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?



Are polynomial sections for unramified principal series generated by the spherical vector?
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](http://www-users.math.umn.edu/~garrett/m/v/iwahori_fixed.pdf), 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](https://web.math.pmf.unizg.hr/~gmuic/vita/analytic_continuation.pdf), 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.


