I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated.

Let us denote $$y:=\begin{pmatrix} y_1&&\\&\ddots&\\&&y_n\end{pmatrix},\quad y_i>0,$$ and $$\mu:=\{\mu_1,\dots,\mu_n\}\in\mathbb{C}^n,\quad \Re(\mu_i)\ge 0.$$ Let $W_\mu(y)$ be the spherical Whittaker function on $\mathrm{GL}_n(\mathbb{R})$ with spectral parameter $\mu$, with [usual (Stade's) normalization][1]. Let $y$ be not in the positive Weyl chamber, i.e. there exists at least one $1\le i\le n-1$ such that $y_i\ge y_{i+1}$. Let $\delta$ be the modular character. For above kind of $y$ I am looking for an upper bound of $\delta^{-1/2}(y)W_\mu(y)$ which is strong when $y_1$ is small, preferably of the form $$\delta^{-1/2}(y)W_\mu(y)\ll_\mu y_1^{A},$$ where $A=A(\Re(\mu))$ is positive and depends on $\Re(\mu)$. When $n=2$ and $y_1\ge y_2$ this sort of bound is obvious, for e.g. $$W_\mu(y)=(y_1y_2)^{\frac{\mu_1+\mu_2}{2}}\delta^{1/2}(y)K_{\frac{\mu_1-\mu_2}{2}}(2\pi y_1/y_2)\ll_\mu y_1^{\Re(\frac{\mu_1+\mu_2}{2})}\delta^{1/2}(y).$$

Thanks in advance! [1]: https://projecteuclid.org/download/pdf_1/euclid.dmj/1077297295

I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated.

Let us denote $$y:=\begin{pmatrix} y_1&&\\&\ddots&\\&&y_n\end{pmatrix},\quad y_i>0,$$ and $$\mu:=\{\mu_1,\dots,\mu_n\}\in\mathbb{C}^n,\quad \Re(\mu_i)\ge 0.$$ Let $W_\mu(y)$ be the spherical Whittaker function on $\mathrm{GL}_n(\mathbb{R})$ with spectral parameter $\mu$, with usual (Stade's) normalization. Let $y$ be not in the positive Weyl chamber, i.e. there exists at least one $1\le i\le n-1$ such that $y_i\ge y_{i+1}$. Let $\delta$ be the modular character. For above kind of $y$ I am looking for an upper bound of $\delta^{-1/2}(y)W_\mu(y)$ which is strong when $y_1$ is small, preferably of the form $$\delta^{-1/2}(y)W_\mu(y)\ll_\mu y_1^{A},$$ where $A=A(\Re(\mu))$ is positive and depends on $\Re(\mu)$. When $n=2$ and $y_1\ge y_2$ this sort of bound is obvious, for e.g. $$W_\mu(y)=(y_1y_2)^{\frac{\mu_1+\mu_2}{2}}\delta^{1/2}(y)K_{\frac{\mu_1-\mu_2}{2}}(2\pi y_1/y_2)\ll_\mu y_1^{\Re(\frac{\mu_1+\mu_2}{2})}\delta^{1/2}(y).$$

Thanks in advance!

I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated.

Let us denote $$y:=\begin{pmatrix} y_1&&\\&\ddots&\\&&y_n\end{pmatrix},\quad y_i>0,$$ and $$\mu:=\{\mu_1,\dots,\mu_n\}\in\mathbb{C}^n,\quad \Re(\mu_i)\ge 0.$$ Let $W_\mu(y)$ is the spherical Whittaker function with spectral parameter $\mu$, with [usual (Stade's) normalization][1]. Let $y$ be not in the positive Weyl chamber, i.e. there exists at least one $1\le i\le n-1$ such that $y_i\ge y_{i+1}$. Let $\delta$ be the modular character. For above kind of $y$ I am looking an upper bound of $\delta^{-1/2}(y)W_\mu(y)$ which is strong when $y_1$ is small, preferably of the form $$\delta^{-1/2}(y)W_\mu(y)\ll_\mu y_1^{A},$$ where $A=A(\Re(\mu))$ is positive and possibly depends on $\Re(\mu)$. When $n=2$ and $y_1\ge y_2$ this sort of bound is obvious, for e.g. $$W_\mu(y)=(y_1y_2)^{\frac{\mu_1+\mu_2}{2}}\delta^{1/2}(y)K_{\frac{\mu_1-\mu_2}{2}}(2\pi y_1/y_2)\ll_\mu y_1^{\Re(\frac{\mu_1+\mu_2}{2})}\delta^{1/2}(y).$$

Thanks in advance! [1]: https://projecteuclid.org/download/pdf_1/euclid.dmj/1077297295

I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated.

Let us denote $$y:=\begin{pmatrix} y_1&&\\&\ddots&\\&&y_n\end{pmatrix},\quad y_i>0,$$ and $$\mu:=\{\mu_1,\dots,\mu_n\}\in\mathbb{C}^n,\quad \Re(\mu_i)\ge 0.$$ Let $W_\mu(y)$ be the spherical Whittaker function on $\mathrm{GL}_n(\mathbb{R})$ with spectral parameter $\mu$, with [usual (Stade's) normalization][1]. Let $y$ be not in the positive Weyl chamber, i.e. there exists at least one $1\le i\le n-1$ such that $y_i\ge y_{i+1}$. Let $\delta$ be the modular character. For above kind of $y$ I am looking for an upper bound of $\delta^{-1/2}(y)W_\mu(y)$ which is strong when $y_1$ is small, preferably of the form $$\delta^{-1/2}(y)W_\mu(y)\ll_\mu y_1^{A},$$ where $A=A(\Re(\mu))$ is positive and depends on $\Re(\mu)$. When $n=2$ and $y_1\ge y_2$ this sort of bound is obvious, for e.g. $$W_\mu(y)=(y_1y_2)^{\frac{\mu_1+\mu_2}{2}}\delta^{1/2}(y)K_{\frac{\mu_1-\mu_2}{2}}(2\pi y_1/y_2)\ll_\mu y_1^{\Re(\frac{\mu_1+\mu_2}{2})}\delta^{1/2}(y).$$

Thanks in advance! [1]: https://projecteuclid.org/download/pdf_1/euclid.dmj/1077297295