On the functional equations of Siegel Eisenstein series of an odd prime level $p$

Let $p$ be an odd prime. In this paper we write down the functional equations of Siegel Eisenstein series of degree $n$, level $p$ with quadratic or trivial characters. First we study the $U(p)$ action in the space of Siegel Eisenstein series to get $U(p)$-eigen functions. Secondly we show that the functional equations for $U(p)$-eigen Eisenstein series are quite simple and easy to write down. Consequently the matrix that represents functional equations of Siegel Eisenstein series are given by the products of simple matrices.