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We prove the expected algebraicity property for the critical values ofcharacter twists of the standard $L$-function associated to vector-valuedholomorphic Siegel cusp forms of archimedean type $(k_1, k_2, \ldots, k_n)$,where $k_n \geq n+1$ and all $k_i$ are of the same parity. For the proof, weuse an explicit integral representation to reduce to arithmetic properties ofdifferential operators on vector-valued nearly holomorphic Siegel cusp forms.We establish these properties via a representation-theoretic approach.

The special values of the standard $L$-functions for $\mathrm{GSp}_{2n} \times \mathrm{GL}_1$