (G_1, X_1)\) associated to a semisimple algebraic group \(G_1\) and a Hermitian symmetric domain \(X_1\).