Inducing Whittaker Functions from Higher Ranks

Abstract

We construct a family of Whittaker functions for $SL(m,\mathbb{Z})$ induced directly from Whittaker functions for $SL(n,\mathbb{Z})$, for any $2 \leq m<n$. Given Jacquet's Whittaker function $W_{\alpha,N}^{(n)}$ on the generalized upper half-plane $\mathfrak{h}^n$, we show that the function $V_{\alpha,N}^{(m)}:\mathfrak{h}^m\to\mathbb{C}$ defined by restricting $W_{\alpha,N}^{(n)}$ to the block-diagonal embedding $\mathfrak{h}^m\hookrightarrow\mathfrak{h}^n$ is a Whittaker function for $SL(m,\mathbb{Z})$, provided the Langlands parameters $\alpha=(\alpha_i)_{1\leq i\leq n}$ satisfy $\sum_{i=1}^m\alpha_i = m(m-n)/2$. Under this condition, the induced function carries Langlands parameters $\bigl(\alpha_i+\frac{n-m}{2}\bigr)_{1\leq i\leq m}$ and inherits the first $m-1$ entries of the character tuple of $W_{\alpha,N}^{(n)}$. This result complements the propagation formulas of Ishii and Stade, which relate Whittaker functions on $GL(n,\mathbb{R})$ to those on $GL(n-1,\mathbb{R})$ and $GL(n-2,\mathbb{R})$. In contrast, our construction passes directly from $GL(n,\mathbb{R})$ to $GL(m,\mathbb{R})$ for any $m < n$ in a single step.