# Comptes Rendus
Mathématique

Eknath Ghate and Mihir Sheth

On non-admissible irreducible modulo *p* representations of GL₂(Q*p*²)

Volume 358, issue 5 (2020), p. 627-632

Published online: 14 September 2020

https://doi.org/10.5802/crmath.85

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Number Theory and Reductive Group Theory / *Théorie des nombres et théorie des groupes réductifs*

# On non-admissible irreducible modulo $p$ representations of $\text{GL}_2(\mathbb{Q}_{p^2})$

## *Sur les représentations irréductibles non-admissibles modulo $p$ de $\text{GL}_2(\mathbb{Q}_{p^2})$*

Eknath Ghate<sup>a</sup> and Mihir Sheth<sup>a, a</sup>

<sup>a</sup> School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai - 400005, India.

E-mails: eghate@math.tifr.res.in, mihir@math.tifr.res.in.

**Abstract.** We use a Diamond diagram attached to a 2-dimensional reducible split mod $p$ Galois representation of $\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_{p^2})$ to construct a non-admissible smooth irreducible mod $p$ representation of $\text{GL}_2(\mathbb{Q}_{p^2})$ following the approach of Daniel Le.

**Résumé.** Nous utilisons un diagramme de Diamond attaché à une représentation galoisienne mod $p$ simple réductible de dimension 2 de $\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_{p^2})$ pour construire une représentation mod $p$ non-admissible irréductible lisse de $\text{GL}_2(\mathbb{Q}_{p^2})$ en suivant l'approche de Daniel Le.

**2020 Mathematics Subject Classification.** 22E50, 11S37.

**Funding.** The authors acknowledge support of the Department of Atomic Energy, Government of India, under project number 12-R&D-TFR-5.01-0500.

Manuscript received 12th March 2020, accepted 9th June 2020.

## 1. Introduction

Let $p$ be a prime number, $\mathbb{Q}_p$ be the field of $p$-adic numbers, and $\overline{\mathbb{F}}_p$ be an algebraic closure of the finite field $\mathbb{F}_p$ of cardinality $p$. The study of the admissibility of smooth irreducible representations of connected reductive $p$-adic groups goes back to Harish-Chandra (see [6]). Building upon his work, Jacquet proved that every such representation over the field of complex numbers is admissible (see [8], see also [3]). This result was extended by Vignéras to smooth irreducible representations over any algebraically closed field of characteristic not equal to $p$ (cf. [12]). In

<footnote>* Corresponding author.</footnote>
the note [1], the authors ask whether this is true for smooth irreducible representations over algebraically closed fields of characteristic $p$. It is known that every smooth irreducible representation of $GL_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ is admissible (see [2]). However, Daniel Le recently constructed non-admissible smooth irreducible $\overline{\mathbb{F}}_p$-linear representations of $GL_2(F)$, for $F$ a finite unramified extension of $\mathbb{Q}_p$ of degree at least 3 and for $p > 2$, providing a negative answer to the question raised above (see [9]). In this paper, we follow Le's approach and construct non-admissible irreducible representations of $GL_2(\mathbb{Q}_{p^2})$ where $\mathbb{Q}_{p^2}$ is the unramified extension of $\mathbb{Q}_p$ of degree 2. These results support the viewpoint of Breuil and Paškūnas that the mod $p$ (and $p$-adic) representation theory of $GL_2(F)$ becomes more complicated as soon as $F \neq \mathbb{Q}_p$ (see [5], see also [11]).

Let $G = GL_2(\mathbb{Q}_{p^2})$, $K = GL_2(\mathbb{Z}_{p^2})$, and $\Gamma = GL_2(\overline{\mathbb{F}}_{p^2})$, where $\mathbb{Z}_{p^2}$ is the ring of integers of $\mathbb{Q}_{p^2}$ with residue field $\overline{\mathbb{F}}_{p^2}$. Fix an embedding $\overline{\mathbb{F}}_{p^2} \hookrightarrow \overline{\mathbb{F}}_p$. Let $I$ and $I_1$ denote the Iwahori and the pro-$p$ Iwahori subgroups of $K$ respectively, and $K_1$ denote the first principal congruence subgroup of $K$. Write $N$ for the normalizer of $I$ (and of $I_1$) in $G$. As a group, $N$ is generated by $I$, the center $Z$ of $G$, and by the element $\Pi = \begin{pmatrix} 0 & 1 \\ p & 0 \end{pmatrix}$. All representations considered in this paper from now on are over $\overline{\mathbb{F}}_p$-vector spaces. For a character $\chi$ of $I$, $\chi^s$ denotes its $\Pi$-conjugate sending $g$ in $I$ to $\chi(\Pi g \Pi^{-1})$.

A weight is a smooth irreducible representation of $K$. The $K$-action on such a representation factors through $\Gamma$ and thus any weight is described by a 2-tuple $(r_0, r_1) \otimes \det^m := \text{Sym}^{r_0} \overline{\mathbb{F}}_p^{-2} \otimes (\text{Sym}^{r_1} \overline{\mathbb{F}}_p^{-2})^{\text{Frob}} \otimes \det^m$ of integers with $0 \le r_0, r_1 \le p-1$ together with a determinant twist for some $0 \le m < p^2-1$ (see [4, Lemma 2.16 and Proposition 2.17]). Given a weight $\sigma$, its subspace $\sigma^{I_1}$ of $I_1$-invariants has dimension 1. If $\chi_\sigma$ denotes the corresponding smooth character of $I$ and $\chi_\sigma \neq \chi_\sigma^s$, then there exists a unique weight $\sigma^s$ such that $\chi_{\sigma^s} = \chi_\sigma^s$ (see [10, Theorem 3.1.1]).

A *basic 0-diagram* is a triplet $(D_0, D_1, r)$ consisting of a smooth $KZ$-representation $D_0$, a smooth $N$-representation $D_1$ and an $IZ$-equivariant isomorphism $r: D_1 \xrightarrow{\sim} D_0^{I_1}$ with the trivial action of $p$ on $D_0$ and $D_1$. Given such a diagram such that $D_0^{K_1}$ has finite dimension, the smooth injective $K$-envelope $\text{inj}_K D_0$ admits a non-canonical $N$-action which glues together with the $K$-action to give a smooth $G$-action on $\text{inj}_K D_0$ (see [5, Theorem 9.8]). The $G$-subrepresentation of $\text{inj}_K D_0$ generated by $D_0$ is smooth admissible and its $K$-socle equals the $K$-socle $\text{soc}_K D_0$ of $D_0$.

From now on, assume that $p$ is odd. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_{p^2}) \to \text{GL}_2(\overline{\mathbb{F}}_p)$ be a continuous generic Galois representation such that $p$ acts trivially on its determinant and $\mathcal{D}(\rho)$ be the set of weights, called *Diamond weights*, associated to $\rho$ as described in [5, Section 11]. Breuil and Paškūnas attach a family of basic 0-diagrams $(D_0(\rho), D_1(\rho), r)$, called *Diamond diagrams*, to $\rho$ such that $\text{soc}_K D_0(\rho) = \bigoplus_{\sigma \in \mathcal{D}(\rho)} \sigma$ (see [5, Theorem 13.8]).

For a finite unramified extension $F$ of $\mathbb{Q}_p$ of degree at least 3, Le uses a Diamond diagram attached to an *irreducible* $\rho : \text{Gal}(\overline{\mathbb{Q}}_p/F) \to \text{GL}_2(\overline{\mathbb{F}}_p)$ to construct an infinite dimensional diagram which gives rise to a non-admissible smooth irreducible representation of $\text{GL}_2(F)$ (see [9]). His strategy does not work for a Diamond diagram attached to an irreducible Galois representation of $\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_{p^2})$ because such a diagram does not have suitable $\Pi$-action dynamics. However, for $F = \mathbb{Q}_{p^2}$, we observe that a Diamond diagram attached to a *reducible split* $\rho$ has an indecomposable subdiagram with suitable $\Pi$-action dynamics so that Le's method can be used to obtain a non-admissible irreducible representation of $G = \text{GL}_2(\mathbb{Q}_{p^2})$.

## 2. Reducible Diamond diagram

Let $\omega_2$ be Serre's fundamental character of level 2 for the fixed embedding $\overline{\mathbb{F}}_{p^2} \to \overline{\mathbb{F}}_p$, and let $\rho : \text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_{p^2}) \to \text{GL}_2(\overline{\mathbb{F}}_p)$ be a continuous reducible split generic Galois representation. The restriction of $\rho$ to the inertia subgroup is, up to a twist by some character, isomorphic to

$$ \left( \begin{array}{cc} \omega_2^{r_0+1+(r_1+1)p} & 0 \\ 0 & 1 \end{array} \right) $$
for some $0 \le r_0, r_1 \le p-3$, not both equal to 0 or equal to $p-3$ (see [4, Corollary 2.9 (i)] and [5, Definition 11.7 (i)]). Define the weight

$$ \sigma := (r_0 + 1, p - 2 - r_1) \otimes \det^{p-1+r_1 p}. $$

Then the set of Diamond weights for $\rho$ is given by

$$ \mathcal{D}(\rho) = \{(r_0, r_1), \sigma, \sigma^s, (p-3-r_0, p-3-r_1) \otimes \det^{r_0+1+(r_1+1)p}\} $$

(see [5, Lemma 11.2 or Section 16, Example (ii)]). Fix a Diamond diagram $(D_0(\rho), D_1(\rho), r)$ attached to $\rho$, and identify $D_1(\rho)$ with $D_0(\rho)^{I_1}$ as IZ-representations via $r$. There is a direct sum decomposition $D_0(\rho) = \bigoplus_{v \in \mathcal{D}(\rho)} D_{0,v}(\rho)$ of K-representations with $\mathrm{soc}_K D_{0,v}(\rho) = v$ (see [5, Proposition 13.4]).

Now define

$$ D_0 := D_{0,\sigma}(\rho) \oplus D_{0,\sigma^s}(\rho) \quad \text{and} \quad D_1 := D_0^{I_1}. $$

It follows from [5, Theorem 15.4 (ii)] that $(D_0, D_1, r)$ is an indecomposable subdiagram of $(D_0(\rho), D_1(\rho), r)$. Set

$$ \tau := (r_0 + 2, r_1) \otimes \det^{p-2+(p-1)p} \quad \text{and} \quad \tau' := (p-1-r_0, p-3-r_1) \otimes \det^{r_0+(r_1+1)p}. $$

The graded pieces of the socle filtrations of $D_{0,\sigma}(\rho)$ and $D_{0,\sigma^s}(\rho)$, with the convention that we ignore a weight if a negative entry appears, are as follows (see [5, Theorem 14.8 or Section 16, Example (ii)]):

$$ D_{0,\sigma}(\rho) : \sigma \quad\rightarrow\quad \tau \oplus \tau^s \quad\rightarrow\quad (p-4-r_0, r_1-1) \otimes \det^{r_0+2} $$

$$ D_{0,\sigma^s}(\rho) : \sigma^s \quad\rightarrow\quad \tau' \oplus \tau'^s \quad\rightarrow\quad (r_0-1, p-4-r_1) \otimes \det^{(r_1+2)p}. $$

We have from [5, Corollary 14.10] that

$$ D_1 = \chi_\sigma \oplus \chi_\tau \oplus \chi_\tau^s \oplus \chi_\sigma^s \oplus \chi_{\tau'} \oplus \chi_{\tau'}^s. \quad (1) $$

For an IZ-representation V and an IZ-character $\chi$, we write $V^\chi$ for the $\chi$-isotypic part of V.

### 3. An infinite dimensional diagram and the construction

Let $D_0(\infty) := \bigoplus_{i \in \mathbb{Z}} D_0(i)$ be the smooth KZ-representation with component-wise KZ-action, where there is a fixed isomorphism $D_0(i) \cong D_0$ of KZ-representations for every $i \in \mathbb{Z}$. Following [9], we denote the natural inclusion $D_0 \hookrightarrow D_0(i) \hookrightarrow D_0(\infty)$ by $\iota_i$, and write $v_i := \iota_i(v)$ for $v \in D_0$ for every $i \in \mathbb{Z}$. Let $D_1(\infty) := D_0(\infty)^{I_1}$. We define a $\Pi$-action on $D_1(\infty)$ as follows. Let $\lambda = (\lambda_i) \in \prod_{i \in \mathbb{Z}} \overline{\mathbb{F}}_p$. For all integers $i \in \mathbb{Z}$, define

$$ \Pi v_i := \begin{cases} (\Pi v)_i & \text{if } v \in D_1^{\chi_\sigma}, \\ (\Pi v)_{i+1} & \text{if } v \in D_1^{\chi_\tau}, \\ \lambda_i (\Pi v)_i & \text{if } v \in D_1^{\chi_{\tau'}}. \end{cases} $$

This uniquely determines a smooth N-action on $D_1(\infty)$ such that $p = \Pi^2$ acts trivially on it. Thus we get a basic 0-diagram $D(\lambda) := (D_0(\infty), D_1(\infty))$, can with the above actions where can is the canonical inclusion $D_1(\infty) \to D_0(\infty)$.

**Theorem 1.** There exists a smooth representation $\pi$ of G such that

(i) $(\pi|_{KZ}, \pi|_N, \mathrm{id})$ contains $D(\lambda)$,

(ii) $\pi$ is generated by $D_0(\infty)$ as a G-representation, and

(iii) $\mathrm{soc}_K \pi = \mathrm{soc}_K D_0(\infty)$.
**Proof.** Let $\Omega$ be the smooth injective $K$-envelope of $D_0$ equipped with the $KZ$-action such that $p$ acts trivially. The smooth injective $I$-envelope $\text{inj}_I D_1$ of $D_1$ appears as an $I$-direct summand of $\Omega$. Let $e$ denote the projection of $\Omega$ onto $\text{inj}_I D_1$. There is a unique $N$-action on $\text{inj}_I D_1$ compatible with that of $I$ and compatible with the action of $N$ on $D_1$. By [5, Lemma 9.6], there is a non-canonical $N$-action on $(1-e)(\Omega)$ extending the given $I$-action. This gives an $N$-action on $\Omega$ whose restriction to $IZ$ is compatible with the action coming from $KZ$ on $\Omega$.

Now let $\Omega(\infty) := \bigoplus_{i \in \mathbb{Z}} \Omega(i)$ with component-wise $KZ$-action where there is a fixed isomorphism $\Omega(i) \cong \Omega$ of $KZ$-representations for every $i \in \mathbb{Z}$. We wish to define a compatible $N$-action on $\Omega(\infty)$. As before, denote the natural inclusion $\Omega \stackrel{\sim}{\to} \Omega(i) \hookrightarrow \Omega(\infty)$ by $\iota_i$, and write $v_i := \iota_i(v)$ for $v \in \Omega$. Let $\Omega_\chi$ denote the smooth injective $I$-envelope of an $I$-character $\chi$. Thus, from (1), we have $e(\Omega) = \text{inj}_I D_1 = \Omega_{\chi_\sigma} \oplus \Omega_{\chi_\tau} \oplus \Omega_{\chi_i^s} \oplus \Omega_{\chi_\sigma^s} \oplus \Omega_{\chi_{i'}}^s \oplus \Omega_{\chi_{i'}}^s$. If $v \in (1-e)(\Omega)$, we define $\Pi v_i := (\Pi v)_i$ for all integers $i$. Otherwise, we define $\Pi v_i := (\Pi v)_i$ if $v \in \Omega_{\chi_\sigma}$, $\Pi v_i := (\Pi v)_{i+1}$ if $v \in \Omega_{\chi_\tau}$, and $\Pi v_i := \lambda_i (\Pi v)_i$ if $v \in \Omega_{\chi_{i'}}$. By demanding that $\Pi^2$ acts trivially, this defines a smooth $N$-action on $\Omega(\infty)$ which is compatible with the $N$-action on $D_1(\infty)$, and whose restriction to $IZ$ is compatible with the action coming from $KZ$ on $\Omega(\infty)$. By [10, Corollary 5.5.5], we have a smooth $G$-action on $\Omega(\infty)$. We then take $\pi$ to be the $G$-representation generated by $D_0(\infty)$ inside $\Omega(\infty)$. If follows easily from the construction that $\pi$ satisfies the properties (i), (ii) and (iii). $\square$

**Theorem 2.** If $\lambda_i \neq \lambda_0$ for all $i \neq 0$, then any smooth representation $\pi$ of $G$ satisfying the properties (i), (ii), and (iii) of Theorem 1 is irreducible and non-admissible.

**Proof.** Let $\pi' \subseteq \pi$ be a non-zero subrepresentation of $G$. By property (iii), we have either $\text{Hom}_K(\sigma, \pi') \neq 0$ or $\text{Hom}_K(\sigma^s, \pi') \neq 0$. We consider the case $\text{Hom}_K(\sigma, \pi') \neq 0$; the other case is treated analogously. There exists a non-zero $(c_i) \in \bigoplus_{i \in \mathbb{Z}} \overline{\mathbb{F}}_p$ such that

$$ \left( \sum_i c_i l_i \right) (D_{0,\sigma}(\rho)) \cap \pi' \neq 0. $$

We claim that

$$ \left( \sum_i c_i l_{i+j} \right) (D_0) \subset \pi' \quad \text{for all } j \in \mathbb{Z}. \qquad (2) $$

We first show that $(\sum_i c_i l_i)(D_{0,\sigma^s}(\rho)) \subset \pi'$. Note that $(\sum_i c_i l_i)(D_{0,\sigma}(\rho)) \cap \pi' \neq 0$ is equivalent to $(\sum_i c_i l_i)(\sigma) \subset \pi'$. Since $(\sum_i c_i l_i)(D_1^{\chi_\sigma}) \subset \pi'$ and $\pi'$ is stable under the $\Pi$-action, we have $(\sum_i c_i l_i)(D_1^{\chi_\sigma^s}) \subset \pi'$. By Frobenius reciprocity, we have a non-zero $K$-equivariant map

$$ \text{Ind}_I^K \left( \left( \sum_i c_i l_i \right) \left( D_1^{\chi_\sigma^s} \right) \right) \to \pi' \qquad (3) $$

whose image is $(\sum_i c_i l_i)(I(\delta(\sigma), \sigma^s))$, where $\delta$ is the bijection on the set of Diamond weights $\mathcal{D}(\rho)$ defined in [5, Section 15], and $I(\delta(\sigma), \sigma^s)$ is the $K$-subrepresentation of $D_{0,\delta(\sigma)}(\rho)$ with cosocle $\sigma^s$ (and socle $\delta(\sigma)$). In our setting, $\delta$ maps $\sigma$ to $\sigma^s$ and vice versa (see [5, Lemma 15.2]). Thus $I(\delta(\sigma), \sigma^s) = \sigma^s$ and so $(\sum_i c_i l_i)(\sigma^s) \subset \pi'$. Let $R(\sum_i c_i l_i)(\sigma)$ be the $K$-subrepresentation of the compact induction $\text{c-Ind}_{KZ}^G(\sum_i c_i l_i)(\sigma)$ defined in [5, Section 17]. By [5, Lemmas 17.1, 17.4 and 17.8], we have

$$ \text{Ind}_I^K \left( \left( \sum_i c_i l_i \right) \left( D_1^{\chi_\sigma^s} \right) \right) \subset R \left( \left( \sum_i c_i l_i \right) (\sigma) \right), $$

and by Frobenius reciprocity, there is a non-zero map

$$ \text{c-Ind}_{KZ}^{G} \left( \left( \sum_i c_i l_i \right) (\sigma) \right) \to \pi' \qquad (4) $$

which restricts to the map (3). So the image $Q$ of $R(\sum_i c_i l_i)(\sigma)$ in $\pi'$ under the map (4) contains $(\sum_i c_i l_i)(\sigma^s)$. Since $\text{soc}_K Q \subset \text{soc}_K \pi = \text{soc}_K D_0(\infty)$ and the Jordan-Hölder factors of
$R(\sum_i c_i l_i)(\sigma)$ are multiplicity free (see [5, Lemma 17.11]), $\text{soc}_K Q$ is isomorphic to a subrep-
resentation of the direct sum of the weights in $\mathcal{D}(\rho)$. Therefore by [5, Lemma 19.5], $\text{soc}_K Q =$
$(\sum_i c_i l_i)(\sigma^s)$, and by [5, Lemma 19.7], $Q$ contains a copy of the $K$-representation $D_{0,\sigma^s}(\rho)$. But
$(\sum_i c_i l_i)(D_{0,\sigma^s}(\rho))$ is the unique $K$-subrepresentation of $\pi$ isomorphic to $D_{0,\sigma^s}(\rho)$ and with $K$-
socle $(\sum_i c_i l_i)(\sigma^s)$. Thus $(\sum_i c_i l_i)(D_{0,\sigma^s}(\rho)) = Q \subset \pi'$.

Now, since $(\sum_i c_i l_i)(\sigma^s) \subset \pi'$, a symmetric argument shows that $(\sum_i c_i l_i)(D_{0,\sigma}(\rho)) \subset \pi'$. Thus

$$
\left( \sum_i c_i l_i \right) (D_0) \subset \pi'.
$$

Therefore

$$
\left( \sum_i c_i l_i \right) (D_1^{\chi_r}) \subset \pi' \quad \text{and} \quad \left( \sum_i c_i l_i \right) (D_1^{\chi_r^s}) \subset \pi'.
$$

Since $\pi'$ is stable under the Π-action, we have

$$
\left( \sum_i c_i l_{i+1} \right) \left( D_1^{\chi_r^s} \right) \subset \pi' \quad \text{and} \quad \left( \sum_i c_i l_{i-1} \right) \left( D_1^{\chi_r^s} \right) \subset \pi'.
$$

In particular,

$$
\left( \sum_i c_i l_{i+1} \right) (D_{0,\sigma}(\rho)) \cap \pi' \neq 0 \quad \text{and} \quad \left( \sum_i c_i l_{i-1} \right) (D_{0,\sigma}(\rho)) \cap \pi' \neq 0.
$$

By the same arguments as above, we find that

$$
\left( \sum_i c_i l_{i+1} \right) (D_0) \subset \pi' \quad \text{and} \quad \left( \sum_i c_i l_{i-1} \right) (D_0) \subset \pi'.
$$

The claim (2) is now proved by repeatedly using the Π-action.

For $(d_i) \in \bigoplus_{i \in \mathbb{Z}} \overline{\mathbb{F}}_p$, let $\#(d_i)$ denote the number of non-zero $d_i$'s. Among all the non-zero elements $(c_i)$ of $\bigoplus_{i \in \mathbb{Z}} \overline{\mathbb{F}}_p$ for which $(\sum_i c_i l_i)(D_0) \subset \pi'$, we pick one with $\#(c_i)$ minimal. We may also assume that $c_0 \neq 0$ using (2). We now show that $\#(c_i) = 1$. Assume to the contrary that $\#(c_i) > 1$. Since $(\sum_i c_i l_i)(D_1^{\chi_{\tau'}}) \subset \pi'$ and $\pi'$ is stable under the $\Pi$-action, we have

$$
\left( \sum_i \lambda_i c_i l_i \right) \left( D_1^{\chi_{\tau'}} \right) \subset \pi'.
$$

Since $(\sum_i \lambda_0 c_i l_i)(D_1^{\chi_{\tau'}})$ is also clearly in $\pi'$, subtracting it from the above, we get

$$
\left( \sum_i (\lambda_i - \lambda_0) c_i l_i \right) \left( D_1^{\chi_{\tau'}} \right) \subset \pi'.
$$

Writing $(c'_i) := ((\lambda_i - \lambda_0)c_i)$, we see that

$$
\left( \sum_i c'_i l_i \right) (D_{0,\sigma^s}(\rho)) \cap \pi' \neq 0.
$$

Following the same arguments as in the previous paragraphs, we get that $(\sum_i c'_i l_i)(D_0) \subset \pi'$.
However, the hypothesis $\lambda_i \neq \lambda_0$ for all $i \neq 0$, and the assumption $\#(c_i) > 1$ imply that $(c'_i)$ is non-
zero and $\#(c'_i) = \#(c_i) - 1$ contradicting the minimality of $\#(c_i)$. Therefore, we have $c_0 l_0(D_0) \subset \pi'$.
So $l_0(D_0) \subset \pi'$. Using (2) again, we get that $\bigoplus_{j \in \mathbb{Z}} l_j(D_0) = D_0(\infty) \subset \pi'$. By property (ii), we have
$\pi' = \pi$.

The non-admissibility of $\pi$ is clear because $\pi^{K_1} \supseteq \text{soc}_K \pi$ and $\text{soc}_K \pi$ is not finite dimensional
by property (iii). $\square$
**Remark 3.** If the diagram $(D_0(\rho), D_1(\rho), r)$ is defined over $\mathbb{F}_{p^2}$ and $(\lambda_i) \in \prod_{i \in \mathbb{Z}} \mathbb{F}_{p^2}^\times$, then the representation $\pi$ in Theorem 1 has a model $\pi_0$ over $\mathbb{F}_{p^2}$. Furthermore, $\pi_0$ is absolutely irreducible and non-admissible if the $(\lambda_i)$ satisfy the hypothesis of Theorem 2. In fact, for any field $C$ containing $\mathbb{F}_{p^2}$, the methods of this paper produce an absolutely irreducible non-admissible smooth $C$-representation $C \otimes_{\mathbb{F}_{p^2}} \pi_0$ of $G$.

Now let $C$ be an arbitrary field of characteristic $p$ with algebraic closure $\overline{C}$. From the discussion in the previous paragraph, the representation $\overline{C} \otimes_{\mathbb{F}_{p^2}} \pi_0$ is a smooth irreducible $\overline{C}$-representation which has a model $C' \otimes_{\mathbb{F}_{p^2}} \pi_0$ over $C'$, where $C' = C\mathbb{F}_{p^2} \subset \overline{C}$. By [7, Lemma II.5], there exists a smooth irreducible $C$-representation $\pi_C$ such that $\overline{C} \otimes_{\mathbb{F}_{p^2}} \pi_0$ is a $\overline{C}$-subrepresentation of $\overline{C} \otimes_C \pi_C$. Since $\overline{C} \otimes_{\mathbb{F}_{p^2}} \pi_0$ is non-admissible, $\overline{C} \otimes_C \pi_C$ is non-admissible and hence $\pi_C$ is non-admissible by [7, Lemma III.1(ii)]. Thus we obtain a smooth irreducible non-admissible representation of $G$ over any field $C$ of characteristic $p$.

## Acknowledgments

We thank Daniel Le, Sandeep Varma, M.-F. Vignéras and the referee for useful comments on earlier versions of this paper. The second author thanks Anand Chitrao for helpful discussions on diagrams.

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