An analogue of Siegel’s $\phi$-operator for automorphic forms for $\textrm {GL}_ n(\textbf {Z})$

Abstract:

If $\mathcal {S}{P_n}$ is the symmetric space of $n \times n$ positive matrices, $Y \in \mathcal {S}{P_n}$ can be decomposed into \[ Y = \left ( {\begin {array}{*{20}{c}} 1 & 0 \\ x & I \\ \end {array} } \right )\left ( {\begin {array}{*{20}{c}} {{v^{ - 1}}} & 0 \\ 0 & {{v^{ - 1/(n - 1)}}W} \\ \end {array} } \right )\left ( {\begin {array}{*{20}{c}} 1 & {{T_x}} \\ 0 & I \\ \end {array} } \right ),\] where $W \in \mathcal {S}{P_{n - 1}}$ . By letting $v \to \infty$ we obtain the $\phi$-operator that attaches to every automorphic form for $G{L_n}(\mathbb {Z})$, $f(Y)$, an automorphic form for $G{L_{n - 1}}(\mathbb {Z})$, $f|\phi (W)$.