2.1 Unitary group

We define an integral unitary similitude group, which is associated to the following data. If $\Psi _n$ denotes the $n \times n$ matrix with $1$ ’s on the anti-diagonal and $0$ ’s elsewhere, then let $J_n$ denote the following element of $\operatorname {GL}_{2n}(\mathbb {Z})$ :

$$ \begin{align*}J_n = \left(\begin{array}{cc}0 & \Psi_n \\-\Psi_n & 0\end{array}\right).\end{align*} $$

Let $\mathcal {D}_F^{-1}$ denote the inverse different of $\mathcal {O}_F$ , and define the $2n$ -dimensional lattice $\Lambda = (\mathcal {D}_F^{-1})^{n} \oplus \mathcal {O}_F^n$ . Let G be the group scheme over $\mathbb {Z}$ defined by

$$ \begin{align*}G(R) = \{(g,\mu) \in \operatorname{Aut}_{\mathcal{O}_F \otimes_{\mathbb{Z}} R}(\Lambda \otimes_{\mathbb{Z}} R) \times R^\times : {}^tg J_n{}^cg = \mu J_n\}\end{align*} $$

for any ring R. Over $\mathbb {Z}[1/\operatorname {Disc}(F/\mathbb {Q})]$ , it is a quasi-split connected reductive group which splits over $\mathcal {O}_{\widetilde {F}}[1/\operatorname {Disc}(F/\mathbb {Q})]$ , where $\widetilde {F}$ denotes the normal closure of $F/\mathbb {Q}$ . Let $\nu : G \rightarrow \operatorname {GL}_1$ be the multiplier character sending $(g,\mu ) \mapsto \mu $ .

If $R = \Omega $ is an algebraically closed field of characteristic 0, then

$$ \begin{align*}G \times \operatorname{Spec} \Omega \cong \{ (\mu,g_\tau) \in \mathbb{G}_m \times \operatorname{GL}_{2n}^{\operatorname{Hom}(F,\Omega)} : g_{\tau c} = \mu J_n {}^tg_{\tau}^{-1} J_n \quad \forall \tau \in \operatorname{Hom}(F,\Omega)\}.\end{align*} $$

Fix the lattice $\Lambda _{(n)} \cong (\mathcal {D}_F^{-1})^n$ consisting of elements of $\Lambda $ whose last n coordinates are equal to 0, and define $\Lambda _{(n)}^{\prime } \cong \mathcal {O}_F^n$ consisting of elements of $\Lambda $ whose first n coordinates are equal to $0$ . Let $P^+_{(n)}$ denote the subgroup of G preserving $\Lambda _{(n)}$ . Write $L_{(n),\operatorname {lin}}$ for the subgroup of $P_{(n)}^+$ consisting of elements with $\nu = 1$ which preserve $\Lambda _{(n)}^{\prime }$ , and write $L_{(n),\operatorname {herm}}$ for the subgroup of $P_{(n)}^+$ which act trivially on $\Lambda /\Lambda _{(n)}$ and preserve $\Lambda _{(n)}^{\prime }$ . Then $L_{(n),\operatorname {lin}} \cong \operatorname {RS}^{\mathcal {O}_F}_{\mathbb {Z}} \operatorname {GL}_n$ and $L_{(n),\operatorname {herm}} \cong \mathbb {G}_m$ , and we can define $L_{(n)} := L_{(n),\operatorname {lin}} \times L_{(n),\operatorname {herm}}$ .

Finally, let $G(\mathbb {A}^\infty )^{\operatorname {ord},\times } := G(\mathbb {A}^{p,\infty }) \times P^+_{(n)}(\mathbb {Z}_p)$ , and $G(\mathbb {A}^{\infty })^{\operatorname {ord}} = G(\mathbb {A}^{p,\infty }) \times \varsigma _p^{\mathbb {Z}_{\geq 0}} P_{(n)}^+(\mathbb {Z}_p),$ where $\varsigma _p \in L_{(n),\operatorname {herm}}(\mathbb {Q}_p) \cong \mathbb {Q}_p^\times $ denotes the unique element with multiplier $p^{-1}$ .

2.2 Level structure

If $N_2 \geq N_1 \geq 0$ are integers, then let $U_p(N_1,N_2)$ be the subgroup of elements of $G(\mathbb {Z}_p)$ which ${\operatorname {mod}} p^{N_2}$ lie in $P^+_{(n)}(\mathbb {Z}/p^{N_2}\mathbb {Z})$ and map to 1 in $L_{(n),\operatorname {lin}}(\mathbb {Z}/p^{N_1}\mathbb {Z})$ . If $U^p$ is an open compact subgroup of $G(\mathbb {A}^{p,\infty })$ , we write $U^p(N_1, N_2)$ for $U^p \times U_p(N_1,N_2)$ .

If $N \geq 0$ is an integer, we write $U_p(N)$ for the kernel of the map $P^{+}_{(n)}(\mathbb {Z}_p) \rightarrow L_{(n),\operatorname {lin}}(\mathbb {Z}/p^N\mathbb {Z})$ . In addition, $U_p(N)$ will also denote the image of this kernel inside $L_{(n),\operatorname {lin}}(\mathbb {Z}_p)$ .

2.3 Shimura variety

Fix a neat open compact subgroup U (as defined in section 0.6 of Pink [Reference Pink16]), and let S be a locally noetherian scheme over $\mathbb {Q}$ . Recall from §3.1 in [Reference Harris, Lan, Taylor and Thorne10] that a polarized G-abelian scheme with U-level structure is an abelian scheme A over S of relative dimension $n\cdot [F:\mathbb {Q}]$ along with the following data:

• • An embedding $\imath : F \hookrightarrow \operatorname {End}^0(A)$ such that $\operatorname {Lie} A$ is locally free of rank n over $F \otimes _{\mathbb {Q}} \mathcal {O}_S$ .

• • A polarization $\lambda : A \rightarrow A^\vee $

• • U-level structure $[\eta ]$ .

For more precise definitions, see §3.1.1 of [Reference Harris, Lan, Taylor and Thorne10]. Denote by $X_{U}$ the smooth quasi-projective scheme over $\mathbb {Q}$ which represents the functor that sends a locally noetherian scheme $S/\mathbb {Q}$ to the set of quasi-isogeny classes of polarized G-abelian schemes with U-level structure. Let $[(A^{\operatorname {univ}},\imath ^{\operatorname {univ}},\lambda ^{\operatorname {univ}},[\eta ^{\operatorname {univ}}])]$ denote the universal equivalence class of polarized G-abelian varieties with U-level structure. Allowing U to vary, the inverse system $\{X_{U}\}$ has a right $G(\mathbb {A}^\infty )$ action, with finite etale transition maps $g: X_{U} \rightarrow X_{U^{\prime }}$ whenever $U^{\prime } \supset g^{-1}Ug$ .

For each U, denote by $\Omega ^1_{A^{\operatorname {univ}}/X_{U}}$ the sheaf of relative differentials on $A^{\operatorname {univ}}$ . Let $\Omega _U$ denote the Hodge bundle (i.e., the pullback by the identity section of $\Omega ^1_{A^{\operatorname {univ}}/X_U}$ ). It is locally free of rank $n\cdot [F:\mathbb {Q}]$ and does not depend on $A^{\operatorname {univ}}$ .

For each neat open compact subgroup $U \subset G(\mathbb {A}^\infty )$ , there is a normal projective scheme $X^{\operatorname {min}}_{U}$ over $\operatorname {Spec} \mathbb {Q}$ together with a $G(\mathbb {A}^\infty )$ -equivariant dense open embedding

$$ \begin{align*}j_U: X_U \hookrightarrow X^{\operatorname{min}}_U,\end{align*} $$

which is known as the minimal compactification of $X_U$ . Let the boundary be denoted by $\partial X^{\operatorname {min}}_U = X^{\operatorname {min}}_U \smallsetminus j_{U} X_{U}$ . The inverse system $\{X^{\operatorname {min}}_U\}$ also has a right $G(\mathbb {A}^\infty )$ -action. Furthermore, there is a normal projective flat $\mathbb {Z}_{(p)}$ scheme $\mathcal {X}^{\operatorname {min}}_{U}$ whose generic fiber is $X^{\operatorname {min}}_U$ . We will denote the ample line bundle on $\mathcal {X}^{\operatorname {min}}_{U}$ constructed in Propositions 2.2.1.2 and 2.2.3.1 in Lan [Reference Lan14] by $\omega _U$ . Its pullback to $X_U$ is identified with $\wedge ^{n[F:\mathbb {Q}]}\Omega _U$ , and the system $\{\omega _U\}$ over $\{\mathcal {X}_{U}^{\operatorname {min}}\}$ has an action of $G(\mathbb {A}^{p,\infty } \times \mathbb {Z}_p)$ . If we let $\overline {X}_U^{\operatorname {min}} = \mathcal {X}_U^{\operatorname {min}} \otimes _{\mathbb {Z}_{(p)}} \mathbb {F}_p$ , there is a canonical $G(\mathbb {A}^{p,\infty })$ -invariant section $\operatorname {Hasse}_U \in H^0(\overline {X}_U^{\operatorname {min}}, \omega _U^{\otimes (p-1)})$ constructed in Corollaries 6.3.1.7-8 in [Reference Lan14] satisfying

$$ \begin{align*}g^\ast \operatorname{Hasse}_{g^{-1}Ug} = \operatorname{Hasse}_U \quad \forall g \in G(\mathbb{A}^{p,\infty} \times \mathbb{Z}_p).\end{align*} $$

Denote by $\overline {X}^{\operatorname {min} n-\operatorname {ord}}_U$ the zero locus in $\overline {X}^{\operatorname {min}}_U$ of $\operatorname {Hasse}_U$ .

Proof. The nonzero locus over $\overline {X}^{\operatorname {min}}_U$ is associated to the sheaf of algebras

$$ \begin{align*}\left(\oplus_{i=0}^\infty \omega_U^{\otimes (p-1)ai}\right)/(\operatorname{Hasse}^a_U - 1) \quad \forall a \in \mathbb{Z}_{>0}.\end{align*} $$

Over $\mathbb {F}_p$ , it is associated to the algebra

$$ \begin{align*}\left(\oplus_{i=0}^\infty H^0(\overline{X}^{\operatorname{min}}_U,\omega^{\otimes(p-1)ai})\right)/(\operatorname{Hasse}_U^a - 1) \quad \forall a \in \mathbb{Z}_{>0}.\end{align*} $$

We conclude the lemma.

2.4 Ordinary locus

Now let $\mathcal {S}$ denote a locally Noetherian scheme over $\mathbb {Z}_{(p)}$ , and fix a neat open compact subgroup $U^p$ along with two positive integers $N_2 \geq N_1$ . Then the ordinary locus is a smooth quasi-projective scheme $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)}$ over $\mathbb {Z}_{(p)}$ representing the functor which sends $\mathcal {S}$ to the the set of prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized G-abelian schemes with $U^p(N_1,N_2)$ -level structure as defined in §3.1 of [Reference Harris, Lan, Taylor and Thorne10]. It is a partial integral model of $X_{U^p(N_1,N_2)}$ . Let $[\mathcal {A}^{\operatorname {univ}},\imath ^{\operatorname {univ}},\lambda ^{\operatorname {univ}},[\eta ^{\operatorname {univ}}]]/\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)}$ denote the universal equivalence class of ordinary prime-to-p quasi-polarized G-abelian schemes with $U^p(N_1,N_2)$ -level structure up to quasi-isogeny. Finally, let $\overline {X}^{\operatorname {ord}}_{U^p(N_1,N_2)} = \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)} \otimes _{\mathbb {Z}_{(p)}} \mathbb {F}_p$ , which forms an inverse system each with a right $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -action. Furthermore, the map

$$ \begin{align*}\varsigma_p: \overline{X}^{\operatorname{ord}}_{U^p(N_1,N_2+1)} \rightarrow \overline{X}^{\operatorname{ord}}_{U^p(N_1,N_2)}\end{align*} $$

is the absolute Frobenius map composed with the forgetful map $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2+1)} \rightarrow \mathcal {X}^{\operatorname {ord}}_{U^p(N_2,N_2)}$ for any $N_2 \geq N_1 \geq 0$ . If $N_2> 0$ , then $\varsigma _p$ defines a finite flat map

$$ \begin{align*}\varsigma_p: \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2+1)} \rightarrow \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2)}\end{align*} $$

with fibers of degree $p^{n^2[F^+:\mathbb {Q}]}$ (see §3.1 of [Reference Harris, Lan, Taylor and Thorne10]).

For each $U^p(N_1,N_2)$ such that $U^p$ is neat, there is a partial minimal compactification of the ordinary locus $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)}$ denoted by $\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)}$ . By Theorem 6.2.1.1 in [Reference Lan14], this compactification of the ordinary locus is a normal quasi-projective scheme over $\mathbb {Z}_{(p)}$ together with a dense open $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -equivariant embedding

$$ \begin{align*}j_{U^p(N_1,N_2)}: \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2)} \hookrightarrow \mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)}.\end{align*} $$

Its generic fiber is $X^{\operatorname {min}}_{U^p(N_1,N_2)}$ , but unlike $\mathcal {X}^{\operatorname {min}}_{U^p(N_1,N_2)}$ , it is not proper. Furthermore, by Proposition 6.2.2.1 in [Reference Lan14], the induced action of $g \in G(\mathbb {A}^\infty )^{\operatorname {ord}}$ on $\{\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)}\}$ is quasi-finite. Write $\partial \mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)} = \mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)} - j_{U^p(N_1,N_2)} \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)}$ for the boundary, and let $\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)}$ be the formal completion along the special fiber of $\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)}$ . Note that by Corollary 6.2.2.8 and Example 3.4.5.5 in [Reference Lan14], the natural map

$$ \begin{align*}\mathfrak{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2^{\prime})} \stackrel{\sim}{\longrightarrow} \mathfrak{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)}\end{align*} $$

is an isomorphism, and so we will drop $N_2$ from notation. Define

$$ \begin{align*}\overline{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)} = \mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)} \otimes_{\mathbb{Z}_{(p)}} \mathbb{F}_p.\end{align*} $$

For each $U^p(N_1,N_2)$ , note that there are $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ -equivariant open embeddings

$$ \begin{align*}\mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)} \hookrightarrow \mathcal{X}^{\operatorname{min}}_{U^p(N_1,N_2)}.\end{align*} $$

This induces a map on the special fibers

(2.1) $$ \begin{align}\overline{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)} \hookrightarrow \overline{X}^{\operatorname{min}}_{U^p(N_1,N_2)} \backslash \overline{X}^{\operatorname{min},n-\operatorname{ord}}_{U^p(N_1,N_2)},\end{align} $$

which is both an open and closed embedding by Proposition 6.3.2.2 of [Reference Lan14]. Note that only when the level is prime-to-p is the nonzero locus of $\operatorname {Hasse}_{U^p(N_1,N_2)}$ isomorphic to the special fiber of the minimally compactified ordinary locus. When $N_2> 0$ , the map in (2.1) is not an isomorphism.

2.5 Toroidal compactifications

We now introduce toroidal compactifications of $X_U$ and $\mathcal {X}_{U^p(N_1,N_2)}$ , which are parametrized by neat open compact subgroups of $G(\mathbb {A}^\infty )$ and certain cone decompositions defined in [Reference Lan14] and [Reference Harris, Lan, Taylor and Thorne10]. Let $\mathcal {J}^{\operatorname {tor}}$ be the indexing set of pairs $(U,\Delta )$ defined in Proposition 7.1.1.21 in [Reference Lan14] or on pages 169–170 in [Reference Harris, Lan, Taylor and Thorne10], where U is a neat open compact subgroup and $\Delta $ is a U-admissible cone decomposition as defined in §5.2 of [Reference Harris, Lan, Taylor and Thorne10]. We will not recall the definition here as it is not necessary for any argument.

If $(U,\Delta ) \in \mathcal {J}^{\operatorname {tor}}$ , then by Theorem 1.3.3.15 of [Reference Lan14], there is a smooth projective scheme $X_{U,\Delta }$ and a divisor with simple normal crossings $\partial X_{U,\Delta } \subset X_{U,\Delta }$ equipped with an isomorphism

$$ \begin{align*}j_{U,\Delta}: X_{U} \stackrel{\sim}{\longrightarrow} X_{U,\Delta} \smallsetminus \partial X_{U,\Delta}\end{align*} $$

and a projection $\pi _{\operatorname {tor}/\operatorname {min}}: X_{U,\Delta } \rightarrow X^{\operatorname {min}}_U$ such that the following diagram commutes:

$$ \begin{align*} \begin{aligned} X_{U} &\hookrightarrow X_{U,\Delta} \\ \downarrow & \ \ \ \ \ \ \downarrow \\ X_U & \hookrightarrow X_U^{\operatorname{min}}. \end{aligned} \end{align*} $$

The collection $\{X_{U,\Delta }\}_{\mathcal {J}^{\operatorname {tor}}}$ becomes a system of schemes with a right $G(\mathbb {A}^\infty )$ -action via the maps $\pi _{(U,\Delta )/(U^{\prime },\Delta ^{\prime })}: X_{U,\Delta } \rightarrow X_{U^{\prime },\Delta ^{\prime }}$ whenever $(U,\Delta ) \geq (U^{\prime },\Delta ^{\prime })$ (see page 166 of [Reference Harris, Lan, Taylor and Thorne10] for the definition of $\geq $ in this context).

If $(U^p(N_1,N_2),\Delta ) \in \mathcal {J}^{\operatorname {tor}}$ , then by Theorem 7.1.4.1 of [Reference Lan14], there is a smooth quasi-projective scheme $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ and a divisor with simple normal crossings $\partial \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta } \subset \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ equipped with an isomorphism

$$ \begin{align*}j^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta}: \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2)}\stackrel{\sim}{\longrightarrow} \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta} \smallsetminus \partial \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta};\end{align*} $$

$$ \begin{align*}\pi^{\operatorname{ord}}_{\operatorname{tor}/\operatorname{min}}: \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta} \rightarrow \mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)}\end{align*} $$

such that the following diagram commutes:

$$ \begin{align*} \begin{aligned} \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2)} &\hookrightarrow \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta} \\ \downarrow & \ \ \ \ \ \ \downarrow \\ \mathcal{X}^{\operatorname{ord}}_{U^p)N_1,N_2)}& \hookrightarrow \mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)}. \end{aligned} \end{align*} $$

The collection $\{\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }\}_{\mathcal {J}^{\operatorname {tor}}}$ becomes a system of schemes with a right $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -action via the maps $\pi _{(U^p(N_1,N_2),\Delta )/(U^{p^{\prime }}(N_1^{\prime },N_2^{\prime }),\Delta ^{\prime })}: \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta } \rightarrow \mathcal {X}^{\operatorname {ord}}_{U^{p^{\prime }}(N_1^{\prime },N_2),\Delta ^{\prime }}$ whenever $(U^p(N_1,N_2),\Delta ) \geq (U^{p^{\prime }}(N_1^{\prime },N_2),\Delta ^{\prime })$ (see page 167 of [Reference Harris, Lan, Taylor and Thorne10] for the definition of $\geq $ in this context).

3.1 Automorphic bundles on compactifications of the Shimura variety

Recall from the previous section that we have a locally free sheaf $\Omega _U$ on $X_U$ , which is the pullback by the identity section of the sheaf of relative differentials from $A^{\operatorname {univ}}$ , the universal abelian variety over $X_U$ . On $\mathcal {X}^{\operatorname {min}}_U$ , the normal integral model of the minimal compactification of $X_U$ , there is an ample line bundle $\omega _U$ whose pullback to $X_U$ is identified with $\wedge ^{n[F:\mathbb {Q}]} \Omega _U$ .

Any universal abelian variety $A^{\operatorname {univ}}/X_{U}$ extends to a semi-abelian variety $A_\Delta /X_{U,\Delta }$ (see remarks 1.1.2.1 and 1.3.1.4 of [Reference Lan14]). Define $\Omega _{U,\Delta }$ as the pullback by the identity section of the sheaf of relative differentials on $A_{\Delta }$ . Note that when restricting to the Shimura variety $X_U$ , the sheaf $\left .\Omega _{U,\Delta }\right |{}_{X_U}$ is canonically isomorphic to $\Omega _U$ . Let $\mathcal {O}_{X_{U,\Delta }}(||\nu ||)$ denote the structure sheaf with $G(\mathbb {A}^\infty )$ -action twisted by $||\nu ||$ .

Let $\mathcal {E}^{\operatorname {can}}_{U,\Delta }$ denote the principal $L_{(n)}$ -bundle on $X_{U,\Delta }$ , defined as follows: For a Zariski open W, $\mathcal {E}^{\operatorname {can}}_{U,\Delta }(W)$ is the set of pairs of isomorphisms

$$ \begin{align*}\xi_0: \left.\mathcal{O}_{X_{U,\Delta}}(||\nu||)\right|_W \stackrel{\sim}{\longrightarrow} \mathcal{O}_W \qquad \text{and} \qquad \xi_1: \Omega_{U,\Delta} \stackrel{\sim}{\longrightarrow} \operatorname{Hom}_{\mathbb{Q}}(V/V_{(n)},\mathcal{O}_W),\end{align*} $$

where $V = \Lambda \otimes \mathbb {Q} = F^{2n}$ and $V_{(n)} = \Lambda _{(n)} \otimes \mathbb {Q} \cong F^{n}$ . There is an action of $h \in L_{(n)}$ on $\mathcal {E}^{\operatorname {can}}_{U,\Delta }$ by

$$ \begin{align*}h(\xi_0,\xi_1) = (\nu(h)^{-1} \xi_0, \xi_1 \circ h^{-1}).\end{align*} $$

The inverse system $\{\mathcal {E}^{\operatorname {can}}_{U,\Delta }\}$ has an action of $G(\mathbb {A}^\infty )$ .

Let R be any $\mathbb {Q}$ -algebra. Fix a representation $\rho $ of $L_{(n)}$ on a finite, locally free R-module $W_{\rho }$ . Define the locally free sheaf $\mathcal {E}^{\operatorname {can}}_{U,\Delta ,\rho }$ over $X_{U,\Delta } \times \operatorname {Spec} R$ as follows: For a Zariski open W, let $\mathcal {E}^{\operatorname {can}}_{U,\Delta ,\rho }(W)$ be the set of $L_{(n)}(\mathcal {O}_W)$ -equivariant maps of Zariski sheaves of sets,

$$ \begin{align*}\left.\mathcal{E}^{\operatorname{can}}_{U,\Delta}\right|_W \rightarrow W_\rho \otimes_R \mathcal{O}_W.\end{align*} $$

With fixed $\rho $ , the system of sheaves $\{\mathcal {E}^{\operatorname {can}}_{U,\Delta ,\rho }\}$ has a $G(\mathbb {A}^\infty )$ -action. If $\operatorname {Std}$ denotes the representation over $\mathbb {Z}$ of $L_{(n)}$ on $\Lambda /\Lambda _{(n)}$ , then let $\omega _{U,\Delta } := \mathcal {E}^{\operatorname {can}}_{U,\Delta ,\wedge ^{n[F:\mathbb {Q}]}\operatorname {Std}^\vee }.$ We will write $\mathcal {I}_{\partial X_{U,\Delta }}$ for the ideal sheaf in $\mathcal {O}_{X_{U,\Delta }}$ , defining the boundary $\partial X_{U,\Delta }$ . Define the subcanonical extension

$$ \begin{align*}\mathcal{E}^{\operatorname{sub}}_{U,\Delta,\rho} = \mathcal{E}^{\operatorname{can}}_{U,\Delta,\rho} \otimes \mathcal{I}_{\partial X_{U,\Delta}}.\end{align*} $$

Recall the projection $\pi _{\operatorname {tor}/\operatorname {min}}: X_{U,\Delta } \rightarrow X^{\operatorname {min}}_U$ , and define $\mathcal {E}^{\operatorname {sub}}_{U,\rho } = \pi _{\operatorname {tor}/\operatorname {min}\ast } \mathcal {E}^{\operatorname {sub}}_{U,\Delta ,\rho }.$ The coherent sheaves defined on $X^{\operatorname {min}}_U$ are independent of the choice of $\Delta $ . If we fix $\rho $ , there is an action of $G(\mathbb {A}^\infty )$ on the system $\{\mathcal {E}^{\operatorname {sub}}_{U,\rho }\}$ indexed by neat open compact subgroups.

Now let $\rho _0$ be a representation of $L_{(n)}$ on a finite locally free $\mathbb {Z}_{(p)}$ -module. By Definition 8.3.5.1 of [Reference Lan14], there is a system of coherent sheaves associated to $\rho _0$ over $\{\mathcal {X}^{\operatorname {min}}_U\}$ with $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ -action whose pullback to $\{X^{\operatorname {min}}_U\}$ is $G(\mathbb {A}^\infty )$ -equivariantly identified with $\{\mathcal {E}^{\operatorname {sub}}_{U,\rho _0 \otimes \mathbb {Q}}\}$ . We will also refer to these sheaves by $\mathcal {E}^{\operatorname {sub}}_{U,\rho _0}$ . Note that over $\mathcal {X}^{\operatorname {min}}_U$ ,

$$ \begin{align*}\mathcal{E}^{\operatorname{sub}}_{U,\rho_0} \otimes \omega_{U} \cong \mathcal{E}^{\operatorname{sub}}_{U,\rho_0 \otimes (\wedge^{n[F:\mathbb{Q}]} \operatorname{Std}^\vee)},\end{align*} $$

where $\omega _U$ denotes the ample line bundle defined on $\mathcal {X}^{\operatorname {min}}_U$ .

3.2 Automorphic bundles on the ordinary locus

We now define automorphic vector bundles on the system of integral models of the minimally compactified ordinary locus $\{\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)}\}$ as well as its formal completion along the special fiber $\{\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1)}\}$ . The global sections of these coherent sheaves will consist of what we consider cuspidal p-adic automorphic forms. We first recall some definitions of sheaves defined on $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ .

Any universal abelian variety $\mathcal {A}^{\operatorname {univ}}/\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)}$ extends uniquely to a semi-abelian variety $\mathcal {A}_\Delta /\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ by Remarks 1.1.2.1 and 1.3.1.4 of [Reference Lan14]. Define $\Omega ^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ as the pullback by the identity section of the sheaf of relative differentials on $\mathcal {A}_{\Delta }$ . The inverse system $\{\Omega ^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }\}$ has an action of $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ . There is also a natural map

$$ \begin{align*}\varsigma_p: \varsigma_p^{\ast}\Omega^{\operatorname{ord}}_{U^p(N_1,N_2-1),\Delta} \rightarrow \Omega^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta}.\end{align*} $$

Denote by $\mathcal {O}_{\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }}(||\nu ||)$ the structure sheaf $\mathcal {O}_{\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }}$ with $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -action twisted by $||\nu ||$ (recall that $\{\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }\}$ has a right $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -action).

Let $\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta }$ denote the principal $L_{(n)}$ -bundle on $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ in the Zariski topology defined as follows: For a Zariski open W, $\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta }(W))$ is the set of pairs of isomorphisms

$$ \begin{align*}\xi_0:\left.\mathcal{O}_{\mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta}}(||\nu||)\right|_W \stackrel{\sim}{\longrightarrow} \mathcal{O}_W \qquad \text{and} \qquad \xi_1:\Omega^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta} \stackrel{\sim}{\longrightarrow} \operatorname{Hom}_{\mathbb{Z}}(\Lambda/\Lambda_{(n)}, \mathcal{O}_W).\end{align*} $$

(Recall that $\Lambda _{(n)}$ is the sublattice of $\Lambda = (\mathcal {D}_F^{-1})^n \oplus \mathcal {O}_F^n$ consisting of elements whose last n coordinates are equal to $0$ .) There is an action of $h \in L_{(n)}$ on $\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta }$ by

$$ \begin{align*}h(\xi_0,\xi_1) = (\nu(h)^{-1} \xi_0, \xi_1 \circ h^{-1} ).\end{align*} $$

The inverse system $\{\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta }\}$ has an action of $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ . Let R be a $\mathbb {Z}_{(p)}$ -algebra. Fix a representation $\rho $ of $L_{n,(n)}$ on a finite, locally free R-module $W_\rho $ . Denote the canonical extension to $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta ,\rho } \times \operatorname {Spec} R$ of the automorphic vector bundle on $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2)}$ associated to $\rho $ by $\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta ,\rho }$ , which is defined as follows: For any Zariski open W, $\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta ,\rho }(W)$ is the set of $L_{(n)}(\mathcal {O}_W)$ -equivariant maps of Zariski sheaves of sets

$$ \begin{align*}\left.\mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2),\Delta}\right|_W \rightarrow W_\rho \otimes_{R} \mathcal{O}_W.\end{align*} $$

When $\rho $ is fixed, the system of sheaves $\{\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta ,\rho }\}$ has an action of $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ . Furthermore, the inverse of $\varsigma _p^{\ast }$ gives a map

$$ \begin{align*}(\varsigma^{\ast}_p)^{-1}:{\varsigma_p}_{\ast} \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2),\Delta,\rho} \stackrel{\sim}{\longrightarrow} \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2-1),\Delta,\rho} \otimes_{\mathcal{O}_{\mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2-1),\Delta}}} {\varsigma_{p}}_{\ast}\mathcal{O}_{\mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta}}.\end{align*} $$

Composing $(\varsigma _p^{\ast })^{-1}$ with $1 \otimes \operatorname {tr}_{\varsigma _p}: \mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2 -1),\Delta ,\rho } \otimes {\varsigma _p}_{\ast } \mathcal {O}_{\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }} \rightarrow \mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2 -1),\Delta ,\rho }$ gives a $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ -equivariant map

$$ \begin{align*}\operatorname{tr}_F: {\varsigma_p}_{\ast} \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2),\Delta,\rho} \rightarrow \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2-1),\Delta,\rho}\end{align*} $$

satisfying $\operatorname {tr}_F \circ \varsigma _p^{\ast } = p^{n^2[F^+:\mathbb {Q}]}$ . If $\operatorname {Std}$ denotes the representation over $\mathbb {Z}$ of $L_{(n)}$ on $\Lambda /\Lambda _{(n)}$ , then let $\omega _{U^p(N_1,N_2),\Delta } := \mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2),\Delta ,\wedge ^{n[F:\mathbb {Q}]}\operatorname {Std}^\vee }$ denote the pullback of $\omega _{U}$ to $\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ . We will write $\mathcal {I}_{\partial \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }}$ for the ideal sheaf in $\mathcal {O}_{\mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }}$ defining the boundary $\partial \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }$ . Define the subcanonical extension as

$$ \begin{align*}\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2),\Delta,\rho} = \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2),\Delta,\rho} \otimes \mathcal{I}_{\partial \mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta}}.\end{align*} $$

Again, the inverse of $\varsigma _p^{\ast }$ gives a map

$$ \begin{align*}(\varsigma^{\ast}_p)^{-1}:{\varsigma_p}_{\ast} \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2),\Delta,\rho} \stackrel{\sim}{\longrightarrow} \mathcal{E}^{\operatorname{ord},\operatorname{can}}_{U^p(N_1,N_2-1),\Delta,\rho} \otimes_{\mathcal{O}_{\mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2-1),\Delta}}} {\varsigma_{p}}_{\ast}\mathcal{I}_{\partial\mathcal{X}^{\operatorname{ord}}_{U^p(N_1,N_2),\Delta}}.\end{align*} $$

Composing $(\varsigma _p^{\ast })^{-1}$ with $1 \otimes \operatorname {tr}_{\varsigma _p}: \mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2-1),\Delta ,\rho } \otimes {\varsigma _p}_{\ast }\mathcal {I}_{\partial \mathcal {X}^{\operatorname {ord}}_{U^p(N_1,N_2),\Delta }} \rightarrow \mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U^p(N_1,N_2-1),\Delta ,\rho }$ gives another $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ -equivariant map

$$ \begin{align*}\operatorname{tr}_F: {\varsigma_p}_{\ast}\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2),\Delta,\rho} \rightarrow \mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2-1),\Delta,\rho}\end{align*} $$

satisfying $\operatorname {tr}_F \circ \varsigma _p^{\ast } = p^{n^2[F^+:\mathbb {Q}]}$ and compatible with the analogous map defined on $\{\mathcal {E}^{\operatorname {ord},\operatorname {can}}_{U,\Delta ,\rho }\}_{U}$ .

Denote the pushforward by $\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\rho } = \pi ^{\operatorname {ord}}_{\operatorname {tor}/\operatorname {min}\ast } \mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\Delta ,\rho }.$ These coherent sheaves defined on $\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)}$ are independent of the choice of $\Delta $ by Proposition 1.4.3.1 and Lemma 8.3.5.2 in [Reference Lan14]. Note that

$$ \begin{align*}\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2),\rho} \otimes \omega_{U^p(N_1,N_2)} \cong \mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2),\rho \otimes (\wedge^{n[F:\mathbb{Q}]} \operatorname{Std}^\vee)},\end{align*} $$

and by Lemma 5.5 in [Reference Harris, Lan, Taylor and Thorne10], the pullback of $\mathcal {E}^{\operatorname {sub}}_{U^p(N_1,N_2),\rho }$ to $\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2),\rho }$ is $\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\rho }.$

Abusing notation, denote the pullback of $\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\rho }$ to $\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1)}$ by $\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1),\rho }$ . It is independent of $N_2$ , and thus, $\operatorname {tr}_F$ induces a $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ -equivariant map

$$ \begin{align*}\operatorname{tr}_F: {\varsigma_p}_{\ast}\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1),\rho} \rightarrow \mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1),\rho}\end{align*} $$

over $\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1)}$ , and also induces an endomorphism on global sections.

4.1 Weights

For each integer $0 \leq i \leq n$ , let $\Lambda _{(i)}$ denote the elements of $\Lambda $ for which the last $2n - i$ coordinates are zero, and let $B_n$ denote the Borel of G preserving the chain $\Lambda _{(n)}\supset \Lambda _{(n-1)} \supset ... \supset \Lambda _{(0)}.$ Let $T_n$ denote the subgroup of diagonal matrices of G.

Let $X^{\ast }(T_{n/\Omega }) := \operatorname {Hom}(T_n \times \operatorname {Spec} \Omega , \mathbb {G}_m \times \operatorname {Spec} \Omega )$ , and denote by $\Phi _n \subset X^{\ast }(T_{n/\Omega })$ the set of roots of $T_n$ on $\operatorname {Lie} G$ . The subset of positive roots with respect to $B_n$ will be denoted $\Phi ^+_n$ , and $\Delta _n$ will denote the set of simple positive roots. For any ring $R\subset \mathbb {R}$ , let $X^\ast (T_{n/\Omega })^+_R$ denote the subset of elements $X^\ast (T_n/\Omega ) \otimes _{\mathbb {Z}} R$ which pair nonnegatively with the simple coroots $\check {\alpha } \in X_{\ast }(T_n/\Omega ) = \operatorname {Hom}(\mathbb {G}_m \times \operatorname {Spec} \Omega , T_n \times \operatorname {Spec} \Omega )$ corresponding to the elements of $\alpha \in \Delta _n$ .

Let $\Phi _{(n)} \subset \Phi _n$ denote the set of roots of $T_n$ on $\operatorname {Lie} L_{(n)}$ , and set $\Phi _{(n)}^+ = \Phi _{(n)} \cap \Phi _n^+$ as well as $\Delta _{(n)} = \Delta _n \cap \Phi _{(n)}$ . If $R \subset \mathbb {R}$ is a subring, then $X^{\ast }(T_{n/\Omega })^+_{(n),R}$ will denote the subset of $X^{\ast }(T_{n/\Omega })_{(n)} \otimes _{\mathbb {Z}} R$ consisting of elements which pair nonnegatively with the simple coroot $\check {\alpha } \in X_{\ast }(T_{n/\Omega })_{(n)}$ corresponding to each $\alpha \in \Delta _{(n)}$ .

Recall that $L_{(n)} \times \operatorname {Spec} \Omega \cong \operatorname {GL}_1 \times \operatorname {GL}_n^{\operatorname {Hom}(F,\Omega )}$ , which induces an identification

$$ \begin{align*}T_n \times \operatorname{Spec} \Omega \cong \operatorname{GL}_1 \times (\operatorname{GL}_1^n)^{\operatorname{Hom}(F,\Omega)},\end{align*} $$

and hence, $X^{\ast }(T_{n/\Omega }) \cong \mathbb {Z} \bigoplus (\mathbb {Z}^n)^{\operatorname {Hom}(F,\Omega )}.$ Under this isomorphism, the image of $X^{\ast }(T_{n/\Omega })^+_{(n)}$ is the set

Furthermore, $X^{\ast }(T_{n/\Omega })^+$ is identified with

Denote by $\operatorname {Std}$ the representation of $L_{(n)}$ on $\Lambda /\Lambda _{(n)}$ over $\mathbb {Z}$ . Note that the representation $\wedge ^{n[F:\mathbb {Q}]} \operatorname {Std}^{\vee }$ is irreducible with highest weight . If $\rho $ is an irreducible algebraic representation of $L_{(n)}$ over $\overline {\mathbb {Q}}_p$ , then its highest weight lies in $X^{\ast }(T_{n/\overline {\mathbb {Q}}_p})_{(n)}^+$ and uniquely up to isomorphism identifies $\rho $ . Thus, for any $\underline {b} \in X^{\ast }(T_{n/\overline {\mathbb {Q}}_p})_{(n)}^+$ , let $\rho _{\underline {b}}$ denote the $L_{(n)}$ -representation over $\overline {\mathbb {Q}}_p$ with highest weight $\underline {b}$ .

Define the set of classical highest weights $X^{\ast }(T_{n/\overline {\mathbb {Q}}_p})_{\operatorname {cl}}^+$ as any $\underline {b} = (b_0,(b_{\tau ,i})_{\tau \in \operatorname {Hom}(F,\overline {\mathbb {Q}}_p)}) \in X^{\ast }(T_{n/\overline {\mathbb {Q}}_p})^+_{(n)}$ such that $b_{\tau ,1} + b_{\tau c, 1} \leq -2n$ .

We next turn to local components of automorphic representations (i.e., smooth representations of $G(\mathbb {Q}_\ell )$ when $\ell \neq p$ ). We relate them to smooth representations of $\operatorname {GL}_{2n}(\mathbb {Q}_\ell )$ via local base change defined below.

4.2 Local base change

For a rational prime $\ell \neq p$ , denote the primes of $F^+$ above $\mathbb {Q}$ as $u_1, \cdots , u_r, v_1 \cdots v_s$ , where each $u_i = w_i{}^cw_i$ splits in F and none of the $v_j$ split in F. Note that

$$ \begin{align*}G(\mathbb{Q}_{\ell}) \cong \prod_{i=1}^r \operatorname{GL}_{2n}(F_{w_i}) \times H,\end{align*} $$

where

$$ \begin{align*}H = \left\{ (\mu,g_i) \in \mathbb{Q}_\ell^\times \times \prod_{i=1}^s \operatorname{GL}_{2n}(F_{v_i}) : {}^tg_i J_n {}^c g_i = \mu J_n \quad \forall i\right\}.\end{align*} $$

Here, H contains a product $\prod _{i=1}^s G^1(F_{v_i}^+),$ where $G^1$ denotes the group scheme over $\mathcal {O}_{F^+}$ defined by

$$ \begin{align*}G^1(R) = \{g \in \operatorname{Aut}_{\mathcal{O}_F \otimes_{\mathcal{O}_{F^+}} R}(\Lambda \otimes_{\mathcal{O}_{F^+}} R) : {}^tg J_n {}^cg = J_n\}.\end{align*} $$

Note that $\operatorname {ker} \nu \cong \operatorname {RS}^{\mathcal {O}_{F^+}}_{\mathbb {Z}} G^1.$ If $\Pi $ is an irreducible smooth representation of $G(\mathbb {Q}_\ell )$ , then

$$ \begin{align*}\Pi = \left(\otimes_{i=1}^r \Pi_{w_i}\right) \otimes \Pi_H.\end{align*} $$

Define $\operatorname {BC}(\Pi )_{w_i} := \Pi _{w_i}$ and $\operatorname {BC}(\Pi )_{cw_i} := \Pi _{w_i}^{c,\vee }$ . This does not depend on the choice of $w_i$ . We call $\Pi $ unramified at $v_i$ if $v_i$ is unramified over $F^+$ and

$$ \begin{align*}\Pi^{G^1(\mathcal{O}_{F^+,v_i})} \neq (0).\end{align*} $$

Let $B^1$ denote the Borel subgroup of $G^1$ consisting of upper triangular matrices and $T^1$ the torus subgroup consisting of diagonal matrices.

If $\Pi $ is unramified at $v_i$ , then there is a character $\chi $ of $T^1(F_{v_i}^+)/T^1(\mathcal {O}_{F^+,v_i})$ such that $\left .\Pi \right |{}_{G^1(F_{v_i}^+)}$ and $\operatorname {n-Ind}_{B^1(F_{v_i}^+)}^{G^1(F_{v_i}^+)} \chi $ share an irreducible subquotient with a $G^1(\mathcal {O}_{F^+,v_i})$ -fixed vector. Define a map between the torus of diagonal matrices of $\operatorname {GL}_{2n}(F_{v_i})$ and $G^1(\mathcal {O}_{F^+,v_i})$ :

(4.1) $$ \begin{align} \mathbb N: T_{\operatorname{GL}_{2n}}(F_{v_i}) &\rightarrow T^1(F_{v_i}^+),\qquad\qquad \end{align} $$

(4.2) $$ \begin{align}\qquad\quad\!\!\! \left(\begin{array}{ccc}t_1 & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & t_{2n}\end{array}\right) & \mapsto \left(\begin{array}{ccc}t_1/{}^c t_{2n} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & t_{2n}/{}^ct_1\end{array}\right). \end{align} $$

We define $\operatorname {BC}(\Pi )_{v_i}$ to be the unique subquotient of

$$ \begin{align*}\operatorname{n-Ind}_{B_{\operatorname{GL}_{2n}}(F_{v_i})}^{\operatorname{GL}_{2n}(F_{v_i})} \chi \circ \mathbb{N}\end{align*} $$

with a $\operatorname {GL}_{2n}(\mathcal {O}_{F,v_i})$ -fixed vector, where $B_{\operatorname {GL}_{2n}}(F_{v_i})$ denote the Borel subgroup of upper triangular matrices.

Lemma 4.1 (Lemma 1.1 in [Reference Harris, Lan, Taylor and Thorne10]).

Suppose that $\psi \otimes \pi $ is an irreducible smooth representation of

$$ \begin{align*}L_{(n)}(\mathbb{Q}_\ell) \cong L_{(n),\operatorname{herm}}(\mathbb{Q}_\ell) \times L_{(n),\operatorname{lin}}(\mathbb{Q}_\ell) = \mathbb{Q}_\ell^\times \times \operatorname{GL}_n(F_\ell).\end{align*} $$

1. 1. If v is unramified over $F^+$ and $\pi _v$ is unramified, then $\operatorname {n-Ind}_{P_{(n)}(\mathbb {Q}_q)}^{G(\mathbb {Q}_q)}(\psi \otimes \pi )$ has a subquotient $\Pi $ which is unramified at v. Moreover, $\operatorname {BC}(\Pi )_v$ is the unramified irreducible subquotient of $\operatorname {n-Ind}_{B_{\operatorname {GL}_{2n}}(F_v)}^{\operatorname {GL}_{2n}(F_v)} (\pi _v^{c,\vee } \otimes \pi _v)$ .

2. 2. If v is split over $F^+$ and $\Pi $ is an irreducible subquotient of the normalized induction $\operatorname {n-Ind}_{P_{(n)}(\mathbb {Q}_q)}^{G(\mathbb {Q}_q)} (\psi \otimes \pi )$ , then $\operatorname {BC}(\Pi )_v$ is an irreducible subquotient of $\operatorname {n-Ind}_{B_{\operatorname {GL}_{2n}}(F_v)}^{\operatorname {GL}_{2n}(F_v)} (\pi _{{}^cv}^{c,\vee } \otimes \pi _v)$ .

Note that in both cases, $\operatorname {BC}(\Pi _v)$ does not depend on v.

4.3 Cuspidal automorphic representations

Here, we define automorphic representations on $G(\mathbb {A})$ whose finite parts will be realized in the space of global sections of $\mathcal {E}^{\operatorname {sub}}_{U,\rho }$ on $X^{\operatorname {min}}_{U,\rho }$ . We first recall a few definitions. Let $U(n) \subset \operatorname {GL}_n(\mathbb {C})$ denote the subgroup of matrices g satisfying ${}^tg{}^cg = 1_n$ . Define

$$ \begin{align*}\mathcal{K}_{n,\infty} = (U(n) \times U(n))^{\operatorname{Hom}(F^+,\mathbb{R})} \rtimes S_2,\end{align*} $$

where $S_2$ acts by permuting $U(n) \times U(n)$ . We can embed $\mathcal {K}_{n,\infty }$ in

$$ \begin{align*}G(\mathbb{R}) \, \subset \, \, \mathbb R^\times \times \prod_{\tau \in \operatorname{Hom}(F^+,\mathbb R)} GL_{2n}(F \otimes_{F^+,\tau} \mathbb R)\end{align*} $$

via the map sending

$$ \begin{align*}(g_{\tau},h_{\tau})_{\tau \in\operatorname{Hom}(F^+,\mathbb{R})} \mapsto \left( 1, \left( \begin{array}{cc} {(g_\tau+h_\tau)/2} & {(g_\tau-h_\tau)\Psi_n/2i} \\ {\Psi_n(g_\tau-h_\tau)/2i} & {\Psi_n(g_\tau+h_\tau)\Psi_n/2} \end{array} \right)_{\tau \in \operatorname{Hom}(F^+,\mathbb{R})} \right),\end{align*} $$

and sending the nontrivial element of $S_2$ to $\left (-1,\left (\begin {smallmatrix} -1_n & 0 \\ 0 & 1_n \end {smallmatrix}\right )_{\tau \in \operatorname {Hom}(F^+,\mathbb {R})}\right ).$ This forces $\mathcal {K}_{n,\infty }$ to be a maximal compact subgroup of $G(\mathbb {R})$ such that $\mathcal {K}_{n,\infty } \cap P_{(n)}(\mathbb {R})$ is a maximal compact of $L_{(n)}(\mathbb {R})$ . Let $\mathfrak {g} = (\operatorname {Lie} G(\mathbb {R}))_{\mathbb {C}}$ , and denote by $A_n$ the image of $\mathbb {G}_m$ in G via the embedding $t \mapsto t \cdot 1_{2n}$ . We define a cuspidal automorphic representation of $G(\mathbb {A})$ to be an irreducible admissible $G(\mathbb {A}^\infty ) \times (\mathfrak {g}, \mathcal {K}_{n,\infty })$ -submodule of the space of cuspidal automorphic forms on the double coset space $G(\mathbb {Q})\backslash G(\mathbb {A})/A_n(\mathbb {R})^0$ . Furthermore, a square-integrable automorphic representation of $G(\mathbb {A})$ is the twist by a character on $\mathbb {Q}^\times \backslash \mathbb {A}^\times /\mathbb {R}_{>0}^\times $ of an irreducible admissible $G(\mathbb {A}^\infty ) \times (\mathfrak {g},\mathcal {K}_{n,\infty })$ -module that occurs discretely in the space of square integrable automorphic forms on $G(\mathbb {A})\backslash G(\mathbb {A})/A_n(\mathbb {R})^0$ .

Now let $\mathfrak {l} = (\operatorname {Lie} L_{(n)}(\mathbb {R}))_{\mathbb {C}}$ , and let $A_{(n)}$ denote the maximal split torus in the center of $L_{(n)}$ . A cuspidal automorphic representation of $L_{(n)}(\mathbb {A})$ is an irreducible admissible $L_{(n)}(\mathbb {A}^\infty ) \times (\mathfrak {l}, \mathcal {K}_{n,\infty } \cap L_{(n)}(\mathbb {R}))$ -submodule of the space of cuspidal automorphic forms of $L_{(n)}(\mathbb {A})$ on the double coset space $L_{(n)}(\mathbb {Q})\backslash L_{(n)}(\mathbb {A})/A_{(n)}(\mathbb {R})^0$ .

For a number field K and any positive integer m, let $\mathcal {K}_{K,\infty }$ denote a maximal compact subgroup of $\operatorname {GL}_m(K_{\infty })$ , and let $\mathfrak {g}\mathfrak {l} = (\operatorname {Lie} \operatorname {GL}_m(K_{\infty }))_{\mathbb {C}}$ . Define a cuspidal automorphic representation of $\operatorname {GL}_m(\mathbb {A}_K)$ as an irreducible admissible $\operatorname {GL}_m(\mathbb {A}^\infty _K) \times (\mathfrak {g}\mathfrak {l}, \mathcal {K}_{K,\infty })$ -submodule of the space of cuspidal automorphic forms on the double coset space $\operatorname {GL}_m(K)\backslash \operatorname {GL}_m(\mathbb {A}_K)/\mathbb {R}^\times _{>0}$ . Finally, by a square-integrable automorphic representation of $\operatorname {GL}_m(\mathbb {A}_K)$ , we shall mean the twist by a continuous character on $K^\times /\mathbb {A}_K^\times /\mathbb {R}_{>0}^\times $ of an irreducible admissible $\operatorname {GL}_m(\mathbb {A}_K^\infty ) \times (\mathfrak {g}\mathfrak {l},\mathcal {K}_{K,\infty })$ -module that occurs discretely in the space of square integrable automorphic forms on $\operatorname {GL}_m(\mathbb {A}_K)$ .

We will now relate the finite parts of these automorphic representations to the global sections of the automorphic bundles defined previously.

4.4 Global sections of automorphic bundles over the Shimura variety

Let $\rho $ be a representation of $L_{(n)}$ on a finite $\mathbb {Q}$ -vector space. Define the admissible $G(\mathbb {A}^\infty )$ -module

$$ \begin{align*}H^0(X^{\operatorname{min}},\mathcal{E}^{\operatorname{sub}}_{\rho}) = \lim_{\stackrel{\rightarrow}{U}} H^0(X^{\operatorname{min}}_U,\mathcal{E}^{\operatorname{sub}}_{U,\rho}).\end{align*} $$

Note that for any neat open compact U, $H^0(X^{\operatorname {min}},\mathcal {E}^{\operatorname {sub}}_{\rho })^{U} = H^0(X^{\operatorname {min}}_U,\mathcal {E}^{\operatorname {sub}}_{U,\rho })$ (see Lemma 5.5 of [Reference Harris, Lan, Taylor and Thorne10] or Proposition 8.3.6.9 of [Reference Lan14].

Proposition 4.2 (Corollary 5.12 in [Reference Harris, Lan, Taylor and Thorne10]).

Suppose that $\underline {b} \in X^{\ast }(T_{n/\overline {\mathbb {Q}}_p})_{\operatorname {cl}}^+$ , and $\rho _{\underline {b}}$ is the irreducible representation of $L_{(n)}$ with highest weight $\underline {b}$ . Then $H^0(X^{\operatorname {min}}, \mathcal {E}^{\operatorname {sub}}_{\rho _{\underline {b}}})$ is a semisimple $G(\mathbb {A}^\infty )$ module, and if $\Pi $ is an irreducible subquotient of $H^0(X^{\operatorname {min}},\mathcal {E}^{\operatorname {sub}}_{\rho _{\underline {b}}})$ , then there is a continuous representation

$$ \begin{align*}R_p(\Pi): G_F \rightarrow \operatorname{GL}_{2n}(\overline{\mathbb{Q}}_p),\end{align*} $$

which is de Rham above p and has the following property: Suppose that $v \nmid p$ is a prime of F above a rational prime $\ell $ such that

• • either $\ell $ splits in $F_0$ ,

• • or F and $\Pi $ are unramified above $\ell $ ;

then

$$ \begin{align*}\left.\operatorname{WD}(R_p(\Pi)\right|{}_{G_{F_v}})^{\operatorname{Frob}-ss} \cong \operatorname{rec}_{F_v}(\operatorname{BC}(\Pi_\ell)_v|\operatorname{det}|_v^{(1-2n)/2}),\end{align*} $$

where $\ell $ is the rational prime below v.

Proof. Each irreducible subquotient $\Pi $ of $H^0(X^{\operatorname {min}},\mathcal {E}^{\operatorname {sub}}_{\rho _{\underline {b}}})$ is the finite part of a cohomological cuspidal $G(\mathbb {A})$ -automorphic representation $\pi $ by Lemma 5.11 in [Reference Harris, Lan, Taylor and Thorne10], and furthermore, $H^0(X^{\operatorname {min}},\mathcal {E}^{\operatorname {sub}}_{\rho _{\underline {b}}})$ is a semisimple $G(\mathbb {A}^\infty )$ -module. For such $\pi $ , by Shin [Reference Shin20] and Moeglin-Waldspurger [Reference Moeglin and Waldspurger15], there is a decomposition into positive integers

and cuspidal conjugate self-dual automorphic representations $\tilde {\pi }_i$ of $\operatorname {GL}_{m_i}(\mathbb {A}_F)$ such that for each $i \in [1,r]$ , $\tilde {\pi }_i ||\operatorname {det}||^{(m_i + n_i - 1)/2}$ is cohomological and satisfies the following at all primes v of F which are split over $F^+$ :

$$ \begin{align*}\pi_v = \boxplus_{i=1}^r \boxplus_{j=0}^{n_i - 1} \tilde{\pi}_{i,v} |\operatorname{det}|_v^{(n_i-1)/2-j}.\end{align*} $$

These $\tilde {\pi }_i$ are automorphic representations which have Galois representations associated to them satisfying full local-global compatibility – results due to many people including [Reference Chenevier and Harris9, Reference Shin19, Reference Caraiani6, Reference Barnet-Lamb, Geraghty, Harris and Taylor2] (for a summary, see [Reference Barnet-Lamb, Gee, Geraghty and Taylor1]).

We will refer to irreducible subquotients of $H^0(X^{\operatorname {min}},\mathcal {E}^{\operatorname {sub}}_{\rho _{\overline {b}}})$ as classical cuspidal G-automorphic forms of weight $\rho _{\underline {b}}$ .

4.5 p-adic (cuspidal) G-automorphic forms

Now let $\rho $ be a representation of $L_{(n)}$ on a finite locally free $\mathbb {Z}_{(p)}$ -module. Let $H^0(\mathfrak {X}^{\operatorname {ord},\operatorname {min}},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{\rho })$ denote the smooth $G(\mathbb {A}^{\infty })^{\operatorname {ord}}$ -module defined as

$$ \begin{align*}H^0(\mathfrak{X}^{\operatorname{ord},\operatorname{min}},\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{\rho}) := \displaystyle \lim_{\stackrel{\rightarrow}{U^p,N_1}} H^0(\mathfrak{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1)}, \mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1),\rho}).\end{align*} $$

For each positive integer r, define

$$ \begin{align*}H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}},\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{\rho} \otimes \mathbb{Z}/p^r\mathbb{Z}) := \displaystyle\lim_{\stackrel{\rightarrow}{U^p(N_1,N_2)}} H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)},\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2),\rho} \otimes \mathbb{Z}/p^r\mathbb{Z}).\end{align*} $$

It is a smooth $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -module of p-adic cuspidal G-automorphic forms of weight $\rho $ , with the property that

$$ \begin{align*}H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}},\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{\rho} \otimes \mathbb{Z}/p^r\mathbb{Z})^{U^p(N_1,N_2)} = H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)},\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{U^p(N_1,N_2),\rho} \otimes \mathbb{Z}/p^r\mathbb{Z}).\end{align*} $$

Note that mod $p^M$ , and there is a $G(\mathbb {A}^\infty )^{\operatorname {ord}}$ -equivariant embedding

$$ \begin{align*}H^0(\mathfrak{X}^{\operatorname{ord},\operatorname{min}},\mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{\rho}) \otimes_{\mathbb{Z}_p} \mathbb{Z}/p^M \mathbb{Z} \hookrightarrow H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}}, \mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{\rho} \otimes \mathbb{Z}/p^M \mathbb{Z}).\end{align*} $$

Fix a neat open compact subgroup $U^p \subset G(\mathbb {A}^{p,\infty })$ and integers $N_2 \geq N_1 \geq 0$ , and recall that there is a canonical section $\operatorname {Hasse}_U \in H^0(\overline {X}^{\operatorname {min}}_U, \omega _U^{\otimes (p-1)})$ which is $G(\mathbb {A}^{p,\infty } \times \mathbb {Z}_p)$ -invariant. Let $\widetilde {\operatorname {Hasse}}_U$ denote the noncanonical lift of $\operatorname {Hasse}_U$ over an open subset of $\mathcal {X}^{\operatorname {min}}_U$ . For each positive integer M, the powers $\widetilde {\operatorname {Hasse}}_U^{p^{M-1}} {\operatorname {mod}} p^M$ are canonical despite the noncanonical choice of $\widetilde {\operatorname {Hasse}}_U$ , and hence, they glue with each other and give a canonical $G(\mathbb {A}^{\infty ,p} \times \mathbb {Z}_p)$ -invariant section $\operatorname {Hasse}_{U,M}$ of $\omega ^{\otimes (p-1)p^{M-1}}$ over $\mathcal {X}^{\operatorname {min}}_U \times \operatorname {Spec} \mathbb {Z}/p^M\mathbb {Z}$ .

Fix $\rho $ a representation of $L_{(n)}$ on a finite free $\mathbb {Z}_{(p)}$ -module. Then for each integer i, define the $G(\mathbb {A}^\infty )^{\operatorname {ord},\times }$ -equivariant map,

$$ \begin{align*} H^0(\mathcal{X}^{\operatorname{min}}_{U^p(N_1,N_2)}, \mathcal{E}^{\operatorname{sub}}_{\rho} \otimes \omega_U^{ip^{M-1}(p-1)}) &\rightarrow H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)}, \mathcal{E}^{\operatorname{ord},\operatorname{sub}}_{\rho} \otimes \mathbb{Z}/p^M\mathbb{Z}), \\f &\mapsto (\left.f\right|{}_{\mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2)}})/\operatorname{Hasse}^i_{U^p(N_1,N_2),M}. \end{align*} $$

Using the map defined above, Harris-Lan-Taylor-Thorne [Reference Harris, Lan, Taylor and Thorne10] prove the following density theorem relating p-adic and classical cuspidal automorphic forms.

Lemma 4.3 (Lemma 6.1 in [Reference Harris, Lan, Taylor and Thorne10]).

Let $\rho $ be an irreducible representation of $L_{(n)}$ on a finite free $\mathbb {Z}_p$ -module. The induced map

$$ \begin{align*}\bigoplus_{j = r}^\infty H^0(\mathcal{X}^{\operatorname{min}}_{U^p(N_1,N_2)},\mathcal{E}^{\operatorname{sub}}_{U^p(N_1,N_2),\rho \otimes (\wedge^{n[F:\mathbb{Q}]} \operatorname{Std}^{\vee})^{jp^{M-1}(p-1)}}) \rightarrow H^0(\mathcal{X}^{\operatorname{ord},\operatorname{min}}_{U^p(N_1,N_2),\rho}, \mathcal{E}^{\operatorname{sub},\operatorname{ord}}_{U^p(N_1,N_2),\rho}\otimes \mathbb{Z}/p^M\mathbb{Z})\end{align*} $$

is surjective for any integer r.

6.1 At unramified primes

We recall the definition of the unramified Hecke algebra. Fix a neat open compact subgroup $U^p = G(\widehat {\mathbb {Z}}^Q) \times U_{Q^p} \subset G(\mathbb {A}^{p,\infty })$ . Suppose that v is a place of F above a rational prime $\ell \notin S$ , and let $i \in \mathbb {Z}$ .

By work of Bernstein-Deligne [Reference Bernstein and Deligne4] building on Satake, there is an element $T_v^{(i)} \in \mathbb {Q}[G(\mathbb {Z}_\ell )\backslash G(\mathbb {Q}_\ell )/G(\mathbb {Z}_\ell )]$ such that if $\Pi _\ell $ is an unramified representation of $G(\mathbb {Q}_\ell )$ , then its eigenvalue on $\Pi _\ell ^{G(\mathbb {Z}_\ell )}$ is equal to

$$ \begin{align*}\operatorname{tr} \operatorname{rec}_{F_v}(\operatorname{BC}(\Pi_\ell)_v)|\operatorname{det}|_v^{(1-2n)/2}(\operatorname{Frob}_v^i).\end{align*} $$

(For more details on this construction, see pages 196–197 in [Reference Harris, Lan, Taylor and Thorne10].) If v is an unramified prime of F which splits over $F^+$ , then we can write the Hecke operator $T_v^{(1)}$ as the double coset

$$ \begin{align*}G(\mathbb{Z}_\ell)\left(\begin{array}{cccc}1 & & & \\ & \ddots & & \\ & & 1 & \\ & \ & & \varpi_v\end{array}\right)G(\mathbb{Z}_\ell),\end{align*} $$

where $\varpi _v$ denotes a uniformizer of $F_v$ .

For each unramified prime v of F and each integer $i \in \mathbb {Z}$ , there exists an integer $d_v^{(i)} \in \mathbb {Z}$ such that

$$ \begin{align*}d_{v}^{(i)} T_v^{(i)} \in \mathbb{Z}[G(\mathbb{Z}_\ell)\backslash G(\mathbb{Q}_\ell))/ G(\mathbb{Z}_\ell)].\end{align*} $$

Let $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p} := \mathbb {Z}_p[G(\widehat {\mathbb {Z}}^{Q})\backslash G(\mathbb {A}^{Q})/G(\widehat {\mathbb {Z}}^Q)]$ denote the abstract unramified Hecke algebra. Let $N_1$ and $N_2$ be two integers $N_2 \geq N_1 \geq 0$ , and let $\rho $ be a representation of $L_{(n)}$ over $\mathbb {Z}_{(p)}$ . The Hecke algebra $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p}$ has an action on the classical and p-adic spaces $H^0(\mathcal {X}^{\operatorname {min}}_{U^p(N_1,N_2)},\mathcal {E}^{\operatorname {sub}}_{U^p(N_1,N_2),\rho })$ , $H^0(\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1)},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1),\rho })$ and $H^0(\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\rho } \otimes \mathbb {Z}/p^M\mathbb {Z})$ induced from the action of $G(\mathbb {A}^S)$ . Denote by $\mathbb {T}^{\mathrm {ur}}_{U^p(N_1,N_2),\rho }$ the image of $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p}$ in the endomorphism algebra

$$ \begin{align*}\operatorname{End}_{\mathbb{Z}_p}(H^0(\mathcal{X}^{\operatorname{min}}_{U^p(N_1,N_2)},\mathcal{E}^{\operatorname{sub}}_{U^p(N_1,N_2),\rho})).\end{align*} $$

Furthermore, if $W \subset H^0(\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1)},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1),\rho })$ (respectively, if $W \subset H^0(\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\rho } \otimes \mathbb {Z}/p^M\mathbb {Z})$ ) is a finitely-generated $\mathbb {Z}_p$ -submodule invariant under the action of the algebra $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p}$ , then let $\mathbb {T}^{\operatorname {ord},\mathrm {ur}}_{U^p(N_1,N_2),\rho }(W)$ (respectively, let $\mathbb {T}^{\operatorname {ord},\mathrm {ur}}_{U^p(N_1,N_2),\rho ,M}(W)$ ) denote the image of $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p}$ in $\operatorname {End}_{\mathbb {Z}_p}(W)$ .

For each v, let $\tilde {T}_v^{(i)}$ denote the image of $d_{v}^{(i)} T_v^{(i)}$ in any $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p}$ -algebra $\mathbb {T}$ via the canonical map $\mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p} \rightarrow \mathbb {T}$ .

6.2 At primes which are split in $F_0$

Suppose that $v \in \underline {\mathcal {S}}_{\operatorname {spl}} \sqcup \underline {\mathcal {S}}^c_{\operatorname {spl}}$ is a place of F above a rational prime $\ell $ , and let $\sigma _v$ denote an element of $W_{F_v}$ , the Weil group of $F_v$ . Let $\mathcal {B}$ denote a fixed Bernstein component; it is a subcategory of the smooth representations of $\operatorname {GL}_{2n}(F_v)$ . Every component $\mathcal {B}$ is uniquely associated to an inertial equivalence class $(M,\omega )$ , where M denotes a Levi subgroup of $\operatorname {GL}_{2n}(F_v)$ and $\omega $ is a supercuspidal representation of M. (Recall that two inertial classes $(M,\omega )$ and $(M^{\prime },\omega ^{\prime })$ are equivalent if there exists $g \in G$ and an unramified character $\chi $ of $M^{\prime }$ such that $M = g^{-1}Mg$ and $\omega ^{\prime } = \chi \otimes \omega (g\cdot g^{-1}).$ ) Then, $\mathcal {B}$ is defined to be the full subcategory of smooth representations of $\operatorname {GL}_{2n}(F_v)$ consisting of those representations all of whose irreducible subquotients have inertial support equivalent to $(M,\omega )$ . This implies that there exists some $(M^{\prime },\omega ^{\prime }) \sim (M,\omega )$ such that $\pi $ occurs as a composition factor of the parabolic induction $\operatorname {Ind}_{P_M}^{\operatorname {GL}_{2n}(F_v)}(\omega ^{\prime })$ where $\omega ^{\prime }$ is an irreducible supercuspidal representation and $P_M$ is a parabolic subgroup of $\operatorname {GL}_{2n}(F_v)$ with Levi M.

Let $\mathfrak {z}_{\mathcal {B}} = \mathfrak {z}_{[M,\omega ]}$ denote the Bernstein center of $\mathcal {B}$ , which is the image under the idempotent $e_{\mathcal {B}}$ associated to $\mathcal {B}$ of

$$ \begin{align*}\lim_{\stackrel{\longleftarrow}{ K}} \mathcal{Z}(\mathbb{C}[K\backslash\operatorname{GL}_{2n}(F_v)/K]),\end{align*} $$

the inverse limit over open compact subgroup K of the centers of the complex Hecke algebra for $\operatorname {GL}_{2n}(F_v).$

Proposition 6.1 (Proposition 3.11 in Chenevier [Reference Chenevier8]).

For an inertial equivalence class $[M,\omega ]$ , there is a representative $(M,\omega )$ which can be defined over $\overline {\mathbb {Q}}$ . Let $E \subset \overline {\mathbb {Q}}$ denote a sufficiently large finite-degree normal field over which $\omega $ , $\operatorname {rec}(\omega )$ , $\mathcal {B}_{[M,\omega ]}$ , $\mathfrak {z}_{[M,\omega ]}$ are all defined over E. Let $E[\mathcal {B}_{[M,\omega ]}]$ denote the affine coordinate ring of the variety associated to $\mathcal {B}_{[M,\omega ]}$ . Then there exists a unique pseudocharacter of dimension $2n$

$$ \begin{align*}T^{\mathcal{B}} = T^{[M,\omega]}: W_{F_v} \rightarrow E[\mathcal{B}] = \mathfrak{z}_{\mathcal{B}}\end{align*} $$

such that for all irreducible smooth representations $\pi $ of $\mathcal {B}$ and $\sigma _v \in W_{F_v}$ ,

$$ \begin{align*}T^{\mathcal{B}}(\sigma_v)(\pi) = \operatorname{tr} \operatorname{rec}_{F_v}(\pi)(\sigma_v).\end{align*} $$

For a Bernstein component $\mathcal {B}$ and $\sigma \in W_{F_v}$ , let $T_{v,\mathcal {B},\sigma }$ denote the twist of $T^{\mathcal {B}}(\sigma )$ such that $T_{v,\mathcal {B},\sigma }(\pi ) = \operatorname {tr} \operatorname {rec}_{F_v}(\pi |\operatorname {det}|_v^{(1-2n)/2})(\sigma )$ if $\pi $ is a smooth irreducible representation in $\mathcal {B}$ . Multiplying $T_{v,\mathcal {B},\sigma }$ by $e_{\mathcal {B}}$ if necessary, we may suppose that $T_{v,\mathcal {B},\sigma }$ acts as $0$ on all irreducible $\pi \notin \mathcal {B}$ .

We will denote this compact open subgroup by $K_{\mathfrak {B}} = K_{\mathfrak {B}_v}$ ; note that all irreducible smooth representations inside $\mathcal {B}$ have a fixed vector under $K_{\mathfrak {B}}$ . More generally, for every $\mathcal {B}^{\prime } \subset \mathfrak {B}$ , $\mathfrak {z}_{\mathcal {B}^{\prime }}$ embeds in the center of $\mathcal {H}(\operatorname {GL}_{2n},K_{\mathfrak {B}})_{\mathbb {C}} = \mathbb {C}[K_{\mathfrak {B}}\backslash \operatorname {GL}_{2n}(F_v) /K_{\mathfrak {B}}]$ via multiplication by the characteristic function of $K_{\mathfrak {B}}$ . Let $\mathfrak {z}_{\mathfrak {B}_v} = \mathfrak {z}_{\mathfrak {B}} := \operatorname {im}(\prod _{\mathcal {B}^{\prime } \subset \mathfrak {B}} \mathfrak {z}_{\mathcal {B}^{\prime }} \hookrightarrow \mathcal {H}(\operatorname {GL}_{2n},K_{\mathfrak {B}})_{\mathbb {C}})$ . Note that $\mathfrak {z}_{\mathfrak {B}}$ is the center of $\mathcal {H}(\operatorname {GL}_{2n},K_{\mathfrak {B}})_{\mathbb {C}}$ .

For $\ell \in S_{\operatorname {spl}}$ , assume $K_\ell $ is an open compact subgroup of $G(\mathbb {Q}_\ell )$ such that under the identification $G(\mathbb {Q}_\ell ) \cong \prod _{\underline {\mathcal {S}}_{\operatorname {spl}} \ni v \mid \ell } \operatorname {GL}_{2n}(F_v),$ we can decompose

$$ \begin{align*}K_\ell = \prod_{\underline{\mathcal{S}}_{\operatorname{spl}} \ni v \mid \ell} K_{\mathfrak{B}_v}.\end{align*} $$

If $v \in \underline {\mathcal {S}}_{\operatorname {spl}}$ divides the rational prime $\ell $ and $\mathfrak {B}$ is a Bernstein component, then for any $\sigma \in W_{F_v}$ , we can find an element of $\mathfrak {z}_{\mathfrak {B}}$ , which we will denote by $T_{v,\mathfrak {B},\sigma }$ , such that its eigenvalue on the $K_{\ell }$ -fixed vectors of an irreducible representation $\pi $ of $G(\mathbb {Q}_\ell )$ in $\mathcal {B}$ is

$$ \begin{align*} \operatorname{tr} \operatorname{rec}_{F_v} (\pi_v|\operatorname{det}|_v^{(1-2n)/2})(\sigma).\end{align*} $$

(However, if $\pi _v \notin \mathfrak {B}$ , then $\pi ^{K_\ell }$ is trivial and $T_{v,\mathfrak {B},\sigma }$ acts as 0.) This element $T_{v,\mathfrak {B},\sigma }$ is the image in $\mathfrak {z}_{\mathfrak {B}}$ of $\prod _{\mathcal {B}^{\prime } \subset \mathfrak {B}} T_{v,\mathcal {B}^{\prime },\sigma } \in \prod _{\mathcal {B}^{\prime }\subset \mathfrak {B}} \mathfrak {z}_{\mathcal {B}^{\prime }}$ . It is independent of $\pi $ . Furthermore, for each $\varphi \in \operatorname {Aut}(\mathbb {C})$ , we have that ${}^{\varphi }\mathfrak {B} = \mathfrak {B}$ and additionally,

$$ \begin{align*}{}^{\varphi} \operatorname{rec}_{F_v}(\pi_v|\operatorname{det}|_v^{(1-2n)/2}) \cong \operatorname{rec}_{F_v}({}^\varphi(\pi_v|\operatorname{det}|_v^{(1-2n)/2})).\end{align*} $$

Thus, we have that ${}^\varphi T_{v,\mathfrak {B},\sigma } = T_{v,\mathfrak {B},\sigma }$ , and so $T_{v,\mathfrak {B},\sigma } \in \mathbb {Q}[K_\ell \backslash G(\mathbb {Q}_\ell )/K_\ell ].$

Define

$$ \begin{align*}\mathfrak{z}_\ell^0 := \prod_{\underline{\mathcal{S}}_{\operatorname{spl}} \ni v \mid \ell} (\mathfrak{z}_{\mathfrak{B}_v} \cap \mathbb{Z}[K_{\mathfrak{B}}\backslash \operatorname{GL}_{2n}(F_v) /K_{\mathfrak{B}}]).\end{align*} $$

Then $\mathfrak {z}_\ell ^0$ lies in the center of $\mathbb {Z}[K_\ell \backslash G(\mathbb {Q}_\ell )/K_\ell ]$ . Note that for any element $T \in \mathfrak {z}_{\mathfrak {B}} \cap \mathbb {Q}[K_\ell G(\mathbb {Q}_\ell )/K_{\ell }]$ , there exists a nonzero integer $d(T) \in \mathbb {Z} $ such that $d(T)T\in \mathfrak {z}_\ell ^0,$ where $v \mid \ell $ . Thus, we can choose $d(T_{v,\mathfrak {B},\sigma }) \in \mathbb {Z} \smallsetminus \{0\}$ such that

$$ \begin{align*}d(T_{v,\mathfrak{B},\sigma})T_{v,\mathfrak{B},\sigma} \in \mathbb{Z}[K_\ell\backslash G(\mathbb{Q}_\ell)/K_\ell],\end{align*} $$

so $d(T_{v,\mathfrak {B},\sigma })T_{v,\mathfrak {B},\sigma } \in \mathfrak {z}_\ell ^0$ .

For each $v \in \underline {\mathcal {S}}_{\operatorname {spl}}$ , fix a Bernstein component $\mathcal {B}_v$ , and let $\mathfrak {B}_v$ be the disjoint union as defined in Theorem 6.2. We will make the further assumption that $U^p = \prod _{\ell \neq p} U_\ell $ is a neat open compact subgroup of $G(\mathbb {A}^{p,\infty })$ such that

(6.1) $$ \begin{align} U_\ell = \prod_{\substack{v \in \underline{\mathcal{S}}_{\operatorname{spl}} \\ v \mid \ell}} K_{\mathfrak{B}_v}. \end{align} $$

Let $\mathcal {H}_{\operatorname {spl}, \mathbb {Z}_p} := \left (\bigotimes _{\ell \in S_{\operatorname {spl}}} \mathfrak {z}_\ell ^0\right )$ be the abstract ramified Hecke algebra. For any two integers $N_2 \geq N_1 \geq 0$ and any algebraic representation $\rho $ of $L_{(n)}$ over $\mathbb {Z}_{(p)}$ , recall that the classical space $H^0(\mathcal {X}^{\operatorname {min}}_{U^p(N_1,N_2)},\mathcal {E}^{\operatorname {sub}}_{U^p(N_1,N_2),\rho })$ has an action of $G(\mathbb {A}^{p,\infty })$ which induces an action of $\mathcal {H}_{\operatorname {spl},\mathbb {Z}_p}$ , and similarly, the p-adic spaces $H^0(\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{\rho },\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{\rho })$ and $H^0(\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{\rho },\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{\rho } \otimes \mathbb {Z}/p^M\mathbb {Z})$ have an action of $G(\mathbb {A}^{p,\infty })$ , which similarly induces an action of $\mathcal {H}_{\operatorname {spl},\mathbb {Z}_p}$ . Let $\mathbb {T}^p_{U^p(N_1,N_2),\rho }$ denote the image of $\mathcal {H}^p_{\mathbb {Z}_p} := \mathcal {H}^{\mathrm {ur}}_{\mathbb {Z}_p} \otimes \mathcal {H}_{\operatorname {spl},\mathbb {Z}_p}$ in

$$ \begin{align*}\operatorname{End}_{\mathbb{Z}_p}(H^0(\mathcal{X}^{\operatorname{min}}_{U^p(N_1,N_2)},\mathcal{E}^{\operatorname{sub}}_{U^p(N_1,N_2),\rho})).\end{align*} $$

Furthermore, if $W \subset H^0(\mathfrak {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1)},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1),\rho })$ (resp., $W \subset H^0(\mathcal {X}^{\operatorname {ord},\operatorname {min}}_{U^p(N_1,N_2)},\mathcal {E}^{\operatorname {ord},\operatorname {sub}}_{U^p(N_1,N_2),\rho } \otimes \mathbb {Z}/p^M\mathbb {Z})$ ) is a finitely generated $\mathbb {Z}_p$ -submodule invariant under the action of the algebra $\mathcal {H}^p_{\mathbb {Z}_p}$ , then let $\mathbb {T}^{\operatorname {ord},p}_{U^p(N_1,N_2),\rho }(W)$ (resp. $\mathbb {T}^{\operatorname {ord},p}_{U^p(N_1,N_2),\rho ,M}(W)$ ) denote the image of $\mathcal {H}^p_{\mathbb {Z}_p}$ in $\operatorname {End}_{\mathbb {Z}_p}(W)$ .

For each $v \in \underline {\mathcal {S}}_{\operatorname {spl}} \sqcup \underline {\mathcal {S}}^c_{\operatorname {spl}}$ , let $\tilde {T}_{v,\mathfrak {B},\sigma }$ denote the image of $d(T_{v,\mathfrak {B},\sigma })T_{v,\mathfrak {B},\sigma }$ in any $\mathcal {H}^p_{\mathbb {Z}_p}$ -algebra $\mathbb {T}$ .