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 $2\le m<n$. Given Jacquet's Whittaker function $W_{\hat{I}\pm ,N}^{(n)}$ on the generalized upper half-plane ${\mathfrak{h}}^n$, we show that the function $V_{\hat{I}\pm ,N}^{(m)}:{\mathfrak{h}}^m\to \mathbb{C}$ defined by restricting $W_{\hat{I}\pm ,N}^{(n)}$ 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 $\hat{I}\pm =(\hat{I}{\pm }_i{)}_{1\le i\le n}$ satisfy ${\sum }_{i=1}^m\hat{I}{\pm }_i=m(m-n)/2$. Under this condition, the induced function carries Langlands parameters $(\hat{I}{\pm }_i+\frac{n-m}{2}{)}_{1\le i\le m}$ and inherits the first $m-1$ entries of the character tuple of $W_{\hat{I}\pm ,N}^{(n)}$. 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.

9 pages