On certain Iwahori--Hecke modules of $GL_3$ in characteristic $p$

TL;DR

This paper demonstrates that a finiteness property of Iwahori--Hecke modules observed in $GL_2$ does not extend to $GL_3$, revealing the existence of infinite codimension submodules in certain invariants.

Contribution

It shows that the finiteness results for Iwahori--Hecke modules in $GL_2$ do not hold for $GL_3$, highlighting a fundamental difference in module structure.

Findings

Existence of non-zero Iwahori--Hecke submodules of infinite codimension in $GL_3$

Failure of finiteness results analogous to $GL_2$ case

Contrasts with known properties of $GL_2$ modules

Abstract

In this note, we show that the natural analogue of certain finiteness result of Barthel--Livn$\acute{\text{e}}$ on $GL_2$ fails for $GL_3$. More precisely, within the pro-$p$-Iwahori invariants of a maximal compact induction of $GL_3$, we show there exist non-zero Iwahori--Hecke submodules of \emph{infinite} codimension.

Equations

$$G=\bigcup _{\omega \in W_0,\vec{a}\in {\mathbb{Z}}^{N-1}}IZ\omega t_{\vec{a}}I$$

$$G=\bigcup _{\vec{a}\in {\mathbb{Z}}^{N-1}}KZ\cdot t_{\vec{a}}\cdot I_1,$$

$$t_{\vec{a}}I{t}_{\vec{a}}^{-1}\cap K\subset {\omega }^{-1}I\omega$$

$$\omega t_{\vec{a}}{\omega }^{-1}I\omega t_{\vec{a}}^{-1}{\omega }^{-1}\cap K\subset I.$$

$$t_{\vec{b}}\cdot \gamma$$

$$={\omega }_2\text{diag}(1,{\varpi }^{b_1},\cdots ,{\varpi }^{b_{N-1}})\text{diag}(\varpi ,I_{N-1})$$

$$={\omega }_2{\varpi }^{b_{N-1}}\text{diag}({\varpi }^{-b_{N-1}},{\varpi }^{b_1-b_{N-1}},\cdots ,{\varpi }^{b_{N-2}-b_{N-1}},1)\text{diag}(\varpi ,I_{N-1})$$

$$={\omega }_2{\varpi }^{b_{N-1}}\text{diag}({\varpi }^{1-b_{N-1}},{\varpi }^{b_1-b_{N-1}},\cdots ,{\varpi }^{b_{N-2}-b_{N-1}},1).$$

$$f\mid T_g=\sum _{i\in I_1/(I_1\cap g^{-1}{I}_{1}g)}i{g}^{-1}\cdot f,$$

$$I_1/I_1\cap {\omega }_1{I}_1{\omega }_1\cong \left(\begin{array}{ccc}1& k_F& 0\\ 0& 1& 0\\ 0& 0& 1_{N-2}\end{array}\right).$$

Full text

1. 1 Introduction

2. 2 Notations and preliminaries

3. 2.1 General notations

4. 2.2 Weights

5. 3 The Iwahori–Hecke algebra $\mathcal{H}(IZ,\chi)$

6. 3.1 The Iwahori–Hecke case

7. 3.2 The semi-regular case for $N=3$

8. 3.3 The regular case for $N=3$

9. 4 The pro-$p$-Iwahori invariants of $\text{ind}^{G}_{KZ}\sigma$

10. 4.1 The Iwasawa–Iwahori decomposition of $G$

11. 4.2 A decomposition of $\mathbb{Z}^{N-1}$

12. 4.3 A basis of $(\textnormal{ind}^{G}_{KZ}\sigma)^{I_{1}}$

13. 5 The $\mathcal{H}(IZ,\chi_{\sigma})$-module $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$

14. 5.1 The Iwahori–Hecke case

15. 5.2 The semi-regular case for $N=3$

16. 5.3 The regular case for $N=3$

17. 6 Proof of Theorem 1.1

1 Introduction

In their pioneering work ([BL94], [BL95]), Barthel–Livn$\acute{\text{e}}$ gave a classification of irreducible smooth $\overline{\mathbf{F}}_{p}$-representations (with central characters) of $GL_{2}$ over a non-archimedean local field $F$ of residue characteristic $p$. One feature of their work is they proved the existence of Hecke eigenvalues without the assumption of admissibility, where a key ingredient in their argument is to show any non-zero Iwahori–Hecke submodule of the pro-$p$-Iwahori invariants of a maximal compact induction is finite codimensional. The goal of this note is to demonstrate the analogue of such an ingredient fails for $G=GL_{N}(F)(N\geq 3)$.

Let $\mathfrak{o}_{F}$ be the ring of integers of $F$. We fix a uniformizer $\varpi$ of $F$. Let $K$ be the maximal compact open subgroup $GL_{N}(\mathfrak{o}_{F})$, and $I$ (resp, $I_{1}$) be the standard (resp, pro-$p$-) Iwahori subgroup of $G$. Let $Z$ be the center of $G$.

Let $\sigma$ be an irreducible smooth $\overline{\mathbf{F}}_{p}$-representation of $K$. Denote also by $\sigma$ its extension to $KZ$ on which $\varpi$ acts trivially. Let $\chi_{\sigma}$ be the character of $I$ on the line $\sigma^{I_{1}}$, extended to $IZ$ by requiring $\chi(\varpi)=1$. Our main result is as follows:

Theorem 1.1**.**

(Corollary 6.3) Assume $N=3$. Then there are non-zero $\mathcal{H}(IZ,\chi_{\sigma})$-submodules of $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$ of infinite codimension.

Remark 1.2**.**

This note grows out of an unsuccessful attempt to show the existence of Hecke eigenvalues for $p$-modular representations of $G$. Such a problem is listed as [AHHV17, Question 8], and to our knowledge it is only answered positively for $GL_{2}$ ([BL94]) and $U(2,1)$ ([Xu18]).

2 Notations and preliminaries

2.1 General notations

Let $F$ be a non-archimedean local field, with ring of integers $\mathfrak{o}_{F}$ and maximal ideal $\mathfrak{p}_{F}$, and let $k_{F}$ be its residue field of characteristic $p$ and order $q$. Fix a uniformizer $\varpi$ in $F$. Let $G=GL_{N}(F)$ ($N\geq 3$), $K=GL_{N}(\mathfrak{o}_{F})$, $Z\cong F^{\times}$ the center of $G$. Let $K_{1}$ be the kernel of the reduction map $\emph{red}:K\rightarrow GL_{N}(k_{F})$. Denote the latter group by $\mathbb{G}$. Let $\mathbb{B}=\mathbb{T}\ltimes\mathbb{U}$ be the standard Borel subgroup of $\mathbb{G}$, with the subgroup $\mathbb{T}$ of diagonal matrices and the upper unipotent radical $\mathbb{U}$. Let $I$ (resp, $I_{1}$) be the (resp, pro-$p$-) Iwahori subgroup of $G$, defined as the inverse image of $\mathbb{B}$ (resp, $\mathbb{T}$) in $K$ via red.

Denote by $\omega_{1}$ and $\omega_{2}$ the following elements in $G$:

$\omega_{1}=\begin{pmatrix}\beta&0\\ 0&I_{N-2}\end{pmatrix}$, $\omega_{2}=\begin{pmatrix}0&I_{N-1}\\ 1&0\end{pmatrix}$,

where $\beta$ is the $2\times 2$ matrix $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$.

We identify the finite Weyl group $W_{0}$ of $G$ with the group of permutation matrices. Note that $W_{0}$ is generated by $\omega_{1}$ and $\omega_{2}$.

Put $\gamma=\omega_{2}\cdot\text{diag}(\varpi,I_{N-1})$. Recall that the normalizer of $I_{1}$ in $G$ is generated by $I$ and $\gamma$.

In this note, all representations are smooth over $\overline{\mathbf{F}}_{p}$. As $I_{1}$ is a pro-$p$-sylow subgroup of $I$, any character $\chi$ of $I$ taking values in $\overline{\mathbf{F}}^{\times}_{p}$ factors through the finite abelian group $I/I_{1}\cong\mathbb{T}$. Hence the group $W_{0}$ acts on the set of characters of $I$, and the conjugate of $\chi$ by an $\omega\in W_{0}$ is denoted as $\chi^{\omega}$.

2.2 Weights

Let $\sigma$ be an irreducible smooth representation of $K$. As $K_{1}$ is pro-$p$ and normal in $K$, $\sigma$ factors through the finite group $\mathbb{G}$, i.e., $\sigma$ is the inflation of an irreducible representation of $\mathbb{G}$. Conversely, any irreducible representation of $\mathbb{G}$ inflates to an irreducible smooth representation of $K$. We may therefore identify irreducible smooth representations of $K$ with irreducible representations of $\mathbb{G}$, and we shall call them weights of $K$ or $\mathbb{G}$ from now on.

For a weight $\sigma$ of $K$, it is well-known that $\sigma^{I_{1,K}}$ is one-dimensional ([CE04, Theorem 6.12]).

3 The Iwahori–Hecke algebra $\mathcal{H}(IZ,\chi)$

Let $\chi$ be a character of $I$, extended to $IZ$ by requiring $\chi(\varpi)=1$. By [BL94, Proposition 5], the algebra $\mathcal{H}(IZ,\chi):=\text{End}_{G}(\text{ind}^{G}_{IZ}\chi)$ is isomorphic to the convolution algebra $\mathcal{H}_{IZ}(\chi)$ given by:

$\mathcal{H}_{IZ}(\chi)=\{\varphi:G\rightarrow\overline{\mathbf{F}}_{p}\mid\varphi(i_{1}gi_{2})=\chi(i_{1}i_{2})\varphi(g),\forall i_{1},i_{2}\in IZ,g\in G,\text{smooth~{}with~{}compact~{}support~{}modulo}~{}KZ\}$.

Denote by $T_{\varphi}$ the Hecke operator in $\mathcal{H}(IZ,\chi)$ which corresponds to a function $\varphi\in\mathcal{H}_{IZ}(\chi)$, via the aforementioned isomorphism.

For an element $g\in G$, denote by $\varphi_{g}$ the function in $\mathcal{H}_{IZ}(\chi)$ supported on $IZgI$ and satisfying $\varphi(g)=1$. Such condition uniquely determines $\varphi_{g}$ if it exists. We will write $T_{\varphi_{g}}$ as $T_{g}$ for short.

Recall the Iwahori decomposition of $G$:

[TABLE]

Lemma 3.1**.**

There is a non-zero function $\varphi\in\mathcal{H}_{IZ}(\chi)$ supported on $IZ\omega t_{\vec{a}}I$, for some $\omega\in W_{0}$ and $\vec{a}\in\mathbb{Z}^{N-1}$, if and only if:

$\chi=\chi^{\omega}$.

Proof.

The only condition on the value of $\varphi$ at $\omega t_{\vec{a}}$ is: for any $i_{1},i_{2}\in I$ satisfying $i_{1}\omega t_{\vec{a}}=\omega t_{\vec{a}}i_{2}$, the identity $\chi(i_{1})\varphi(\omega t_{\vec{a}})=\varphi(\omega t_{\vec{a}})\chi(i_{2})$ holds. The lemma follows from some simple computation. ∎

For $\omega\in W_{0},\vec{a}\in\mathbb{Z}^{N-1}$, we consider the function $\varphi_{\omega\cdot t_{\vec{a}}}$. Note that $\varphi_{\omega\cdot t_{\vec{a}}}$ is only well-defined under the condition $\chi=\chi^{\omega}$ (Lemma 3.1). By Lemma 3.1 and the Iwahori decomposition (1), the algebra $\mathcal{H}_{IZ}(\chi)$ has a basis $\{\varphi_{\omega\cdot t_{\vec{a}}}\mid\omega\in W(\chi),\vec{a}\in\mathbb{Z}^{N-1}\}$, where $W(\chi):=\{\omega\in W_{0}\mid\chi=\chi^{\omega}\}$. As mentioned above, we will write $T_{\varphi_{\omega\cdot t_{\vec{a}}}}$ as $T_{\omega\cdot t_{\vec{a}}}$ for short.

Corollary 3.2**.**

The set $\{T_{\omega\cdot t_{\vec{a}}}\mid\omega\in W(\chi),\vec{a}\in\mathbb{Z}^{N-1}\}$ consists of a basis of the algebra $\mathcal{H}(IZ,\chi)$.

3.1 The Iwahori–Hecke case

We consider the Iwahori–Hecke case first ([Vig05], [Oll06]).

Proposition 3.3**.**

When $W(\chi)=W_{0}$, the algebra $\mathcal{H}(IZ,\chi)$ is non-commutative and generated by the operators $T_{\gamma}$ and $T_{\omega_{1}}$. More precisely, there is an isomorphism of algebras:

$\mathcal{H}(IZ,\chi)\cong\overline{\mathbf{F}}_{p}[T_{\gamma},T_{\omega_{1}}]/(T^{N}_{\gamma}-1,T^{2}_{\omega_{1}}+T_{\omega_{1}})$.

Remark 3.4**.**

Proposition 3.3 generalizes [BL94, Proposition 11]. Note that the presentation here is different from that in loc.cit, as a different Hecke operator is chosen.

3.2 The semi-regular case for $N=3$

We assume $N=3$.

In this case, we assume $\chi$ is of the form $1\otimes 1\otimes\eta$ for some non-trivial character $\eta$ of $F^{\times}$, so the group $W(\chi)$ is just $\{Id,\omega_{1}\}$. By [Oll06, Proposition 1], that is essentially the only semi-regular case.

Proposition 3.5**.**

([Oll06, Theorem 25]) The Hecke algebra $\mathcal{H}(IZ,\chi)$ is generated by $T_{\omega_{1}},T_{\omega_{1}\cdot t_{(1,0)}},T_{\omega_{1}\cdot t_{(0,-1)}}$. More precisely, the algebra $\mathcal{H}(IZ,\chi)$ is isomorphic to:

$\overline{\mathbf{F}}_{p}[T_{\omega_{1}},T_{\omega_{1}\cdot t_{(1,0)}},T_{\omega_{1}\cdot t_{(0,-1)}}]/(T^{2}_{\omega_{1}}+T_{\omega_{1}},T_{\omega_{1}\cdot t_{(1,0)}}\cdot T_{\omega_{1}\cdot t_{(0,-1)}},T_{\omega_{1}\cdot t_{(0,-1)}}\cdot T_{\omega_{1}\cdot t_{(1,0)}})$.

3.3 The regular case for $N=3$

In this case, the group $W(\chi)$ is trivial. The algebra $\mathcal{H}(IZ,\chi)$ has a linear basis $\{T_{t_{\vec{a}}}\mid\vec{a}\in\mathbb{Z}^{N-1}\}$ (Corollary 3.2).

Proposition 3.6**.**

([Oll06, Proposition 18]) The Hecke algebra $\mathcal{H}(IZ,\chi)$ is commutative and generated by the following six Hecke operators

$T_{t_{(1,0)}},T_{t_{(-1,0)}},T_{t_{(0,1)}},T_{t_{(0,-1)}},T_{t_{(1,1)}},T_{t_{(-1,-1)}}$**

with nine relations:

$T_{t_{(1,0)}}\cdot T_{t_{(-1,0)}}=0,T_{t_{(1,0)}}\cdot T_{t_{(0,1)}}=0,T_{t_{(1,0)}}\cdot T_{t_{(-1,-1)}}=0,$**

$T_{t_{(0,1)}}\cdot T_{t_{(0,-1)}}=0,T_{t_{(0,1)}}\cdot T_{t_{(-1,-1)}}=0,T_{t_{(1,1)}}\cdot T_{t_{(-1,-1)}}=0,$**

$T_{t_{(1,1)}}\cdot T_{t_{(0,-1)}}=0,T_{t_{(1,1)}}\cdot T_{t_{(-1,0)}}=0,T_{t_{(-1,0)}}\cdot T_{t_{(0,-1)}}=0$.

4 The pro-$p$-Iwahori invariants of $\text{ind}^{G}_{KZ}\sigma$

Let $\sigma$ be a weight of $K$. We extend $\sigma$ to a representation of $KZ$ by requiring $\varpi$ to act trivially. Let $\text{ind}_{KZ}^{G}\sigma$ be the smooth representation compactly induced from $\sigma$, i.e., the representation of $G$ with underlying space $S(G,\sigma)$

$S(G,\sigma)=\{f:G\rightarrow\sigma\mid f(kg)=\sigma(k)\cdot f(g),\forall~{}k\in KZ,g\in G,~{}\text{smooth~{}with~{}supp~{}compact~{}modulo}~{}KZ\}$

and $G$ acting by right translation.

In this section, we determine a basis of the $I_{1}$-invariants of a maximal compact induction $\text{ind}^{G}_{KZ}\sigma$. Compare with [Oll15].

4.1 The Iwasawa–Iwahori decomposition of $G$

Recall the following Iwasawa–Iwahori decomposition of $G$:

[TABLE]

where $t_{\vec{a}}$ denotes the diagonal matrix:

$t_{\vec{a}}=\text{diag}(\varpi^{a_{1}},\ldots,\varpi^{a_{n-1}},1)$.

By (2), a function $f$ in $\text{ind}^{G}_{KZ}\sigma$ invariant under the action of $I_{1}$ is uniquely determined by its values on all diagonal matrices $t_{\vec{a}}$ with $\vec{a}\in\mathbb{Z}^{N-1}$.

4.2 A decomposition of $\mathbb{Z}^{N-1}$

Definition 4.1**.**

For an $\omega\in W_{0}$, denote by $S_{\omega}$ the subset of $\mathbb{Z}^{N-1}$ given by:

$S_{\omega}=\{\vec{a}\in\mathbb{Z}^{N-1}\mid(\omega t_{\vec{a}}\cdot I\cdot(\omega t_{\vec{a}})^{-1})\cap K\subseteq I\}$.

Lemma 4.2**.**

We have:

$(1)$. For any $\vec{a}\in S_{\omega}$, we have

$I=K_{1}\cdot(\omega t_{\vec{a}}\cdot I\cdot(\omega t_{\vec{a}})^{-1}\cap K)$.

$(2)$. There is a disjoint decomposition:

$\mathbb{Z}^{N-1}=\bigcup_{\omega\in W_{0}}S_{\omega}$.

Proof.

Denote by $I(\omega,\vec{a})$ the group $\omega t_{\vec{a}}\cdot I\cdot(\omega t_{\vec{a}})^{-1}$. By definition, if $\vec{a}\in S_{\omega}$ then $I(\omega,\vec{a})\cap K$ is contained in $I$.

We start by some general remarks. For a pair $(i,j)$, all $(i,j)$-entries of $I(\omega,\vec{a})$ consist of $\mathfrak{p}^{l^{\omega}_{ij}}_{F}$, for some $l^{\omega}_{ij}\in\mathbb{Z}$. We view $l^{\omega}_{ij}$ as functions of $\vec{a}$. Some simple observations are given first:

a). $l^{\omega}_{ij}=1-l^{\omega}_{ji}$ (for $(i,j)$ with $i\neq j$).

b). If $(i,j)\neq(i^{\prime},j^{\prime})$, then $l^{\omega}_{ij}\neq l^{\omega}_{i^{\prime}j^{\prime}}$ (as functions of $\vec{a}$).

c).  $S_{\omega}=\{\vec{a}\in\mathbb{Z}^{n-1}\mid l^{\omega}_{ij}(\vec{a})\geq 1~{}\text{for}~{}(i,j)~{}\text{with}~{}i>j\}$.

For $(1)$, it suffices to check that $B\cap K\subseteq I(\omega,\vec{a})\cap K$ for $\vec{a}\in S_{\omega}$, as we know $I=K_{1}(B\cap K)$. By $a)$ and $c)$ above we have $l^{\omega}_{ji}=1-l^{\omega}_{ij}\leq 0$ for all $(j,i)$ with $j<i$, as required.

For $(2)$, we show first the decomposition is disjoint. Note that if $\omega\neq\omega^{\prime}$, then there exist one pair $(i,j)$ with $i<j$ and another pair $(i^{\prime},j^{\prime})$ with $i^{\prime}<j^{\prime}$, such that:

$l^{\omega}_{ij}=l^{\omega^{\prime}}_{j^{\prime}i^{\prime}}$,

from which and $a),c)$ above we see $S_{\omega}\cap S_{\omega^{\prime}}=\varnothing$.

Before moving on, we record the set $S_{\omega}$ for a few special $\omega$:

$S_{Id}=\{\vec{a}\in\mathbb{Z}^{n-1}\mid a_{1}\leq a_{2}\leq...\leq a_{n-1}\leq 0\}$,

$S_{\omega_{1}}=\{\vec{a}\in\mathbb{Z}^{n-1}\mid a_{2}+1\leq a_{1}\leq a_{3}\leq...\leq a_{n-1}\leq 0\}$,

$S_{\omega_{2}}=\{\vec{a}\in\mathbb{Z}^{n-1}\mid a_{2}\leq a_{3}\leq...\leq a_{n-1}\leq 0<a_{1}\}$.

Now we prove $S_{\omega}\neq\varnothing$ for any $w\in W_{0}$.

We claim first that $S_{\omega}\neq\varnothing$ implies $S_{\omega\omega^{\prime}}\neq\varnothing$ for any $\omega^{\prime}\in\langle\omega_{2}\rangle$. The observation here is for such an $\omega^{\prime}$ one may find a diagonal matrix $\tilde{t}$ of the form $\text{diag}(\varpi^{a_{1}},...,\varpi^{a_{n}})$ so that $\omega^{\prime}\tilde{t}$ normalizes $I$. More precisely, assume we already have:

[TABLE]

for some $t_{\vec{a}}$. Conjugating the above by $\omega^{\prime}$ we get:

$\omega^{\prime-1}t_{\vec{a}}It^{-1}_{\vec{a}}\omega^{\prime}\cap K\subset\omega^{\prime-1}\omega^{-1}I\omega\omega^{\prime}$.

Here, the left side term can be rewritten as

$\omega^{\prime-1}t_{\vec{a}}\omega^{\prime}\tilde{t}(\tilde{t}^{-1}\omega^{\prime-1}I\omega^{\prime}\tilde{t})\tilde{t}^{-1}\omega^{\prime-1}t_{\vec{a}}^{-1}\omega^{\prime}\cap K$.

Note that $\omega^{\prime-1}t_{\vec{a}}\omega^{\prime}\tilde{t}$ is also a diagonal matrix of the form (uniquely) $\varpi^{l}\cdot t_{\vec{b}}$ for some $l\in\mathbb{Z}$ and $t_{\vec{b}}$. In all we get

$t_{\vec{b}}It^{-1}_{\vec{b}}\cap K\subset\omega^{\prime-1}\omega^{-1}I\omega\omega^{\prime}$

and conclude that $\vec{b}\in S_{\omega\omega^{\prime}}$.

Secondly, a parallel claim holds for $\omega_{1}$: if $\vec{a}\in S_{\omega}$ such that $a_{1}\neq a_{2}$, then $\omega_{1}\cdot\vec{a}=(a_{2},a_{1},...,a_{N-1})\in S_{\omega\omega_{1}}$.

$1)$ $a_{1}>a_{2}$. The condition $\vec{a}\in S_{\omega}$ means that

$t_{\vec{a}}It^{-1}_{\vec{a}}\cap K\subset\omega^{-1}I\omega$.

So by conjugating we have

$\omega_{1}t_{\vec{a}}\omega_{1}(\omega_{1}I\omega_{1})\omega_{1}t^{-1}_{\vec{a}}\omega_{1}\cap K\subset(\omega_{1}\omega)^{-1}I\omega\omega_{1}$.

Note that the group $I$ and $\omega_{1}I\omega_{1}$ are only different at $(12)$ and $(21)$ entries. The assumption here implies the left side group above, i.e., $\omega_{1}I(Id,\vec{a})\omega_{1}\cap K$, strictly contains

$\omega_{1}t_{\vec{a}}\omega_{1}I\omega_{1}t^{-1}_{\vec{a}}\omega_{1}\cap K$.

Combined with the containing the claim in this case follows.

$2)$. $a_{1}<a_{2}$. By conjugating we have the same:

$\omega_{1}t_{\vec{a}}\omega_{1}(\omega_{1}I\omega_{1})\omega_{1}t^{-1}_{\vec{a}}\omega_{1}\cap K\subset(\omega_{1}\omega)^{-1}I\omega\omega_{1}$.

The assumption now implies the $(21)$ (resp., $(12)$)-entries of $\omega_{1}I(Id,\vec{a})\omega_{1}$ is $\mathfrak{o}_{F}$ (resp., $\mathfrak{p}^{a_{2}-a_{1}+1}_{F}$). By the containing above, the $(21)$-entries of $(\omega_{1}\omega)^{-1}I\omega\omega_{1}$ is also $\mathfrak{o}_{F}$, and thus its $(12)$-entries gives us $\mathfrak{p}_{F}$. However, the $(12)$ (resp., $(21)$)-entries of $I(Id,\omega_{1}t_{\vec{a}}\omega_{1})$ is $\mathfrak{p}^{a_{2}-a_{1}}_{F}$ (resp., $\mathfrak{p}_{F}$). This is enough to see (note that $a_{1}<a_{2}$)

$\omega_{1}t_{\vec{a}}\omega_{1}I\omega_{1}t^{-1}_{\vec{a}}\omega_{1}\cap K\subset(\omega_{1}\omega)^{-1}I\omega\omega_{1}$,

as required by the claim.

Recall that the group $W_{0}$ is generated by $\omega_{1}$ and $\omega_{2}$. Starting by the special case $\omega=Id$ above (which clearly contains an $\vec{a}$ such that $a_{1}\neq a_{2}$), the assertion $S_{\omega}\neq\varnothing$ for any $\omega\in W_{0}$ follows from a simple induction argument and the two claims we have just verified.

Finally, we show any $\vec{a}\in\mathbb{Z}^{n-1}$ belongs to some $S_{\omega}$. We find an $\omega$ such that

$\omega t_{\vec{a}}\omega^{-1}=\varpi^{l}t_{\vec{b}}$

for some $l\in\mathbb{Z}$ and $\vec{b}\in S_{Id}$. It is easy to see $l$ and $\vec{b}$ are unique, but in general it is not true for $\omega$. We will generate the right one below.

Put $a_{N}=0$. Recall the diagonal matrix $t_{\vec{a}}$ is

$(\varpi^{a_{1}},\varpi^{a_{2}},...,\varpi^{a_{N}}).$

We write $t_{\vec{a}}$ as $t_{(0)}=(\varpi^{a_{0,1}},\varpi^{a_{0,2}},...,\varpi^{a_{0,N}})$.

For step $1$, we conjugate $t_{(0)}$ by a unique $\omega_{(1)}$ to swap $\varpi^{a_{0,j_{0}}}$ with $\varpi^{a_{0,N}}$ for the $j_{0}$ such that $a_{0,j_{0}}=\text{max}\{a_{0,m}\mid 1\leq m\leq N\}$ and $j_{0}$ is the largest index satisfying that. We obtain a diagonal matrix $t_{(1)}$.

Suppose step $i$ is done, and we get a diagonal matrix $t_{(i)}=(\varpi^{a_{i,1}},\varpi^{a_{i,2}},...,\varpi^{a_{i,N}})$. For step $i+1$, we conjugate $t_{(i)}$ by a unique $\omega_{(i+1)}$ to swap $\varpi^{a_{i,j_{i}}}$ with $\varpi^{a_{i,N-i}}$ for the $j_{i}$ such that $a_{i,j_{i}}=\text{max}\{a_{i,m}\mid 1\leq m\leq N-i\}$ and $j_{i}$ is the largest index satisfying that. We obtain a diagonal matrix $t_{(i+1)}$. Note that if $j_{i}=N-i$, we have nothing to do and $\omega_{(i+1)}=Id$.

We get the desired $\omega$ as the product $\omega_{(N-1)}\cdot\omega_{(N-2)}...\cdot\omega_{(1)}$.

As $\vec{b}\in S_{Id}$, we get

[TABLE]

We show that $\vec{a}\in S_{\omega}$ for the element $\omega$ generated above, i.e., we have

$\omega t_{\vec{a}}It^{-1}_{\vec{a}}\omega^{-1}\cap K\subset I$.

It suffices to compare the group $I$ with $\omega^{-1}I\omega$. We consider the pairs $(j,i)$ with $j<i$ so that conjugating by $\omega^{-1}$ swaps the $(ij)$-entry of a matrix with its $(j^{\prime}i^{\prime})$-entry for some $j^{\prime}<i^{\prime}$. So $(j^{\prime},i^{\prime})$ is also a such pair. We must have $a_{i}\leq a_{j}$ for every such pair. Indeed, if $a_{i}>a_{j}$ happens, the $(ij)$-entries of the left side of (4) is $\mathfrak{o}_{F}$. This contradicts (4). If we have further $a_{i}<a_{j}$ for every such pair, we are done. This is because the $(ij)$-entries of $I(\omega,\vec{a})\cap K$ is $\mathfrak{p}^{a_{j^{\prime}}-a_{i^{\prime}}}_{F}$. However, the situation $a_{j}=a_{i}$ for some pair $(j,i)$ with $j<i$ is already excluded from the process we find $\omega$.

The argument of the Lemma is completed. ∎

For our later purpose, we display the Lemma explicitly for $N=3$.

$\omega=Id=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix},S_{\omega}=\{(a_{1},a_{2})\in\mathbb{Z}^{2}\mid a_{1}\leq a_{2}\leq 0\}$;

$\omega=\omega_{1}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix},S_{\omega}=\{(a_{1},a_{2})\in\mathbb{Z}^{2}\mid a_{2}+1\leq a_{1}\leq 0\}$;

$\omega=\omega_{1}\omega_{2}=\begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix},S_{\omega}=\{(a_{1},a_{2})\in\mathbb{Z}^{2}\mid a_{1}\geq a_{2}+1\geq 2\}$;

$\omega=\omega_{2}\omega_{1}=\begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix},S_{\omega}=\{(a_{1},a_{2})\in\mathbb{Z}^{2}\mid a_{1}\leq 0,a_{2}\geq 1\}$;

$\omega=\omega_{2}=\begin{pmatrix}0&1&0\\ 0&0&1\\ 1&0&0\end{pmatrix},S_{\omega}=\{(a_{1},a_{2})\in\mathbb{Z}^{2}\mid a_{1}\geq 1,a_{2}\leq 0\}$;

$\omega=\omega^{2}_{2}=\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix},S_{\omega}=\{(a_{1},a_{2})\in\mathbb{Z}^{2}\mid a_{2}\geq a_{1}\geq 1\}$.

4.3 A basis of $(\textnormal{ind}^{G}_{KZ}\sigma)^{I_{1}}$

We fix a non-zero vector $v_{0}$ in the line $\sigma^{I_{1}}$. The Iwahori subgroup $I$ acts as a character on $\sigma^{I_{1}}$, for which we denote it by $\chi_{\sigma}$.

By $(2)$ of Lemma 4.2, the Iwasawa–Iwahori decomposition (2) is re-written as

$G=\bigcup_{\omega\in W_{0}}\bigcup_{\vec{a}\in S_{\omega}}KZ\cdot t_{\vec{a}}I_{1}$.

For an $\omega\in W_{0}$, and an $\vec{a}\in S_{\omega}$, denote by $f_{\omega,\vec{a}}$ the $I_{1}$-invariant function in $\text{ind}^{G}_{KZ}\sigma$, supported on the double coset $KZt_{\vec{a}}I_{1}$ and having value $v_{0}$ at the element $\omega t_{\vec{a}}$. Note that $f_{\omega,\vec{a}}(t_{\vec{a}})=\omega^{-1}v_{0}$.

Proposition 4.3**.**

The set of functions $\{f_{\omega,\vec{a}}\mid\omega\in W_{0},\vec{a}\in S_{\omega}\}$ consists of a basis of the $I_{1}$-invariants of $\textnormal{ind}^{G}_{KZ}\sigma$.

Proof.

Let $f$ be an $I_{1}$-invariant function in $\textnormal{ind}^{G}_{KZ}\sigma$, supported on the coset $KZt_{\vec{a}}I_{1}$, for some $\omega\in W_{0}$ and $\vec{a}\in S_{\omega}$. For $g\in K\cap\omega t_{\vec{a}}\cdot I_{1}\cdot(\omega t_{\vec{a}})^{-1}$, we have $f(g\cdot\omega t_{\vec{a}})=\sigma(g)f(\omega t_{\vec{a}})=f(\omega t_{\vec{a}})$. As $K_{1}$ acts trivially on $\sigma$, by $(1)$ of Lemma 4.2 we see $f(\omega t_{\vec{a}})$ is indeed $I_{1}$-invariant, and thus $f$ is proportional to $f_{\omega,\vec{a}}$. By the remarks in subsection 4.1 , the proposition follows. ∎

5 The $\mathcal{H}(IZ,\chi_{\sigma})$-module $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$

In this part, we compute the right action of an Iwahori–Hecke algebra $\mathcal{H}(IZ,\chi)$ on the $(IZ,\chi)$-isotypic of a maximal compact induction.

Proposition 5.1**.**

For an $\omega\in W_{0}$ and $\vec{a}\in S_{\omega}$, we have:

$(1)$.  The group $I$ acts on the function $f_{\omega,\vec{a}}$ as the character $\chi^{\omega}_{\sigma}$.

$(2)$.   One has

$\gamma\cdot f_{\omega,\vec{a}}=f_{\omega\cdot\omega^{-1}_{2},\vec{a}^{\gamma}}$,

where

$\vec{a}^{\gamma}:=(1+a_{2}-a_{1},1+a_{3}-a_{1}\cdots,1+a_{N-1}-a_{1},1-a_{1})$.

Proof.

For $(1)$, it is a simple computation by definition.

For $(2)$, one displays as follows:

[TABLE]

As $f_{\omega,\vec{a}}$ is supported on $KZt_{\vec{a}}I_{1}$, one concludes that the function $\gamma f_{\omega,\vec{a}}$ will be supported on a single coset $KZt_{\vec{b}}I_{1}$, where $\vec{b}$ is that given in the statement. By definition, we check that $\gamma f_{\omega,\vec{a}}(\omega^{\prime}t_{\vec{b}})=v_{0}$, where $\omega^{\prime}=\omega\omega^{-1}_{2}$. Hence the claim. ∎

By $(1)$ of last Proposition, we have

Corollary 5.2**.**

For an $\omega\in W_{0}$, a basis of the $(IZ,\chi^{\omega}_{\sigma})$-isotypic of $\textnormal{ind}^{G}_{KZ}\sigma$ is given by:

$\{f_{\nu,\vec{a}}\mid\nu\in\omega\cdot W(\chi_{\sigma}),\vec{a}\in S_{\nu}\}.$**

5.1 The Iwahori–Hecke case

In this case, the group $W(\chi_{\sigma})=W_{0}$. By Corollary 3.2, we know $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$ is the whole space $(\textnormal{ind}^{G}_{KZ}\sigma)^{I_{1}}$, and the latter has a basis $\{f_{\omega,\vec{a}}\mid\omega\in W_{0},\vec{a}\in S_{\omega}\}$ (Proposition 4.3).

The following proposition generalizes [BL95, Lemma 14, 15] and [BL94, Proposition 17] to $GL_{N}(N\geq 3)$.

Proposition 5.3**.**

Assume that $W(\chi_{\sigma})=W_{0}$.

$(1)$. We have $f_{\omega,\vec{a}}\mid T_{\gamma}=f_{\omega\omega_{2},\gamma\cdot\vec{a}}$, where

$\gamma\cdot\vec{a}:=(1-a_{N-1},a_{1}-a_{N-1},\cdots,a_{N-2}-a_{N-1})$.

$(2)$*. We have $f_{\omega,\vec{a}}\mid T_{\omega_{1}}=\begin{cases}f_{\omega\omega_{1},\omega_{1}\cdot\vec{a}},~{}~{}~{}\text{if}~{}a_{1}>a_{2};\\ c_{\sigma,\omega}f_{\omega,\vec{a}},~{}~{}~{}~{}\text{if}~{}N=3,~{}a_{1}=a_{2};\\ -f_{\omega,\vec{a}},~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}N=3,~{}a_{1}<a_{2}.\\ \end{cases}$

Here, $\omega_{1}\cdot\vec{a}:=(a_{2},a_{1},...,a_{N-1})$, and $c_{\sigma,\omega}$ is a constant related to $\sigma$.*

Proof.

For a $g\in G$, the right action of $T_{g}$ on the space $(\text{ind}^{G}_{KZ}\sigma)^{I_{1}}$ is given by:

[TABLE]

for an $f\in(\text{ind}^{G}_{KZ}\sigma)^{I_{1}}$. In this argument, we will take $g$ as $\gamma$ and $\omega_{1}$.

We treat $(1)$ at first. As $\gamma$ normalizes $I_{1}$, the sum above in this case only contains a single term, and $(1)$ follows by an argument similar to that of $(2)$ of Proposition 5.1.

Now we deal with $(2)$. The above formula (5) specifies to:

$f_{\omega,\vec{a}}\mid T_{\omega_{1}}=\sum_{i\in I_{1}/I_{1}\cap\omega_{1}I_{1}\omega_{1}}i\omega_{1}\cdot f_{\omega,\vec{a}}$

Note first the identification:

[TABLE]

Case $1)$.  $a_{1}>a_{2}$. Formally, we have a coset decomposition

$KZt_{\vec{a}}I_{1}=\cup_{i\in(t^{-1}_{\vec{a}}Kt_{\vec{a}}\cap I_{1})\backslash I_{1}}KZt_{\vec{a}}i$

In the case $a_{1}>a_{2}$, a set of representatives for $(t^{-1}_{\vec{a}}Kt_{\vec{a}}\cap I_{1})\backslash I_{1}$ can be taken so that $i$ has zero $(1,2)$-entry and its $(2,1)$-entry goes through $\mathfrak{p}_{F}/\mathfrak{p}_{F}^{a_{1}-a_{2}}$. Hence for all such $i$, we have $\omega_{1}i\omega_{1}\in I_{1}$.

By the proceeding remark, we see immediately the function $f_{\omega,\vec{a}}\mid T_{\omega_{1}}$ is supported on $KZt_{\omega_{1}\cdot\vec{a}}I_{1}$. To compute

$\sum_{i\in I_{1}/I_{1}\cap\omega_{1}I_{1}\omega_{1}}f_{\omega,\vec{a}}(t_{\omega_{1}\cdot\vec{a}}\cdot i\omega_{1}),$

we separate the sum into two parts:

$\sum_{i\in(I_{1}\setminus I_{1}\cap\omega_{1}I_{1}\omega_{1})/I_{1}\cap\omega_{1}I_{1}\omega_{1}}f_{\omega,\vec{a}}(t_{\omega_{1}\cdot\vec{a}}\cdot i\omega_{1})+\omega_{1}f_{\omega,\vec{a}}(t_{\vec{a}})$

The first sum vanishes: we claim that $t_{\omega_{1}\cdot\vec{a}}\cdot i\omega_{1}\notin KZt_{\vec{a}}I_{1}$ for any $i\in I_{1}\setminus I_{1}\cap\omega_{1}I_{1}\omega_{1}$.

It remains to verify the claim. Recall the following identity:

$\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}1&t\\ 0&1\end{pmatrix}=\begin{pmatrix}1&t^{-1}\\ 0&1\end{pmatrix}\begin{pmatrix}-t^{-1}&0\\ 0&t\end{pmatrix}\begin{pmatrix}1&0\\ t^{-1}&1\end{pmatrix}$

for $t\neq 0$. Using the identification (6) and the above identity, we can indeed prove

$t_{\omega_{1}\cdot\vec{a}}\cdot i\omega_{1}\in KZt_{\omega_{1}\cdot\vec{a}}I_{1}$

for all such $i$. Note that it is this place the condition $a_{1}>a_{2}$ plays a role. As $t_{\vec{a}}\neq t_{\omega_{1}\cdot\vec{a}}$, the claim follows.

Now the last term gives us $\omega_{1}\omega^{-1}v_{0}$. As we already know $\omega\cdot\vec{a}\in S_{\omega\omega_{1}}$ (Lemma 4.2), we conclude that $f_{\omega,\vec{a}}\mid T_{\omega_{1}}=f_{\omega\omega_{1},\omega_{1}\cdot\vec{a}}$.

Case $2)$.  $a_{1}<a_{2}$. When $N=3$, the situation $a_{1}<a_{2}$ happens only for $\omega\in\{Id,\omega_{2}\omega_{1},\omega^{2}_{2}\}$. Explicitly, a set of representatives for $(t^{-1}_{\vec{a}}Kt_{\vec{a}}\cap I_{1})\backslash I_{1}$ can be taken as follows:

for $\omega=Id,a_{1}<a_{2}\leq 0$, $\begin{pmatrix}1&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{a_{2}-a_{1}}&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{-a_{1}}\\ 0&1&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{-a_{2}}\\ 0&0&1\end{pmatrix}$;

for $\omega=\omega_{2}\omega_{1},a_{1}\leq 0,a_{2}\geq 1$, $\begin{pmatrix}1&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{a_{2}-a_{1}}&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{-a_{1}}\\ 0&1&0\\ 0&\mathfrak{p}_{F}/\mathfrak{p}_{F}^{a_{2}}&1\end{pmatrix}$;

for $\omega^{2}_{2},1\leq a_{1}<a_{2}$, $\begin{pmatrix}1&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{a_{2}-a_{1}}&0\\ 0&1&0\\ \mathfrak{p}_{F}/\mathfrak{p}_{F}^{a_{1}}&\mathfrak{p}_{F}/\mathfrak{p}_{F}^{a_{2}}&1\end{pmatrix}$.

In every case, for an $i$ as above, we denote its $(12)$-entry by $t$. We can check that

$KZt_{\vec{a}}i\omega_{1}I_{1}=KZt_{\vec{a}}u(t)\omega_{1}I_{1}$,

where $u(t)$ is the matrix

$\begin{pmatrix}1&t&0\\ 0&1&0\\ 0&0&1\end{pmatrix}$.

Here, we note that $i$ can be written as $u(t)\cdot i^{\prime}$ for some $i^{\prime}\in I_{1}$ such that $\omega_{1}i^{\prime}\omega_{1}\in I_{1}$.

Now for $t\in\mathfrak{p}_{F}$, we see the double coset is $KZt_{\omega_{1}\cdot\vec{a}}I_{1}$. For $t\in\mathfrak{o}_{F}\setminus\mathfrak{p}_{F}$, to verify the claim in Case $1)$ we have indeed shown

$t_{\vec{a}}u(t)\omega_{1}\in KZt_{\vec{a}}I_{1}$.

In all, we see the function $f_{\omega,\vec{a}}\mid T_{\omega_{1}}$ is supported on $KZt_{\omega_{1}\cdot\vec{a}}I_{1}\cup KZt_{\vec{a}}I_{1}$. Thus, it is a linear combination of $f_{\omega,\vec{a}}$ and $f_{\omega\omega_{1},\omega_{1}\cdot\vec{a}}$ (Proposition 4.3). The statement in this case then follows by applying the quadratic relation $T^{2}_{\omega_{1}}=-T_{\omega_{1}}$ and case $1)$.

Case $3)$.  $N=3,a_{1}=a_{2}$. In this case, we can first check that

$KZt_{\vec{a}}I_{1}\omega_{1}I_{1}=KZt_{\vec{a}}I_{1}$,

where, a set of representatives for $(t^{-1}_{\vec{a}}Kt_{\vec{a}}\cap I_{1})\backslash I_{1}$ can be chosen such that $\omega_{1}i\omega_{1}\in I_{1}$. The function $f_{\omega,\vec{a}}\mid T_{\omega_{1}}$ is thus proportional to $f_{\omega,\vec{a}}$, say $c\cdot f_{\omega,\vec{a}}$ for some $c$. By the quadratic relation of $T_{\omega_{1}}$, $c$ can only be [math] or $-1$. However, the exact value of $c$ depends on the weight $\sigma$ and $\omega$, and we denote $c$ by $c_{\sigma,\omega}$. Note in the current case $\omega\in\{Id,\omega^{2}_{2}\}$. We record the equation which determines $c_{\sigma,\omega}$:

$\sum_{t\in k_{F}}~{}\begin{pmatrix}1&t&0\\ 0&1&0\\ 0&0&1\end{pmatrix}\omega_{1}\omega^{-1}v_{0}=c_{\sigma,\omega}\omega^{-1}v_{0}.$

The argument for the proposition is done. ∎

Remark 5.4**.**

The condition $W(\chi_{\sigma})=W_{0}$ plays only the role that the two operators in consideration are well-defined, and it is not used in the argument.

From now on, without causing confusion we will sometimes write a function of the form $c\cdot f_{\omega,\vec{a}}$ as $c\cdot\vec{a}$.

Proposition 5.5**.**

Assume $N=3$ and $W(\chi_{\sigma})=W_{0}$. We have:

$(1)$. For $(a_{1},a_{2})\in S_{Id}$ satisfying $a_{1}\leq a_{2}\leq 0$, we have

$(a_{1}-1,a_{2})=(a_{1},a_{2})\mid(T_{\gamma}T_{\omega_{1}})^{2}=(a_{1},a_{2})\mid T_{t_{(-1,0)}}$;

$(a_{1}-1,a_{2}-1)=(a_{1},a_{2})\mid(T_{\gamma}T_{\omega_{1}}T_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(-1,-1)}}$.

$(2)$. For $(a_{1},a_{2})\in S_{\omega_{1}}$ satisfying $a_{2}+1\leq a_{1}\leq 0$, we have:

$(a_{1},a_{2}-1)=(a_{1},a_{2})\mid(T_{\omega_{1}}T_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(0,-1)}}$;

$(a_{1}-1,a_{2}-1)=(a_{1},a_{2})\mid(T_{\gamma}T_{\omega_{1}}T_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(-1,-1)}}$.

$(3)$. For $(a_{1},a_{2})\in S_{\omega_{1}\omega_{2}}$ satisfying $a_{1}\geq a_{2}+1\geq 2$, we have

$(a_{1}+1,a_{2})=(a_{1},a_{2})\mid(T_{\omega_{1}}T^{2}_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(1,0)}}$;

$(a_{1}+1,a_{2}+1)=(a_{1},a_{2})\mid(T^{2}_{\gamma}T_{\omega_{1}}T^{2}_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(1,1)}}$.

$(4)$. For $(a_{1},a_{2})\in S_{\omega_{2}\omega_{1}}$ satisfying $a_{1}\leq 0,a_{2}\geq 1$, we have

$(a_{1}-1,a_{2})=(a_{1},a_{2})\mid(T_{\gamma}T_{\omega_{1}})^{2}=(a_{1},a_{2})\mid T_{t_{(-1,0)}}$;

$(a_{1},a_{2}+1)=(a_{1},a_{2})\mid(T^{2}_{\gamma}T_{\omega_{1}})^{2}=(a_{1},a_{2})\mid T_{t_{(0,1)}}$.

$(5)$. For $(a_{1},a_{2})\in S_{\omega_{2}}$ satisfying $a_{1}\geq 1,a_{2}\leq 0$, we have:

$(a_{1}+1,a_{2})=(a_{1},a_{2})\mid(T_{\omega_{1}}T^{2}_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(1,0)}}$;

$(a_{1},a_{2}-1)=(a_{1},a_{2})\mid(T_{\omega_{1}}T_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(0,-1)}}$.

$(6)$. For $(a_{1},a_{2})\in S_{\omega^{2}_{2}}$ satisfying $a_{2}\geq a_{1}\geq 1$, we have:

$(a_{1},a_{2}+1)=(a_{1},a_{2})\mid(T^{2}_{\gamma}T_{\omega_{1}})^{2}=(a_{1},a_{2})\mid T_{t_{(0,1)}}$;

$(a_{1}+1,a_{2}+1)=(a_{1},a_{2})\mid(T^{2}_{\gamma}T_{\omega_{1}}T^{2}_{\gamma})^{2}=(a_{1},a_{2})\mid T_{t_{(1,1)}}$.

Proof.

In every item listed above, the assertion for the first equality follows from $(1)$ and Case $1)$ of Proposition 5.3. In the process, we see the operator in the statement is unique (in terms of $T_{\gamma}$ and $T_{\omega_{1}}$). On the other hand, one can check that it equals a $T_{g}$ for some $g\in G$. This can be seen by considering the braid relations on the extended Weyl group. Indeed, in every case, the element $g$ is a diagonal matrix of the form $\varpi t_{\vec{a}}$ for some $\vec{a}\in\mathbb{Z}^{2}$, so by our definition the operator is nothing but $T_{t_{\vec{a}}}$. This gives the second equality. ∎

5.2 The semi-regular case for $N=3$

Suppose $\chi_{\sigma}$ is semi-regular, i.e., the group $W(\chi_{\sigma})$ is of order $2$. We assume $\chi_{\sigma}$ is of the form $1\otimes 1\otimes\eta$ for some non-trivial character $\eta$, whence $W(\chi_{\sigma})=\{Id,\omega_{1}\}$. In this case, Corollary 5.2 says that $(\text{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$ has a basis $\mathcal{V}_{Id}\cup\mathcal{V}_{\omega_{1}}$, where $\mathcal{V}_{Id}=\{f_{Id,\vec{a}}\mid\vec{a}\in S_{Id}\}$ and $\mathcal{V}_{\omega_{1}}=\{f_{\omega_{1},\vec{a}}\mid\vec{a}\in S_{\omega_{1}}\}$. We will abuse the same notation to denote the subspace spanned by them.

Proposition 5.6**.**

For $\vec{a}\in S_{Id}\cup S_{\omega_{1}}$, we have the following:

$(1)$*. $\vec{a}\mid T_{\omega_{1}}=\begin{cases}\omega_{1}\cdot\vec{a},~{}~{}~{}~{}~{}~{}\text{if}~{}a_{1}>a_{2};\\ c\cdot\vec{a},~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}a_{1}=a_{2};\\ -\vec{a},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}a_{1}<a_{2}.\\ \end{cases}$

Here, $\omega_{1}\cdot\vec{a}=(a_{2},a_{1})$, and $c$ is a constant related to $\sigma$.*

$(2)$. For $(a_{1},a_{2})\in S_{Id}\cup S_{\omega_{1}}$, $(a_{1},a_{2})\mid T_{\omega_{1}\cdot t_{(0,-1)}}=(a_{2},a_{1}-1)$.

$(3)$.  $T_{\omega_{1}t_{(1,0)}}$ vanishes on the whole space $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$.

Proof.

In the current case, the operator $T_{\omega_{1}}$ is well-defined (Corollary 3.2). The argument of $(2)$ of Proposition 5.3 works here, as that does not involve the specific information of the weight in consideration. Note that $\omega$ here lies in $\{Id,\omega_{1}\}$. So the first case only happens for $\omega=\omega_{1}$, and the other two cases happen for $\omega=Id$.

For $(2)$, we check first that the function $f_{\omega,\vec{a}}\mid T_{\omega_{1}\cdot t_{(0,-1)}}$ is supported on $KZt_{\omega_{1}\cdot\vec{a}+(0,-1)}I_{1}=KZt_{(a_{2},a_{1}-1)}I_{1}$. Write $\omega_{1}t_{0,-1}$ as $g$ for a moment. We compute that

$f_{\omega,\vec{a}}\mid T_{\omega_{1}\cdot t_{(0,-1)}}(t_{(a_{2},a_{1}-1)})=\sum_{i\in I_{1}/I_{1}\cap g^{-1}I_{1}g}f_{\omega,\vec{a}}(t_{(a_{2},a_{1}-1)}i\cdot t_{(0,1)}\omega_{1})$.

We note the identification

$I_{1}/I_{1}\cap g^{-1}I_{1}g\cong\begin{pmatrix}1&0&0\\ 0&1&k_{F}\\ 0&0&1\end{pmatrix}.$

It remains to check the sum above is equal to $\omega_{1}\omega^{-1}v_{0}$. Such a value is contributed by the term $i=Id$. For $i\neq Id$, we claim that

$t_{(a_{2},a_{1}-1)}i\cdot t_{(0,1)}\omega_{1}\notin KZt_{\vec{a}}I_{1}$.

This can be verified by combing the identification above with argument similar to that of Proposition 5.3. Indeed, we have

$t_{(a_{2},a_{1}-1)}i\cdot t_{(0,1)}\omega_{1}\in KZt_{(2-a_{1},1+a_{2}-a_{1})}I_{1}$,

where we note $a_{1}\leq 0$.

For $(3)$, by estimating its support we firstly show $\vec{a}\mid T_{\omega_{1}t_{(1,0)}}$ lies in $\mathcal{V}_{\omega_{1}}$. Then the assertion follows from $(2)$ and $T_{\omega_{1}t_{(1,0)}}\cdot T_{\omega_{1}\cdot t_{(0,-1)}}=0$ (Proposition 3.5). ∎

Proposition 5.7**.**

We have

$(1)$. For $(a_{1},a_{2})\in S_{Id}$,

$(a_{1}-1,a_{2})=(a_{1},a_{2})\mid T_{\omega_{1}\cdot t_{(0,-1)}}T_{\omega_{1}}=(a_{1},a_{2})\mid T_{t_{(-1,0)}}$,

$(a_{1}-1,a_{2}-1)=(a_{1},a_{2})\mid T^{2}_{\omega_{1}\cdot t_{(0,-1)}}=(a_{1},a_{2})\mid T_{t_{(-1,-1)}}$.

$(2)$.   For $(a_{1},a_{2})\in S_{\omega_{1}}$,

$(a_{1},a_{2}-1)=(a_{1},a_{2})\mid T_{\omega_{1}}T_{\omega_{1}\cdot t_{(0,-1)}}=(a_{1},a_{2})\mid T_{t_{(0,-1)}}$,

$(a_{1}-1,a_{2}-1)=(a_{1},a_{2})\mid T^{2}_{\omega_{1}\cdot t_{(0,-1)}}=(a_{1},a_{2})\mid T_{t_{(-1,-1)}}$.

Proof.

In every item in the statement, the assertion for the first equality simply follow from last Proposition. The assertion for the second equality holds because the two operators coincide, e.g., $T_{\omega_{1}\cdot t_{(0,-1)}}T_{\omega_{1}}=T_{t_{(-1,0)}}$, by considering the braid relation on the extended Weyl group. ∎

5.3 The regular case for $N=3$

Suppose $\chi_{\sigma}$ is regular, i.e., $W(\chi_{\sigma})=Id$. In this case, we know from Corollary 3.2 that a basis of $(\text{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$ is $\{f_{Id,\vec{a}}\mid\vec{a}\in S_{Id}\}$. We have the following:

Proposition 5.8**.**

For $(a_{1},a_{2})\in S_{Id}$, we have

$(1)$.  $(a_{1},a_{2})\mid T_{t_{(-1,0)}}=(a_{1}-1,a_{2}),~{}(a_{1},a_{2})\mid T_{t_{(-1,-1)}}=(a_{1}-1,a_{2}-1)$.

$(2)$. $(a_{1},a_{2})\mid\mathcal{T}=0$ for $\mathcal{T}\in\{T_{t_{(1,0)}},T_{t_{(0,1)}},T_{t_{(0,-1)}},T_{t_{(1,1)}}\}$.

Proof.

We only prove $(a_{1},a_{2})\mid T_{t_{(-1,0)}}=(a_{1}-1,a_{2})$ in detail. First we check that the function $f_{Id,\vec{a}}\mid T_{t_{(-1,0)}}$ is supported on $KZt_{(a_{1}-1,a_{2})}I_{1}$.

We compute

$f_{Id,\vec{a}}\mid T_{t_{(-1,0)}}(t_{(a_{1}-1,a_{2})})=\sum_{i\in I_{1}/I_{1}\cap t_{(1,0)}I_{1}t_{(-1,0)}}~{}f_{Id,\vec{a}}(t_{(a_{1}-1,a_{2})}it_{(1,0)})$.

We note the identification

$I_{1}/I_{1}\cap t_{(1,0)}I_{1}t_{(-1,0)}\cong\begin{pmatrix}1&k_{F}&k_{F}\\ 0&1&0\\ 0&0&1\end{pmatrix}.$

For $i=Id$, we get $f_{Id,\vec{a}}(t_{\vec{a}})=v_{0}$.

For $i\neq Id$, we claim that $t_{(a_{1}-1,a_{2})}it_{(1,0)}\notin KZt_{\vec{a}}I_{1}$.

By the identification above, we write $i$ as

$\begin{pmatrix}1&t_{1}&t_{2}\\ 0&1&0\\ 0&0&1\end{pmatrix}$

for $t_{1},t_{2}\in k_{F}$.

When $t_{1}\neq 0,t_{2}=0$, by a similar argument as that of Proposition 5.3, we have $t_{(a_{1}-1,a_{2})}it_{(1,0)}\in KZt_{(a_{2}+1,a_{1}-1)}I_{1}$.

When $t_{1}\cdot t_{2}\neq 0$, starting from the case above, we show first that $t_{(a_{1}-1,a_{2})}it_{(1,0)}$ shares the same double coset with

$t_{(a_{2}+1,a_{1}-1)}\begin{pmatrix}1&0&\varpi^{-1}t_{2}\\ 0&1&0\\ 0&0&1\end{pmatrix}$.

Then by an argument similar to that of semi-regular case, we can check

$t_{(a_{2}+1,a_{1}-1)}\begin{pmatrix}1&0&\varpi^{-1}t_{2}\\ 0&1&0\\ 0&0&1\end{pmatrix}\in KZt_{(1-a_{2},a_{1}-1-a_{2})}I_{1}$.

When $t_{1}=0,t_{2}\neq 0$, we have already shown in the semi-regular case that

$t_{(a_{1}-1,a_{2})}it_{(1,0)}\in KZt_{(2-a_{1},1+a_{2}-a_{1})}I_{1}$.

In all cases, the condition $a_{1}\leq a_{2}\leq 0$ is essentially used in the argument. The claim is verified. The assertion $(a_{1},a_{2})\mid T_{t_{(-1,0)}}=(a_{1}-1,a_{2})$ is proved.

Now the assertions in $(2)$ about $T_{(1,0)},T_{(1,1)},T_{(0,-1)}$ follows from what we have just proved and Proposition 3.6.

The remaining two assertions can be proved by the same manner and the details are omitted. ∎

6 Proof of Theorem 1.1

Definition 6.1**.**

For an $\vec{a}\in S_{\omega}$, we say a vector $\vec{b}\in S_{\omega}$ is proper to $\vec{a}$ if

$\omega t_{\vec{b}}\cdot I\cdot(\omega t_{\vec{b}})^{-1}\cap K\subseteq\omega t_{\vec{a}}\cdot I\cdot(\omega t_{\vec{a}})^{-1}\cap K$.

Furthermore, we say $\vec{b}$ is strictly proper to $\vec{a}$ if the containing above is strict.

Theorem 6.2**.**

Assume $N=3$. Let $\vec{a}\in S_{Id}$. If $\vec{a}^{\ast}$ is proper to $\vec{a}$ in $S_{Id}$, then there exists (not unique in general) an operator $\mathcal{T}\in\mathcal{H}(IZ,\chi_{\sigma})$ such that

$f_{Id,\vec{a}}\mid\mathcal{T}=f_{Id,\vec{a}^{\ast}}$.

Proof.

For any vector $\vec{a}^{\ast}$ strictly proper to $\vec{a}$ we can obtain the corresponding function $f_{Id,\vec{a}^{\ast}}$ by applying Proposition 5.5,5.7,5.8 repeatedly (but such process might not be unique anymore). ∎

Based on last theorem, we have the following corollary.

Corollary 6.3**.**

There are non-zero $\mathcal{H}(IZ,\chi_{\sigma})$-submodules of $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$ of infinite codimension.

Proof.

Consider the submodule $M$ of $(\textnormal{ind}^{G}_{KZ}\sigma)^{IZ,\chi_{\sigma}}$ generated by a single function $f_{Id,\vec{a}}$, for an $\vec{a}=(a_{1},a_{2})\in S_{Id}$ satisfying $a_{1}<a_{2}$. By Theorem 6.2, the module $M$ contains the subspace $M^{\prime}$ spanned by functions

$\{f_{Id,\vec{b}}\mid\vec{b}~{}\text{is~{}proper~{}to}~{}\vec{a}\}$

We claim that if a function $f\in M$ is supported in the cosets $\bigcup_{\vec{d}\in S_{Id}}KZt_{\vec{d}}I_{1}$, then it lies in $M^{\prime}$. Suppose that $f=f_{Id,\vec{a}}\mid\mathcal{T}$ for some $\mathcal{T}\in\mathcal{H}(IZ,\sigma)$.

We verify the claim case by case.

Case $a)$. $W(\chi_{\sigma})=Id$. By Proposition 3.6 we write $\mathcal{T}$ as a polynomial of $T_{t_{(1,0)}},T_{t_{(-1,0)}},T_{t_{(0,1)}}$ and $T_{t_{(0,-1)}},T_{t_{(1,1)}},T_{t_{(-1,-1)}}$. By Proposition 5.8, we see $\mathcal{T}$ can be replaced by a polynomial of $T_{t_{(-1,0)}}$ and $T_{t_{(-1,-1)}}$.

Case $b)$. $W(\chi_{\sigma})=\{Id,\omega_{1}\}$. We write $\mathcal{T}$ as a polynomial of $T_{\omega_{1}}$, $T_{\omega_{1}\cdot t_{(1,0)}}$, and $T_{\omega_{1}\cdot t_{(0,-1)}}$ (Proposition 3.5). Assume $f$ is supported in the cosets $\bigcup_{\vec{d}\in S_{Id}}KZt_{\vec{d}}I_{1}$. We can firstly eliminate from $\mathcal{T}$ any monomial with the operator $T_{\omega_{1}\cdot t_{1,0}}$ (Proposition 5.6). As a polynomial of $T_{\omega_{1}}$ acts as a scalar on $f_{Id,\vec{a}}$ (Proposition 5.6), we may assume any monomial of it does not appear in $\mathcal{T}$ 111The same remark applies to Case $c)$.. Then $\mathcal{T}$ can be substituted with a polynomial of $T_{\omega_{1}\cdot t_{(0,-1)}}T_{\omega_{1}}$ and $T^{2}_{\omega_{1}\cdot t_{(0,-1)}}$, as anything else would force $f$ to be supported outside $\bigcup_{\vec{d}\in S_{Id}}KZt_{\vec{d}}I_{1}$ (Proposition 5.6).

Case $c)$.  $W(\chi_{\sigma})=W_{0}$. We write $\mathcal{T}$ as a polynomial of $T_{\gamma}$ and $T_{\omega_{1}}$ (Proposition 3.3). If $f$ is supported in the cosets $\bigcup_{\vec{d}\in S_{Id}}KZt_{\vec{d}}I_{1}$, then $\mathcal{T}$ can be replaced by a polynomial of $(T_{\gamma}T_{\omega_{1}})^{2}$ and $(T_{\gamma}T_{\omega_{1}}T_{\gamma})^{2}$; otherwise, anything else would force $f$ to be supported outside $\bigcup_{\vec{d}\in S_{Id}}KZt_{\vec{d}}I_{1}$ (Proposition 5.3). Note here we have assumed $a_{1}<a_{2}$.

In every case, the claim follows by Theorem 6.2.

We now assume $a_{2}<-1$. Consider the subspace $M^{\prime\prime}$ of $(\textnormal{ind}^{G}_{KZ}\sigma)^{I_{1}}$ spanned by

$\{f_{Id,\vec{c}}\mid\vec{c}=(\ast,a_{2}+1),\ast\leq a_{2}+1\}$.

The space $M^{\prime\prime}$ is infinitely dimensional, but by our claim above it has no non-zero intersection with $M$. The argument is done. ∎

Acknowledgements

A major part of this note was done when the author was a postdoc at Warwick Mathematics Institute (Leverhulme Trust RPG-2014-106) and Einstein Institute of Mathematics (ERC 669655), and he would like to thank both institutions for the hospitality.