Exercise 4.7.2 of Daniel Bump's Automorphic Forms and Representations (Page 520)
- (a) Let $F$ be a local field, either Archimedean or non-Archimedean. By a finite function, we mean a function $\phi$ such that the functions $x \mapsto \phi(ax)$ with $a \in F^\times$ span a finite-dimensional vector space. Show that a function is a finite linear combination of functions of the form $\chi(x) \log |x|^n$ where $\chi$ is a quasicharacter and $n \geq 0$. [Hint: Study theorem 4.5.4]
Theorem 4.5.4 Let $\chi_1$ and $\chi_2$ be quasicharacters of $F^\times$ and let $\chi$ and $\chi'$ be quasicharacters of $T(F)$ defined by $$\chi\begin{pmatrix} t_1 & \\ & t_2 \end{pmatrix} := \chi_1(t_1) \chi_2(t_2), \quad \chi'\begin{pmatrix} t_1 & \\ & t_2 \end{pmatrix} := \chi_2(t_1) \chi_1(t_2).$$ Then, the representation of $T(F)$ on the Jacquet module of $\mathcal{B}(\chi_1, \chi_2)$ is equivalent to the following two dimensional complex representation: $$t \mapsto \begin{cases}\begin{pmatrix} \delta^{1/2}\chi(t) & \\ & \delta^{1/2}\chi'(t) \end{pmatrix} &\ \text{if}\ \chi_1 \neq \chi_2;\\ \delta^{1/2} \chi(t) \begin{pmatrix} 1 & v(t_1/t_2) \\ & 1 \end{pmatrix}&\text{if}\ \chi_1 = \chi_2 \end{cases}$$ Here $v : F^\times \to \mathbb{Z} \subset \mathbb{C}$ is the valuation map.
I am not certain of how to use the result from the principal series on finite functions on the one-dimensional group $F^\times$. On the other hand, it seems that Jacquet-Langlands has an alternative method to address this problem. According to Jacquet-Langlands' Automorphic Form for GL(2) Page 139, the authors state the following lemma
Lemma 8.1(Page 139) Let $H := H_0 \times \mathbb{Z}^n \times \mathbb{R}^m$ where $H_0$ is compact. Define the projection $$\xi_i : H \to \mathbb{R}; \quad \xi_i(h_0, x_1, \dots, x_{n+m}) := x_i$$ and for any nonnegative integers $p_1, \dots, p_{n+m}$ and quasicharacter $\chi$ defined on $H$ we define the function $$\chi \prod_{i=1}^{m+n} \xi_i^{p_i}$$ on $H$. Then, these functions are finite and continuous on $H$ and they form a basis of the space of continuous finite functions on $H$.
According to the proof provided, they showed that finite continuous functions can be written on $H$ can be written as product of finite continuous functions of $H_0$, $\mathbb{Z}$ and $\mathbb{R}$. Also, they showed that characters forms a basis of the space of continuous finite functions on $H$, so naturally shouldn't the basis of the space of continuous finite functions on $H$ be product of characters of $H_0$, $\mathbb{Z}$ and $\mathbb{R}$? and not the product of quasicharacter $\chi$ of $H$ with projections $\xi_i$?
Furthermore, the authors remarked on page 142, when $F$ is a local field, and if $H = (F^\times)^n$ finite continuous functions on $H$ can be written as a linear combination of the form $$\prod_{i=1}^n \chi_i(x_i)(\log |x_i|_F)^{m_i}$$ where $m_i$ are non-negative integers.
Again, I am not certain how did the logarithm function appears in the linear combination.
Thank you for reading and helping me with this question.