Iwahori action on the $p$-ordinary line of a principal series representation

Question

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite extension of $\mathbb{Q}_p$. Let $\GL_n(F)$ be the general linear group with diagonal torus $T_n(F)$, upper triangular Borel $B_n(F)$ and Weyl group $W(\GL_n, T_n)$.

Let $\lambda_1, \dotsc, \lambda_n: F^{\times} \rightarrow \mathbb{C}^{\times}$ be $n$ quasicharacters of $F$. We introduce a modified character $$ \lambda^{\natural}: T_n(F) \rightarrow \mathbb{C}^{\times}, \quad \diag[t_1, \dotsc, t_n] \mapsto \prod_{i=1}^{n} \lvert t_i\rvert^{n-i} \lambda_i(t_i) $$ and define an algebraically induced principal series representation $I_{B_n}^{\GL_n}(\lambda) := \Ind_{B_n(F)}^{\GL_n(F)}(\lambda^{\natural})$.

Setup: Let $\pi$ be a generic irreducible admissible representation of $\GL_n(F)$ with Whittaker model $\mathcal{W}(\pi,\psi)$ with respect to a generic character $\psi$ of $U_n(F)$ trivial on $U_n(\mathcal{O}_F)$. We assume that $\pi$ occurs as a subquotient of $I_{B_n}^{\GL_n}(\lambda)$ such that $\lambda^{w}$ for $w \in W(\GL_n, T_n)$ are pairwise distinct.

Hecke story: Consider

• Iwahori subgroups: For any $\alpha \geq \beta \geq 0$, we write $\Iw_{\beta, \alpha}$ for the subgroup of matrices in $\GL_n(\mathcal{O}_F)$ that become upper triangular modulo $\varpi^{\alpha}$, and which lies in $U_n(\mathcal{O}_F/\varpi^{\beta})$ when reduced modulo $\varpi^{\beta}$. We set $\Iw_{\alpha} = \Iw_{0, \alpha}$.
• Define $$\Delta_{F} := T_n(\mathcal{O}_F^{\times}) \cdot \{\varpi^{e}:=\diag[\varpi^{e_1}, \dotsc, \varpi^{e_n}]: e_1 \geq e_2 \geq \cdots \geq e_n \geq 0 \} \subset T_n(F).$$

Then the Iwahori–Hecke algebra $\mathcal{H}(\beta,\alpha)$ is defined as the double coset algebra $\mathcal{H}(\Iw_{\beta, \alpha}, \Iw_{\beta, \alpha} \Delta_{F}\Iw_{\beta, \alpha})$. For $1 \leq \nu \leq n$, we write $\omega_\nu$ for the $\nu$-th fundamental weight with $\nu$ leading $1$'s and $n-\nu$ tailing $0$'s, i.e. $\omega_\nu = (\underbrace{1, \dotsc, 1}_\nu, \underbrace{0, \dotsc, 0}_{n - \nu})$, we define the Hecke operator $V_{\nu} := \Iw_{\beta, \alpha} \varpi^{\omega_{\nu}} \Iw_{\beta, \alpha}$.

Ordinary line: Let $\mathcal{W}(\pi,\psi)_{\ord}$ be the subspace of $\mathcal{W}(\pi,\psi)^{U_n(\mathcal{O}_F)}$ consisting of simultaneous $V_{\nu}$-eigenvector $W$ for $1 \leq \nu < n$ with nonzero eigenvalues (union with $0$). Then Proposition 1.3 of arXiv 1708.02616 says that $\mathcal{W}(\pi,\psi)_{\ord}$ is one dimensional. We call it the ordinary line.

My question is: how does the Iwahori subgroup $\Iw_{\alpha}$ act on this ordinary line?

To be more precise, one checks that $\Iw_{\alpha}$ acts on $\mathcal{W}(\pi,\psi)^{U_n(\mathcal{O}_F)}$ naturally. Since $\mathcal{W}(\pi,\psi)_{\ord}$ is one dimensional, there exists a character $\vartheta: \mathrm{Iw}_{\alpha} \rightarrow \mathbb{C}^{\times}$ such that $$ W(gr) = \vartheta(r) W(g), \quad \forall g \in \GL_n(F), r \in \Iw_{\alpha}. $$ Then how to describe this character $\vartheta$?. More concretely, I wonder:

1. Does $\vartheta$ arise from the subtorus of $\Iw_{\alpha}$? i.e. do there exist quasicharacters $\theta_1, \dotsc, \theta_n: F^{\times} \rightarrow \mathbb{C}^{\times}$ of conductors dividing $\varpi^{\alpha}$ such that $$ \vartheta((r_{ij})_{i,j}) = \prod_{i=1}^{n} \theta_i(r_{ii})? $$ If so, where can I find a reference or is it easy to prove? Or at least for sufficiently large $\alpha$?
2. What is the relation between the quasicharacters $\lambda_i$ and the "Iwahori type" character $\vartheta$ (or $\theta_1, \dotsc, \theta_n$)?

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My attempts to understand these: I am trying to see that for some $\alpha \geq \beta \geq 0$, there is a filtration $$ \mathcal{W}(\pi,\psi)_{\ord} \subseteq \mathcal{W}(\pi,\psi)^{\Iw_{\beta, \alpha}} \subseteq \mathcal{W}(\pi,\psi)^{U_n(\mathcal{O}_F)}.\tag{$\star$}\label{464205_star} $$ Then indeed $\Iw_{\alpha}$ will act via the subtorus, since $\Iw_{\alpha}/\Iw_{\beta, \alpha} \cong T_n(\mathcal{O}_F/\varpi^{\beta})$. But do such $\alpha$ and $\beta$ really exist?

Why I guess like this? I get the clue from Lemma 4.8 of arXiv 2306.07039, where the author proved that when $\lambda_1, \dotsc, \lambda_n$ are all unramified characters, then we can take $\beta = 0$ and $\alpha$ be any positive integer to make the inclusion \eqref{464205_star} come true. But I cannot see how to deal with the general $\lambda^{\natural}$ without the unramified condition.

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Sorry for such a long and naïve post and thank you all for paying attention. :)

Answer 1

As discussed in the comments, the answer to (1) is "yes", at least if the residue field of $F$ is not $\mathbb F_2$, because $\mathrm{Iw}_\alpha$ is the product of its derived subgroup, on which $\vartheta$ is trivial, and $\mathrm{Iw}_\alpha \cap T_n(F)$.

By, and with the notation of, the proof of Proposition 1.3 of Januszewski - Non-abelian $p$-adic Rankin-Selberg $L$-functions and non-vanishing of central $L$-values, since $W_{f_0}(1)$ equals $\operatorname{vol}(U_n(\mathcal O_F))$, we have that $W_{f_0}(r)$ equals $$ \int_{U_n(F)} f_0(w_n u r)\overline\psi(u)\mathrm du = \int_{U_n(F)} f_0((w_n r w_n^{-1})w_n(r^{-1}u r))\overline\psi(u)\mathrm du = (\lambda^{\omega w_n})^\natural(w_n r w_n^{-1})\operatorname{vol}(r U(\mathcal O_F)r^{-1}) = (\lambda^{\omega w_n})^\natural(w_n r w_n^{-1})\operatorname{vol}(U(\mathcal O_F)) = (\lambda^{\omega w_n})^\natural(w_n r w_n^{-1})W_{f_0}(1) $$ (since $r$ normalises $U(\mathcal O_F)$), hence that $\vartheta(r)$ equals $(\lambda^{\omega w_n})^\natural(w_n r w_n^{-1})$, for all $r \in \mathrm{Iw}_\alpha \cap T_n(F)$. This is easy enough to translate into your notation, except that it's not quite an answer to (2), because I don't know what $\omega$ is. (That is to say, I know how it's defined, in Proposition 1.3(i), but I don't know the actual permutation.)