Proof. In the situation of (v), the condition $\pi ^{I(p)} \neq 0$ implies that $\pi $ contains an eigenvector for I and thus is a constituent of a principal series representation. The assertion (v) then follows from (iv). Assertion (vi) in the local setting is the last assertion of [Reference Genestier and LafforgueGLa, Théorème 0.1]; in the global setting, it is [Reference LafforgueLaf18, Proposition 12.5].

Proof. The first two assertions express the standard characterization of ${\hat {G}}$ -local systems as symmetric monoidal functors (see, for example, the discussion on p. 52 of [Reference SimpsonSi]); the third assertion is obvious.

Proof. Both conditions in (a) are checked by composition with a faithful representation $\sigma $ of ${}^LG$ . This reduces the verification of (a) to the case of $GL(n)$ , which is verified in [Reference Taylor and YoshidaTY, Lemma 1.4(3)]. Part (b) for $GL(n)$ is [Reference Taylor and YoshidaTY, Lemma 1.4(4)]. For general groups G, one can cite the argument on p. 10 of [Reference ArthurA84]; Arthur is there classifying Arthur parameters, rather than Weil-Deligne parameters, of real groups, but the proof of uniqueness is the same. More precisely, suppose $\rho $ can be completed to a pure Weil-Deligne parameter $(\rho ,N)$ , and let $\varphi _{\rho ,N}: W_F \times SL(2)$ be as above. Let $\varphi $ be the restriction of $\varphi _{\rho ,N}$ to the $W_F$ factor. Since the image of $\varphi $ is bounded and consists only of semisimple elements, its centralizer $C_\varphi \subset {}^LG$ is reductive. We need to show that the restriction of $\varphi _{\rho ,N}$ to $SL(2)$ , whose image lies in $C_\varphi $ , is unique up to $C_\varphi $ -conjugacy, but this is exactly what Arthur shows in his situation.

Proof. We explain how to deduce this from [Reference Sawin and TemplierST, Lemma 11.4] since this result is not explicitly stated there. Under the irreducibility hypothesis, [Reference Sawin and TemplierST, Lemma 11.4] states that the composition of $\rho $ with any representation of ${}^L G(C)$ is pure of some weight w as a local system on an open subset of X. (The mixedness property assumed in [Reference Sawin and TemplierST] is a consequence of [Reference LafforgueLaf02].) It then follows from [Reference DeligneDe80, Theorem 1.8.4] that the i’th associated graded of the monodromy filtration of the composition of $\rho _z$ with the same representation is pure of weight $w+i$ . In particular, all the Frobenius eigenvalues are q-numbers, and the sum of eigenspaces of Frobenius associated to eigenvalues of size $\leq q^{ \frac {w+i}{2}}$ is the ith member of the monodromy filtration. Part (iii) of the definition of pure Weil-Deligne parameter then follows from the defining property [Reference DeligneDe80, Proposition 1.6.13].

Proof. We shall use the results of [Reference Gan and LomelíGLo] to produce the desired globalizations. Let $x \neq z$ be any other place of K such that $G(K_x)$ is a split reductive group. There are infinitely many such x – just take any x that splits completely in a finite cover of X over which G splits. Thus, it suffices to show

1. (i) There is a supercuspidal representation $\pi '$ of $G(K_x)$ such that ${\mathcal {L}}^{ss}(\pi ')$ is irreducible, and

2. (ii) There is an automorphic representation $\Pi $ of $G_K$ with $\Pi _x {~\overset \sim \longrightarrow ~} \pi '$ and $\Pi _z {~\overset \sim \longrightarrow ~} \pi $ , and with $\Pi _y$ tamely ramified for all $y \neq x, z$ .

Once (i) is given, the existence of $\Pi $ as in (ii) is proved in [Reference Gan and LomelíGLo] by a Poincaré series construction. It thus remains to produce a supercuspidal representation $\pi '$ as in (i).

We let $k' = k(x)$ denote the residue field at x and construct the Kloosterman sheaf $\mathrm {Kl}_{\hat {G}}(\phi ,\chi )$ of [Reference Heinloth, Ngô and YunHNY] over $\mathbb {P}^1_{k'}$ . Let $\pi '$ be the local component at $\infty $ of the corresponding automorphic representation $\pi (\phi ,\chi )$ . Theorem 2 of [Reference Heinloth, Ngô and YunHNY] implies that the restriction to the inertia group at $\infty $ of the local monodromy representation of $\mathrm {Kl}_{\hat {G}}(\phi ,\chi )$ is irreducible. The statement in [Reference Heinloth, Ngô and YunHNY] includes slightly restrictive hypotheses on the prime p for non-simply laced groups, but these hypotheses have since been removed by Xu and Zhu in [Reference Xu and ZhuXZ, Corollary 4.5.8, Remark 4.5.10], under the hypothesis that $G(K_x)$ is split, which we are assuming. However, it follows from construction that $\mathrm {Kl}_{\hat {G}}(\phi ,\chi )$ is the global parameter attached by [Reference LafforgueLaf18] to $\pi (\phi ,\chi )$ . More precisely, [Reference Heinloth, Ngô and YunHNY, Reference Xu and ZhuXZ] computes the local monodromy of $\mathrm { Kl}_{\hat {G}}(\phi ,\chi )$ at unramified places and identifies it with that obtained from the Satake parameters of $\pi (\phi ,\chi )$ . (The construction of $\pi (\phi ,\chi )$ is made explicit in [Reference YunY16] but its existence is implicit in [Reference Heinloth, Ngô and YunHNY].) It then follows that ${\mathcal {L}}^{ss}(\pi ')$ is irreducible, as desired.

Proof. Let $f_{\pi }$ be a pseudo-coefficient of $\pi $ , whose properties are recalled in [Reference Gan and LomelíGLo, §8.1]. In [Reference Gan and LomelíGLo, §8.2], a version of the simple trace formula was formulated as a working hypothesis. Applying Lemma A.1 in the appendix by R. Beuzart-Plessis, it follows as in the proof of [Reference Gan and LomelíGLo, Proposition 8.2] that there is a cuspidal $\Pi $ with $\Pi _x {~\overset \sim \longrightarrow ~} \pi '$ (with $\pi '$ as in Proposition 3.9) for some place x and $Tr(\Pi _z)(f_\pi ) \neq 0$ . This implies (by [Reference Gan and LomelíGLo, Lemma 8.1]) that $\Pi _z$ has the same cuspidal support as $\pi $ , and therefore that

$$ \begin{align*}{\mathcal{L}}^{ss}(\Pi_z) = {\mathcal{L}}^{ss}(\pi)\end{align*} $$

since the semisimple Genestier-Lafforgue parameter is compatible with parabolic induction. Now we do not know, nor care, whether $\Pi _z$ is isomorphic to the discrete series $\pi $ or not. Since ${\mathcal {L}}^{ss}(\Pi _x) = {\mathcal {L}}^{ss}(\pi ')$ is irreducible, so is the global parameter ${\mathcal {L}}(\Pi )$ , and we have produced the desired $\Pi $ .

Proof. It follows from Proposition 3.11 that $Ad\circ {\mathcal {L}}(\pi )$ is a pure Weil-Deligne representation of some weight w. But since G, and therefore $\hat {G}$ , is semisimple, the determinant of $Ad\circ {\mathcal {L}}(\pi )$ is pure of weight $0$ . It follows that w must equal $0$ .

Proof. In fact, Deligne’s result only requires the Frobenius eigenvalues to have the same weight for a single complex embedding; under this condition, $\rho _z(I_z)$ is a finite group. But we have assumed that $\rho _z^{ss}$ is unramified; thus, $\rho _z(I_z)$ is unipotent. It follows that $\rho _z(I_z)$ must be trivial.

Proof. Theorem 4.2 implies that ${\mathcal {V}}$ extends to a tame local system, also called ${\mathcal {V}}$ , over ${\mathbb G}_m$ . By [Reference LafforgueLaf02], Corollary VII.8, ${\mathcal {V}}$ can be written as a (finite) direct sum

$$ \begin{align*}{\mathcal{V}} = \bigoplus_{\chi} {\mathcal{V}}_\chi \otimes \chi,\end{align*} $$

where each ${\mathcal {V}}_\chi $ is a mixed local system and $\chi $ is the pullback to ${\mathbb {P}}^1\setminus |X|$ of a rank $1\ \ell $ -adic local system over $Spec(k)$ – in other words, a continuous $\ell $ -adic character of $Gal(\bar {k}/k)$ . We thus reduce to the case where ${\mathcal {V}}$ is a mixed $\ell $ -adic local system, and since ${\mathcal {V}}$ is assumed semisimple, we may assume ${\mathcal {V}}$ to be irreducible.

Then the purity hypothesis at the points z in the nonempty set Z implies that ${\mathcal {V}}$ is itself pure. Since the tame fundamental group of ${\mathbb G}_{m,\bar {k}}$ is just $\prod _{p' \neq p} {\mathbb Z}_{p'}(1)$ , it follows from the purity that the image J under $\rho $ of the geometric fundamental group of ${\mathbb G}_m$ is finite and necessarily abelian.

Now one knows that the image of the tame inertia group $I_0$ at $0$ in the tame fundamental group of ${\mathbb G}_{m,k}$ is equal to the tame fundamental group of ${\mathbb G}_{m, \bar {k}}$ (i.e., the tame geometric fundamental group of ${\mathbb G}_{m,k}$ ). The same is true for the tame inertia group at $\infty $ . Hence, the hypothesis (b) on the local monodromy at $0$ or $\infty $ implies that the conjugation action of $Gal(\bar {k}/k)$ on J is trivial, and thus, the image under $\rho $ of the arithmetic fundamental group $\pi _1({\mathbb {P}}^1 \setminus |X|,y)$ is also abelian; moreover, the image of $Gal(\bar {k}/k)$ is semisimple by hypothesis. This completes the proof.

Proof. Let $K_z$ denote the completion of K at z, ${\mathcal {O}}_z$ its integer ring, $k_z$ its residue field, which the inclusion of k in K canonically identifies with k. The image of the element $T \in {\mathcal {O}}_z$ in $k_z$ is identified with the element $z \in k^\times $ , so the residue field of the unramified extension ${\mathcal {O}}_z[T^{1/(q-1)}]$ is just $k(z^{1/(q-1)})$ . Both sides of the identity are defined by taking an element of the Galois group, observing that it acts on $T^{1/(q-1)} = z^{1/(q-1)}$ by multiplication by a $(q-1)$ st root of unity and then composing with a fixed embedding of the $(q-1)$ st roots of unity into ${\overline {\mathbb Q}_{\ell }}^\times $ . The only difference is that $(\alpha _a)^{(z)}$ is calculated by first considering the action of the Galois group of K, restricting to the decomposition group, and observing that it factors through the Galois group of the residue field, while $\psi _k(az)$ is calculated by considering the action of the Galois group of the residue field directly, but these agree since the homomorphism from the decomposition group to the Galois group of the residue field is defined to be compatible with the action on elements of the residue field.

Proof. We give two proofs of the lemma.

(i) (Class field theory) It suffices to show that $\operatorname {Frob}_z = \operatorname {Frob}_1 \cdot m_z^{-1} $ in the abelianization of the tame fundamental group of $\mathbb G_m$ . Class field theory gives an isomorphism between this abelianization and the profinite completion of

$$\begin{align*}K^\times \backslash \mathbb A_K^\times / \left( \prod_{\substack{ y \in |\mathbb P^1_k|\\ y \notin \{0,\infty|}} \mathcal O_{K_y}^\times \times \prod_{y \in \{0,\infty\}} U_{K_y, 1} \right), \end{align*}$$

where $U_{K_y, 1}$ denotes the units of $\mathcal O_{K_y}$ that are congruent to $1$ modulo the maximal ideal. Moreover,

• ○ $\operatorname {Frob}_z$ corresponds to the idèle which is a uniformizer at z and the identity at all other places,

• ○ $\operatorname {Frob}_1$ corresponds to the idèle which is a uniformizer at $1$ and the identity at all other places, and

• ○ $m_z$ corresponds to the idele which is z at $0$ and the identity at all other places.

So it suffices to check that the first idèle class is the product of the other two idèle classes. This can be checked by multiplying by $\frac {T- z}{T-1} \in K^\times $ , which is a uniformizer at z, the inverse of a uniformizer at $1$ , and z at $0$ .

(ii) (Kummer theory) We can construct a one-dimensional representation of $Gal(\overline {k}/k)$ on which $\operatorname {Frob}_q$ acts by multiplication by $ \operatorname {tr}(\operatorname {Frob}_1 , \mathcal L)$ . On tensoring $\mathcal L$ with the inverse representation, we reduce to the case where $\operatorname {Frob}_1$ acts trivially on $\mathcal L$ . Since the tame fundamental group of $\mathbb G_m$ is generated by the geometric tame fundamental group and Frobenius, the image of the Galois group acting on $\mathcal L$ must equal the image of the geometric tame fundamental group. The action of the geometric tame fundamental group on $\mathcal L$ must be by a $\operatorname {Frob}_q$ -invariant character, which means it factors through the $\operatorname {Frob}_q$ -coinvariant of the abelianization of the tame geometric fundamental group, which is $\mathbb Z/(q-1)$ . So the action of the geometric fundamental group is by a character of order $q-1$ . The result now follows from Lemma 4.6.

Proof. We give the proof in the $G_c$ case and then describe the modifications necessary for $G_{c,r+}$ .

If (5.2) holds for $g_1$ and $g_2$ , then it holds for $g_1g_2$ , so that it suffices to verify (5.2) for a set of generators of the parahoric $G_c$ . The parahoric $G_c$ is generated by the subgroups $U_\psi $ for those affine functions $\psi $ on the apartment of S such that

$$\begin{align*}\psi ( c) \geq 0. \end{align*}$$

Here, $\psi $ is an affine function of the form $\psi (w) = b \cdot w -s$ for a root $b \in X^*(S)$ and a real number s. If $U_b \subset G$ is the (restricted) root group associated to b, then one has an isomorphism over k:

$$\begin{align*}i_b: U_b {\overset\sim\longrightarrow} \mathrm{Res}_{k_b/k}\mathbb{G}_a, \end{align*}$$

where $k_b$ is the field of definition of the root b and $\mathbb G_a$ is the additive group, and the subgroup $U_\psi $ can be described as

$$\begin{align*}U_{\psi} = \{ u \in U_b(F): v(i_b(u)) \geq s \}, \end{align*}$$

where v is the unique extension of v to $k_b$ .

To obtain (5.2), it suffices to check that the following two statements hold for $g\in U_b$ :

1. (a) If $\rho _{ij} (g) \neq 0$ , then $\lambda _i -\lambda _j = n b$ for some $n\geq 0$ ;

2. (b) $\rho _{ij}(g)$ is a homogeneous polynomial function on $U_b$ defined over k of degree n.

Indeed, if (a) and (b) hold, then we have, for $g \in U_{\psi}$

$$\begin{align*}v (\rho_{ij}(g)) \geq n s \geq ns + n \psi ( c) = n b \cdot c= (\lambda_i -\lambda_j) \cdot c,\end{align*}$$

as desired.

To prove (a) and (b), we use lifting to characteristic $0$ . The group G (which is defined over k) has a lift from the field k to the Witt vectors $W(k)$ , unique up to not-necessarily-unique isomorphism. Since irreducible algebraic representations of G are classified by Galois orbits of highest weights, and the Galois action for k and $W(k)$ are the same, we may lift the representation V from the field k to its ring of Witt vectors $W(k)$ . Both properties in (a) and (b) are preserved by reduction mod p and thus may be checked over the fraction field of $W(k)$ , where $U_b$ is obtained by exponentiating the root subspace of b. In characteristic $0$ , both (a) and (b) follow readily from the representation theory of $SL_2$ .

We now indicate the modifications needed for the statement for $G_{c, r^+}$ . The inequality (5.3) is also stable under matrix multiplication since $r \geq 0$ . Because the subgroups $G_{c,r+}$ are generated by $U_\psi $ for those $\psi $ with $\psi (c)> r$ and the subgroups $T^a_{n \delta }$ for the maximal torus T containing S, for $n> r$ , it suffices to verify (5.3) for these subgroups. Here, a is an (absolute) simple root of T (over $\overline {k}$ ), with associated root subgroup $SL_2$ (over $\overline {k}$ ), $T^a$ is the 1-dimensional torus (over $\overline {k}$ ) which is the image of the coroot

$$\begin{align*}a^{\vee}: \mathbb{G}_m \to T^a \subset T \end{align*}$$

and

$$\begin{align*}T^a_{n \delta} = \{ a^{\vee}(x) \in T^a(F^{ur}) \cap G(F): v(x-1) \geq n \}. \end{align*}$$

We begin with the elements of $U_{\psi }$ as above. That the elements of such $U_\psi $ satisfy (5.3) follow from (a) and (b) above and the additional fact that $\rho _{ij}$ is the constant function $1$ on $U_b$ if $n=0$ . Indeed, these imply that

$$\begin{align*}v (\rho_{ij}(g)) \geq n s \geq ns + n \psi ( c) +nr = n b \cdot c + nr> (\lambda_i -\lambda_j) \cdot c+r \end{align*}$$

if $n>0$ and $v (\rho _{ij}(g)-1) = \infty $ if $n=0$ .

Hence, it remains to show that elements $g\in T^a_{n \delta }$ satisfy (5.3). Because T is the centralizer of S in G, we see that

$$\begin{align*}g \in T^a_{n \delta} \text{ and } \lambda_i \ne \lambda_j \Longrightarrow \rho_{ij}(g)=0. \end{align*}$$

Thus, it suffices to show

$$\begin{align*}v(\rho_{ij}(g)-\delta_{ij}) \geq n> r \quad \text{if}\ \lambda_i = \lambda_j. \end{align*}$$

Now the action of the elements $g \in T^a$ and therefore the matrix entries $\rho _{ij}$ (restricted to $T^a$ ) are given by polynomial functions on $\mathbb G_m$ over $\overline {k}$ , which are equal to $\delta _{ij}$ at the identity element $1$ . Thus, the difference between $\rho _{ij}(g)$ and $\delta_{ij}$ has valuation at least n for $g \in T^a_{n \delta }$ , as desired.

Proof. Assume ${\mathcal {L}}^{ss}(\sigma )$ is unramified; we shall derive a contradiction.

Applying Lafforgue, let

$$\begin{align*}{\mathcal{L}}(\Pi): \mathrm{Gal}(K^{sep}/K) \longrightarrow {^L}G = {\hat{G}} \rtimes W_k \end{align*}$$

be the global Galois representation associated to $\Pi $ as in Theorem 2.1. By property (d), ${\mathcal {L}}(\Pi )_x$ is unramified for all places x of ${\mathbb {P}}^1$ outside of ${\mathbb {P}}^1(k)$ . However, by property (b) and (c), it follows by Theorem 2.3 (iv) that ${\mathcal {L}}(\Pi )^{ss}_0$ and ${\mathcal {L}}(\Pi )^{ss}_{\infty }$ are tamely ramified. For $z \in {\mathbb G}_m(k)$ , with decomposition group $\Gamma _z \subset \pi _1({\mathbb {P}}^1 \setminus |{\mathbb {P}}^1(k)|)$ , it follows by property (a) and Theorem 2.1(iii) that

(7.3) $$ \begin{align} {\mathcal{L}}(\Pi)^{ss}_z {~\overset\sim\longrightarrow~} {\mathcal{L}}^{ss}(\sigma). \end{align} $$

By our hypothesis, ${\mathcal {L}}(\Pi )^{ss}_z$ is thus unramified for all $z \in \mathbb {G}_m(k) = k^{\times }$ .

We shall now examine in greater detail the restriction of ${\mathcal {L}}(\Pi )$ to the local decomposition group $\Gamma _0$ at $0 \in k$ . By property (c) and Theorem 2.3 (iv), one has

$$\begin{align*}{\mathcal{L}}(\Pi)^{ss}_0 : \Gamma_0 = {\mathrm{Gal}}(F^{sep}/F) \longrightarrow {^L}T \longrightarrow {^L}G, \end{align*}$$

where the first map is the L-parameter of the character $\chi $ of $T(F)$ . By the discussion in [Reference Debacker and ReederDR, §4.3], one has

$$\begin{align*}{\mathrm{Hom}}(T(k), \overline{\mathbb{Q}}_l^{\times}) \cong {\mathrm{Hom}}_{\mathrm{Frob}}(I_t , \hat{T}) \cong {\mathrm{Hom}}_{\mathrm{Frob}}(k_n^{\times}, \hat{T}), \end{align*}$$

where $k_n = \mathbb {F}_{q^n}$ is the splitting field of T over k, $I_t$ is the tame inertia group of $\Gamma _0$ and ${\mathrm {Hom}}_{\mathrm {Frob}}$ stands for Frobenius-equivariant homomorphisms. Hence, the restriction of ${\mathcal {L}}(\Pi )^{ss}_0$ to the inertia group (which factors to the tame inertia group) determines and is determined by the restriction $\chi _k$ of $\chi $ to $T(k)$ . The special form of $\chi _k$ given in property (c) and the fact that $\alpha $ is fixed by ${\mathrm {Frob}}$ thus imply that the restriction of $ {\mathcal {L}}(\Pi )^{ss}_0 $ to $I_t$ factors as

(7.4)

Let $V= \otimes _{i=1}^n {\mathfrak {v}}_i$ be the representation of ${^L}G = {\hat {G}} \rtimes W_{k}$ where $\hat {G}$ acts on each ${\mathfrak {v}}_i$ and $W_i$ permutes the factors. The composite $V \circ {\mathcal {L}}(\Pi )^{ss}$ gives rise to a (semisimple) local system ${\mathcal {V}}(\Pi ,V)$ on ${\mathbb {P}}^1 \setminus |{\mathbb {P}}^1(k)|$ over k. However, since $ {\mathcal {L}}(\Pi )^{ss}_z$ is unramified, the purity of ${\mathcal {L}}^{ss}(\sigma )$ and Theorem 4.2 then imply that ${\mathcal {V}}(\Pi ,V)$ extends to a local system on ${\mathbb {G}}_m$ over k.

The representation V is obtained as a tensor product of the representations ${\mathfrak {v}}_i$ of $\hat {G}$ , which do not extend to representations of $\hat {G} \rtimes W_k$ , but do extend to representations of $\hat {G} \rtimes W_{k_n}$ for $k_n$ the degree n extension, since the stabilizer of ${\mathfrak {v}}_i$ in the action of $W_k$ on irreducible factors of the adjoint representation of $\hat {G}$ is $W_{k_n}$ . Thus, we similarly have that the composite ${\mathfrak {v}}_i \circ {\mathcal {L}}(\Pi )^{ss}\mid _{ \mathrm {Gal} (K^{sep} / k_n( t))}$ gives rise to a (semisimple) local system ${\mathcal {V}}(\Pi ,{\mathfrak {v}}_i)$ on ${\mathbb {P}}^1 \setminus |{\mathbb {P}}^1(k)|$ over $k_n$ , which extends to a local system on ${\mathbb G}_m$ over $k_n$ .

The above discussion shows that the local system ${\mathcal {V}}(\Pi ,{\mathfrak {v}}_i)$ on ${\mathbb {G}}_m$ over $k_n$ satisfies the hypotheses of Corollary 4.3. In particular,

1. (a) the local monodromy representations at $0$ and $\infty $ are tamely ramified;

2. (b) the local monodromy representation at $0$ has abelian image.

Corollary 4.3 thus implies that ${\mathcal {V}}(\Pi ,{\mathfrak {v}}_i )$ breaks up as the sum of rank 1 local systems ${\mathcal {L}}_1, \dots , {\mathcal {L}}_N$ over $k_n$ .

Applying Lemma 4.7 to the ${\mathcal {L}}_i$ ’s, and a standard formula for the trace on a tensor product representation of an operator that permutes the tensor factors, one sees that for each $z \in \mathbb {G}_m(k)$ ,

(7.5) $$ \begin{align} \operatorname{tr}(\operatorname{Frob}_z, {\mathcal{V}}(\Pi,V)) = \operatorname{tr}(\operatorname{Frob}_z^n, {\mathcal{V}}(\Pi,{\mathfrak{v}}_1)) = \sum_{j=1}^N \operatorname{tr}(\operatorname{Frob}_z^n, \mathcal L_j) = \sum_{j=1}^N \operatorname{tr}(\operatorname{Frob}_1^n, \mathcal L_j) \cdot \operatorname{tr}(m_z^{-n} , \mathcal L_j).\end{align} $$

Here, it is crucial that we consider the tensor induction instead of the usual induction because the trace of $\operatorname {Frob}_z$ on the usual induction would be identically zero as soon as $n>1$ , and so studying this trace would not be helpful. We take $i=1$ without loss of generality because $\operatorname {tr}(\operatorname {Frob}_z^n, {\mathcal {V}}(\Pi ,{\mathfrak {v}}_i))$ is independent of i.

Let us now evaluate $ \operatorname {tr}(m_z , \mathcal L_j)$ . By (7.4), and observing that the eigenvalues of $\hat {T}$ on ${\mathfrak {v}}_1 $ are the roots of $({\hat {G}}, \hat {T})$ contained in the simple factor corresponding to ${\mathfrak {v}}_1$ , we see that

(7.6) $$ \begin{align} \operatorname{tr}(m_z , \mathcal L_j ) = \mu(z)^{ \langle \alpha, \beta_j \rangle, } \end{align} $$

where $\beta _j$ is either a root of ${\hat {G}}$ (or equivalently a coroot of G) contained in ${\mathfrak {v}}_1$ or the zero element in $X^*(\hat {T})$ .

Now note that

1. (a) $\mu $ is a character of order $q-1$ of $k^{\times }$ ;

2. (b) the function

   $$\begin{align*}z \mapsto \operatorname{tr}(\operatorname{Frob}_z, {\mathcal{V}}(\Pi,{V})) \end{align*}$$
   is a constant function of z, since the local representation at each $z \in \mathbb {G}_m(k)$ is just ${\mathcal {L}}^{ss}(\sigma )$ .

By orthogonality of characters (of $k^{\times }$ ) and (7.5), we thus have

(7.7) $$ \begin{align} 0 &= \sum_{z \in k^\times} \mu(z)^{-2} \cdot \operatorname{tr}(\operatorname{Frob}_z, {\mathcal{V}}(\Pi,{V})) \notag \\ &= \sum_{z \in k^\times} \mu(z)^{-2} \cdot \sum_{j=1}^N \operatorname{tr}(\operatorname{Frob}_1^n, \mathcal L_j) \cdot \mu(z)^{ \langle \alpha, \beta_j \rangle } \notag \\ &= \sum_{j=1}^N \operatorname{tr}(\operatorname{Frob}_1^n, \mathcal L_j) \cdot \sum_{z \in k^\times} \mu(z)^{-2} \cdot \mu(z)^{ \langle \alpha, \beta_j \rangle } \notag \\ &= (q-1) \cdot \sum_{\substack{ j: \\ \langle \alpha, \beta_j \rangle \equiv 2\quad \mod (q-1) }} \operatorname{tr}(\operatorname{Frob}_1^n, \mathcal L_j). \end{align} $$

Furthermore, we have

$$\begin{align*}\langle \alpha, \beta_j \rangle = \langle \sum_{i=1}^n \alpha_i, \beta_j \rangle =\sum_{i=1}^n \langle \alpha_i,\beta_j\rangle = \langle \alpha_1,\beta_j \rangle \end{align*}$$

since for $i \neq 1$ , $\alpha _i$ is a root of a different simple factor from $\beta _j$ and thus is orthogonal to $\beta _j$ .

Now one of the $\beta _j$ ’s is the highest weight $\alpha _1^\vee $ of ${\mathfrak {v}}_1$ , for which one has $\langle \alpha , \alpha _1^{\vee } \rangle = \langle \alpha _1,\alpha _1^\vee \rangle =2$ . For all other $\beta _j$ , one has

$$\begin{align*}-2 \leq \langle \alpha, \beta_j \rangle < 2. \end{align*}$$

Hence, since $\langle \alpha , \beta _j \rangle \in \mathbb {Z}$ , we have

$$\begin{align*}\langle \alpha, \beta_j \rangle \in \{-2,-1,0,1\}. \end{align*}$$

Since we have assumed $q> 5$ , this implies that there is only one term appearing in the sum in (7.7) – namely, the term corresponding to the highest root $\alpha _1^{\vee }$ of ${\hat {G}}$ . Hence, we deduce the desired contradiction

$$\begin{align*}(q-1) \cdot \operatorname{tr}(\operatorname{Frob}_1^n, \mathcal L_j) = 0, \end{align*}$$

which is impossible as $\mathcal L_j$ is a one-dimensional representation.

We have thus completed the proof of Theorem 7.2.

Proof. The construction follows the method of [Reference Gan and LomelíGLo]. Hypothesis (7.10) implies that $\sigma $ has a matrix coefficient $\varphi ^*$ with support in $U^*$ , with the property that $\varphi ^*(1) = 1$ . Using this $U^*$ , let $U = \prod _x U_x \subset G({\mathbb A}_K)$ be the open compact subgroup constructed in Proposition 6.1. Define a function $\varphi = \prod _x \varphi _x: G({\mathbb A}_K) {~\rightarrow ~} {\mathbb C}$ as follows:

1. (a) At every $z \in {\mathbb G}_m(k) \subset {\mathbb {P}}^1(k)$ , $\varphi _z = \varphi ^*$ ;

2. (b) $\varphi _\infty $ is the characteristic function of the group $I_\infty (p)$ of Corollary 6.2;

3. (c) $\varphi _0$ is the character $\chi _k: I_0/I_0(p) \cong T(k) {~\rightarrow ~} {\mathbb C}^\times $ , where $\chi _k$ is as in the proof of Theorem 7.2;

4. (d) For $x \notin |{\mathbb {P}}^1(k)|$ , $\varphi _x$ is the characteristic function of the hyperspecial maximal compact subgroup $G({\mathcal {O}}_x)$ ;

Now define the Poincaré series $P_\varphi : G(K)\backslash G({\mathbf A}_K) {~\rightarrow ~} {\mathbb C}$ :

$$ \begin{align*}P_\varphi(g) = \sum_{\gamma \in G(K)} \varphi(\gamma\cdot g).\end{align*} $$

The sum converges absolutely, as in [Reference Gan and LomelíGLo]. It follows from Corollary 6.2 that

$$ \begin{align*}P_\varphi(1) = \prod_{z \in {\mathbb G}_m(k)} \varphi^*(1) = 1.\end{align*} $$

In particular, $P_\varphi \neq 0$ . The spectral decomposition of $P_{\varphi }$ yields the desired cuspidal automorphic representation $\Pi $ .

Proof. As mentioned at the beginning of the section, we may assume G is simply-connected. Thanks to the results of [Reference FintzenFi], the supercuspidal representation $\sigma $ of $G(F)$ contains an $\mathfrak {s}$ -type by our hypothesis on p. The claim now follows immediately from Theorems 7.2 and 7.9.

Proof. In any case, $\sigma $ is an irreducible constituent of a representation of the form $Ind_{P(F)}^{G(F)} \tau $ for some parabolic subgroup $P(F) = L(F)\cdot N(F) \subset G(F)$ , where $L(F)$ is a Levi subgroup of $P(F)$ , $N(F)$ its unipotent radical, and $\tau $ is a supercuspidal representation of $L(F)$ . Since the Genestier-Lafforgue parametrization is compatible with parabolic induction by Theorem 2.3, it follows that ${\mathcal {L}}^{ss}(\tau )$ is unramified. Since $\tau $ is supercuspidal, it follows from Theorems 7.9 and 7.2 (or more simply Corollary 7.11) that L must be a torus and $\tau $ an unramified character of $L(F)$ .

Proof. Let $\lambda \colon \mathbb G_m \to S$ be a cocharacter which is a negative multiple of $a+b$ . (This exists because $a+b$ is a small rational multiple of a cocharacter, and we may multiply by any negative integer that cancels the denominator of that rational to obtain a cocharacter.) The image of $\lambda $ in $G_{a}/G_{a,0+}$ continues to define a cocharacter of the maximal split torus of $G_{a}/G_{a, 0+}$ . Since $\lambda $ lies in the interior of a Weyl chamber, it defines a Borel subgroup $B_{\lambda }$ over k (under the ‘dynamical’ definition of parabolic subgroups). We shall show that the image of $C \cap G(K)$ in $G_a/G_{a,0+}$ belongs to $B_{\lambda }$ . For this, we need to show that for any $g \in C \cap G(K)$ ,

$$\begin{align*}\lambda(t) \cdot g \cdot \lambda(t)^{-1} \text{ remains bounded as } t \to 0. \end{align*}$$

To verify this, we shall apply Lemma 5.1 to the faithful algebraic representation $\rho \colon G \to \operatorname {Aut}(V)$ of G over k that was fixed in §5.1. Recall that V is equipped with a k-basis $e_1,\dots , e_n$ of V, where $e_i$ is an eigenvector of S with eigencharacter $\lambda _i$ . For any $g \in C \cap G(K)$ and $(i,j)$ such that $\rho _{ij}(g) \ne 0$ , we deduce by the product formula, Lemma 5.1, and the definition of $f(v)$ , that

(8.4) $$ \begin{align} 0 = \sum_v v (\rho_{ij}(g)) \geq \sum_v \lceil f(v) \cdot (\lambda_i -\lambda_j) \rceil = \lceil a \cdot (\lambda_i -\lambda_j) \rceil + \lceil b \cdot (\lambda_i -\lambda_j ) \rceil .\end{align} $$

Now since b is a small perturbation of $-a$ , (8.4) can only be satisfied if $ a \cdot (\lambda _i - \lambda _j) $ is an integer and $b \cdot (\lambda _i -\lambda _j) \leq -a \cdot (\lambda _i - \lambda _j)$ . Indeed, b is a small perturbation of $-a$ , so we may assume that $b \cdot (\lambda _i-\lambda _j)$ is arbitrarily close to $-a \cdot (\lambda _i - \lambda _j)$ . In particular, if $a \cdot (\lambda _i-\lambda _j)$ lies in the interval $(n,n+1)$ for some integer n, then $b \cdot (\lambda _i-\lambda _j)$ lies in the interval $(-n-1,-n)$ , and so the sum of their ceilings is $ n+1 + (-n) = 1>0$ .

With $\lambda $ chosen as in the beginning of the proof, we thus deduce that for $g\in C \cap G(K)$ such that $\rho _{ij}(g) \neq 0$ , we have $(a+b) \cdot (\lambda _i - \lambda _j ) \leq 0$ , and thus,

$$\begin{align*}\lambda\cdot (\lambda_i -\lambda_j) \geq 0. \end{align*}$$

It follows that $\lambda (t) g \lambda (t)^{-1}$ is bounded as $t \to 0$ , since all the nonvanishing coordinates $\rho _{ij}$ of g are eigenvectors under the conjugation action by $\lambda (t)$ with eigenvalue a positive power of t. This implies that the image of g in $G_{a}/G_{a,0+}$ lies in the Borel subgroup $B_{\lambda }$ , as desired.

It remains to show that if $g \in C \cap G(K)$ has trivial image in $G_a/ G_{a, 0+}$ , then g is the identity element. If $\rho _{ij}(g) \neq 0$ , then because $ a \cdot (\lambda _i - \lambda _j) $ is an integer and $b \cdot (\lambda _i -\lambda _j) \leq -a \cdot (\lambda _i - \lambda _j)$ , the right side of (8.4) is $0$ . So every inequality is sharp; that is,

$$\begin{align*}v_0(\rho_{ij}(g)) = a \cdot(\lambda_i - \lambda_j). \end{align*}$$

However, by Lemma 5.1(ii), if $g \in G_{a, 0+}$ , then

$$\begin{align*}v_0(\rho_{ij}(g)) - \delta_{ij})> a \cdot (\lambda_i -\lambda_j). \end{align*}$$

Thus, if $i \neq j$ , we obtain a contradiction from the assumption $\rho _{ij}(g) \neq 0$ . Hence, $\rho _{ij}(g) =0$ for $i \ne j$ . However, if $i=j$ , then $\rho _{ij}(g) \in \mathcal {O}_v$ for every place of K, and thus lies in k, but since $v_0(\rho _{ii}(g) -1)> 0$ , we see that $\rho _{ii}(g) =1$ . Thus, if $g \in G_{0,a+}$ , then $\rho (g)$ is the identity, so that g is the identity since $\rho $ is faithful.

Proof. We construct a Poincaré series as in Theorem 7.9 using a function $\varphi $ which at $\infty $ is a matrix coefficient of $\sigma $ and is a characteristic function of the standard hyperspecial maximal compact $U_v = G(\mathcal {O}_v)$ at all other places v. Now the support of $\varphi _\infty $ is contained in J, and $J \times \otimes _{v \neq \infty }U_v \cap G(K) = \{1\}$ . Thus, the Poincaré series $P_\varphi $ does not vanish at $1$ , and the space of cusp forms it generates contains a cuspidal automorphic representation $\Pi $ that is unramified except at $\infty $ . So ${\mathcal {L}}(\Pi )$ is a parameter with values in ${^L}G$ that is unramified away from $\infty $ . However, by Theorem 7.2, ${\mathcal {L}}^{ss}(\pi )$ is ramified. Since a tamely ramified local system on $\mathbb {A}^1$ is unramified, it follows that the ramification at $\infty $ must be wild.

Proof. Since $\sigma $ is a generic representation of $G(k)$ , one can find a generic character