Inducing Whittaker Functions from Higher Ranks
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914612179369984 |
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| author | Muthuvel, Vishal |
| author_facet | Muthuvel, Vishal |
| contents | 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 $W_{α,N}^{(n)}$ on the generalized upper half-plane $\mathfrak{h}^n$, we show that the function $V_{α,N}^{(m)}:\mathfrak{h}^m\to\mathbb{C}$ defined by restricting $W_{α,N}^{(n)}$ 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 $α=(α_i)_{1\leq i\leq n}$ satisfy $\sum_{i=1}^mα_i = m(m-n)/2$. Under this condition, the induced function carries Langlands parameters $\bigl(α_i+\frac{n-m}{2}\bigr)_{1\leq i\leq m}$ and inherits the first $m-1$ entries of the character tuple of $W_{α,N}^{(n)}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29342 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inducing Whittaker Functions from Higher Ranks Muthuvel, Vishal Number Theory 11F70 (Primary) 11F55, 33C15 (Secondary) 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 $W_{α,N}^{(n)}$ on the generalized upper half-plane $\mathfrak{h}^n$, we show that the function $V_{α,N}^{(m)}:\mathfrak{h}^m\to\mathbb{C}$ defined by restricting $W_{α,N}^{(n)}$ 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 $α=(α_i)_{1\leq i\leq n}$ satisfy $\sum_{i=1}^mα_i = m(m-n)/2$. Under this condition, the induced function carries Langlands parameters $\bigl(α_i+\frac{n-m}{2}\bigr)_{1\leq i\leq m}$ and inherits the first $m-1$ entries of the character tuple of $W_{α,N}^{(n)}$. 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. |
| title | Inducing Whittaker Functions from Higher Ranks |
| topic | Number Theory 11F70 (Primary) 11F55, 33C15 (Secondary) |
| url | https://arxiv.org/abs/2605.29342 |