Are polynomial sections for unramified principal series generated by the spherical vector? Let $F$ be a non-archimedean local field, $G = \operatorname{GL}_n\left(F\right)$, $B$ the standard Borel subgroup, $K = \operatorname{GL}_n\left( \mathcal{O}_F \right)$. We have the Iwasawa decomposition $G = BK$. Let $\pi$ be an unramified principal series $\left( \pi, I\left({\underline{s}}\right) \right) = \operatorname{Ind}_{B}^G\left({\left| \cdot \right|^{s_1}} \boxtimes \dots \boxtimes \left|\cdot\right|^{s_n} \right)$, where $s_1, \dots, s_n \in \mathbb{C}$ and $\underline{s} = \left( s_1, \dots, s_n \right)$. Every element $f \in I\left({\underline{s}}\right)$ is a (right smooth) function $f : G \rightarrow \mathbb{C}$ satisfying $f\left(b g\right) = \delta_B\left(b\right)^{\frac{1}{2}} \chi_{\underline{s}}\left(b\right) f\left( g \right)$, for every $g \in G$ and $b \in B$, where $\chi_{\underline{s}} : B \rightarrow \mathbb{C}^{\times}$ is the character we used for induction.
Thinking of $\underline{s}$ as variables, we have the notion of standard sections (or flat sections): these are functions $f_{\underline{s}} : G \rightarrow \mathbb{C}$ such that  $f_{\underline{s}}\left( bk \right) = \delta_B\left(b\right)^{\frac{1}{2}} \chi_{\underline{s}}\left(b\right) h \left( k \right)$, for every $b \in B$, $k \in K$, where $h : K \rightarrow \mathbb{C}$ is a smooth function that does not depend on $\underline{s}$. In particular, we have the spherical standard section $f_{\underline{s}}^{\circ}\left( bk \right) = \delta_B\left(b\right)^{\frac{1}{2}} \chi_{\underline{s}}\left(b\right)$.
We know that if $q^{s_i} \ne q^{s_j + 1}$ for any $i, j$, then $\pi$ is irreducible. Therefore if we fix such $\underline{s}$, we can write every element of $I \left( \underline{s} \right)$ as a $\mathbb{C}$-linear combination of right translations of $f_{\underline{s}}^{\circ}$.
Question: is it possible to write every standard section $f_{\underline{s}}$ as a $\mathbb{C}\left[ q^{\pm s_1}, \dots, q^{\pm s_n} \right]$-linear combination of right translations of the standard spherical section $f_{\underline{s}}^{\circ}$? If so, is this result also true for other classical $p$-adic groups?
 A: This can't be true, because for some choices of $\underline{s}$, the spherical vector does not generate $I\left( \underline{s} \right)$ (see Paul Garrett - Representations with Iwahori-fixed vectors, page 10). However, we can write $f^{\circ}_{\underline{s}}$ as a $\mathbb{C}\left( q^{-s_1}, \dots, q^{-s_n} \right)$ linear combination of right translations of the normalized spherical standard section. The proof I give is inspired by G. Mui´c, A geometric construction of intertwining operators for reductive p–adic groups, Lemma 5-12.
To see this, let $G = \operatorname{GL}_n\left( F \right)$. Let $U \le K$ be an open compact subgroup. Then $I \left( \underline{s} \right)^U$, the subspace of $U$-fixed vectors, is finite dimensional, for every choice of $\underline{s}$. Let $s_0$ be such that $I \left( \underline{s_0} \right)$ is irreducible. Let $h^1, \dots, h^r$ be a basis for $I \left( \underline{s_0} \right)^U$. This is equivalent to saying that the restrictions of $h^1, \dots, h^r$ to $K$ form a basis for ${C}^{\infty}\left( K / U \right)$ (the space of smooth functions on $K$ which are invariant to $U$ right translations).
We have the standard sections ${h'}_\underline{s}^1, \dots, {h'}_\underline{s}^r \in I\left( \underline{s} \right)^U$, defined by $${h'}_\underline{s}^j \left( b k \right) = \chi_{\underline{s}} \left( b \right) h^j \left( k \right), $$ for $b \in B$ and $k \in K$. These form a basis for $I \left( \underline{s} \right)^U$. Moreover, any standard section which is $U$ invariant, is a $\mathbb{C}$ linear combination of ${h'}_\underline{s}^1, \dots, {h'}_\underline{s}^r$ (as its value is determined by its restriction to $K$, and the restriction to $K$ of these functions is a basis independent of $\underline{s}$).
Since $I \left( \underline{s_0} \right)$ is irreducible, there exist $\left( \varphi_j \right)_{j=1}^r \subseteq C^{\infty}_c\left( G \right)$ (i.e. compactly supported smooth functions on $G$), such that
$$ h^j \left( g \right) = \left(\pi_{\underline{s_0}}\left( \varphi_j \right) f^{\circ}_{\underline{s_0}}\right)\left(g\right) = \int_G \varphi_j\left(g_0 \right) f^{\circ}_{\underline{s_0}} \left( g g_0 \right) dg_0.$$
Since $h^j$ is right $U$ invariant, we may assume that $\varphi_j \in C^{\infty}_c\left( 
U \backslash G \right)$, i.e., that $\varphi_j$ is left $U$ invariant for every $j$.
Define $h^j_{\underline{s}} \in I\left( \underline{s} \right)$ by
$$ h^j_{\underline{s}} \left( g \right) = \left(\pi_{\underline{s}}\left( \varphi_j \right) f^{\circ}_{\underline{s}}\right)\left(g\right) = \int_G \varphi_j\left(g_0 \right) f^{\circ}_{\underline{s}} \left( g g_0 \right) dg_0.$$
$h_{\underline{s}}^j$ lies in the $\mathbb{C}$ span of right translations of $f^{\circ}_{\underline{s}}$: cover the support of $\varphi_j$ with the open covering $\bigcup_{g_0 \in G} {g_0 K}$. Then, since the support of $\varphi_j$ is compact, it admits a finite subcover $\operatorname{supp} \varphi_j \subseteq \bigcup_{i=1}^N g_i K$, and we can write $$h_{\underline{s}}^j \left( g \right) = \sum_{i=1}^N \varphi_j \left( g_i \right) f_{\underline{s}}^{\circ}\left( g g_i \right).$$ Since $\varphi_j$ is $U$ left invariant, this implies that $h_{\underline{s}}^j$ is $U$ right invariant. Therefore the restriction of $h_{\underline{s}}^j$ to $K$ lies in $C^{\infty}\left( K / U \right)$. Since the restrictions $h^1 \restriction_K, \dots, h^r \restriction_K $ form a basis for $C^{\infty}\left( K / U \right)$, there exist $\left( \kappa_{ij}\left( \underline{s} \right) \right) \subseteq \mathbb{C}$, such that
$$ h^j_{\underline{s}} \restriction_K = \sum_{i=1}^r \kappa_{ij}\left( \underline{s} \right)h^i \restriction_K.$$
By the Iwasawa decomposition, we have that for every fixed $g \in G$, the value $f_{\underline{s}}^{\circ}\left( g \right)$ lies in $\mathbb{C} \left[ q^{\pm s_1}, \dots, q^{\pm s_r} \right]$. Since $h^j_{\underline{s}}$ is a $\mathbb{C}$-linear combination of right translations of $f_{\underline{s}}^{\circ}$, we have that for every $k \in K$, the value $h^j_{\underline{s}} \left( k \right)$ lies in $\mathbb{C} \left[ q^{\pm s_1}, \dots, q^{\pm s_r} \right]$. It now follows by inverting a matrix that $\kappa_{ij}\left( \underline{s} \right) \in \mathbb{C} \left[ q^{\pm s_1}, \dots, q^{\pm s_r} \right]$. We get by inverting the matrix of $\left( \kappa_{ij} \left( \underline{s} \right) \right) $ that the restrictions $h^j \restriction_K$ are in the $\mathbb{C}\left( q^{-s_1}, \dots, q^{-s_n} \right)$ span of $h^1_{\underline{s}} \restriction_K,\dots,h^r_{\underline{s}} \restriction_K$. Note that this matrix is invertible for $\underline{s}$ such that the right translations of $f_{\underline{s}}^{\circ}$ span $I \left( \underline{s} \right)$.
This implies that ${h'}_\underline{s}^1, \dots, {h'}_\underline{s}^r$ are in the $\mathbb{C}\left( q^{-s_1}, \dots, q^{-s_n} \right)$ span of $h_{\underline{s}}^1, \dots, h_{\underline{s}}^r$, which is contained in the $\mathbb{C}\left( q^{-s_1}, \dots, q^{-s_n} \right)$ span of $h_{\underline{s}}^1, \dots, h_{\underline{s}}^r$, which is contained in the $\mathbb{C}\left( q^{-s_1}, \dots, q^{-s_n} \right)$ span of right translations of $f_{\underline{s}}^{\circ}$. Therefore we have that all standard sections that are $U$ invariant lie in the space of right translations of $f_{\underline{s}}^{\circ}$. Since this is true for every compact open $U \le K$, we are done.
