2.2.1 Automorphic symbols

By condition (L2), the map

(24)\begin{equation} \iota_\beta : \tilde{S}^{H}_{L_\beta} \rightarrow {S^{G}_K}, \quad [h] \mapsto [\iota(h)\xi t_p^{\beta}], \end{equation}

is well-defined. As $\iota _\beta$ is proper by a well-known result of Borel and Prasad (see, for example, [Reference AshAsh80, Lemma 2.7]) one can consider the pull-back:

(25)\begin{equation} \iota_\beta^{*} : \operatorname{H}^{q}_c({S^{G}_K}, \mathcal{V}_{\mathcal{O}}^{\mu}) \longrightarrow \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \iota_\beta^{*}\mathcal{V}_{\mathcal{O}}^{\mu}). \end{equation}

2.2.2 Twisting

By condition (L1), the map $\iota : \tilde {S}^{H}_{L_\beta } \to {S^{G}_K}$, $[h]\mapsto [\iota (h)]$ is well-defined and proper. As $\xi t_p^{\beta }\in \Lambda _p$, using the $\bullet$-action from (12) one can consider the map

\[ H(\mathbb{A})\times V_{\mathcal{O}}^{\mu}\to H(\mathbb{A})\times V_{\mathcal{O}}^{\mu}, \quad (h,v)\mapsto (h, (\xi t_p^{\beta})\bullet v), \]

inducing a homomorphism of sheaves $\tau _\beta ^{\circ } : \iota _\beta ^{*}\mathcal {V}_{\mathcal {O}}^{\mu } \longrightarrow \iota ^{*}\mathcal {V}_{\mathcal {O}}^{\mu }$, hence a map in cohomology

(26)\begin{equation} \tau_\beta^{\circ} : \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \iota_\beta^{*}\mathcal{V}_{\mathcal{O}}^{\mu}) \longrightarrow \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \iota^{*}\mathcal{V}_{\mathcal{O}}^{\mu}). \end{equation}

Similarly, using the natural action of $G(E)$ on $V_{E}^{\mu }$ instead of the $\bullet$-action one defines a map

(27)\begin{equation} \tau_\beta : \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \iota_\beta^{*}\mathcal{V}_{E}^{\mu}) \longrightarrow \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \iota^{*}\mathcal{V}_{E}^{\mu}), \end{equation}

and $\tau _\beta =\mu ^{\vee }(t_p^{-\beta })\tau _\beta ^{\circ }$, because by (12) one has $(\xi t_p^{\beta }) \bullet v= \mu ^{\vee }(t_p^{\beta }) (\xi t_p^{\beta })\cdot v$ for all $v\in V_{E}^{\mu }$.

2.2.3 Critical maps

For $j_1,j_2 \in \mathbb {Z}$ let $V^{(j_1,j_2)}$ be the $1$-dimensional $H$-representation

\[ (h_1,h_2) \mapsto \operatorname{N}_{F/\mathbb{Q}}(\det(h_1)^{j_1} \det(h_2)^{j_2}). \]

Let $V^{(j_1,j_2)}_{\mathcal {O}}$ be a free rank one $\mathcal {O}$-module on which the previously defined natural $H(\mathbb {Z}_p)$-action is extended to a $H(\mathbb {Q}_p)$-action by letting $p\in \mathbb {Q}_p^{\times }$ act trivially. Note that this action is similar to the $\Lambda _p$-action on $V^{\mu }_{\mathcal {O}}$ defined in § 1.2.

It follows from [Reference Grobner and RaghuramGR14, Proposition 6.3] that $j \in \operatorname {Crit}(\mu )$ (see (4)) if and only if

(28)\begin{equation} \dim(\operatorname{Hom}_H(V^{\mu}, V^{(j,{{\sf w}}-j)}) ) = 1. \end{equation}

Fix a non-zero $\kappa _j \in \operatorname {Hom}_H(V^{\mu }, V^{(j,{{\sf w}}-j)})$ normalized so as to obtain an integral map:

\[ \kappa_j : V^{\mu}_{\mathcal{O}} \rightarrow V^{(j,{{\sf w}}-j)}_{\mathcal{O}}. \]

Denoting by $\mathcal {V}^{(j,{{\sf w}}-j)}_{\mathcal {O}}$ the sheaf on $\tilde {S}^{H}_{L_\beta }$ attached to $V^{(j,{{\sf w}}-j)}_{\mathcal {O}}$ by the construction described in § 1.3, one obtains a homomorphism:

(29)\begin{equation} \kappa_j : \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \iota^{*}\mathcal{V}_{\mathcal{O}}^{\mu}) \longrightarrow \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \mathcal{V}^{(j,{{\sf w}}-j)}_{\mathcal{O}}). \end{equation}

Putting (25), (26) and (29) together, for each $j \in \operatorname {Crit}(\mu )$, we obtain a map:

(30)\begin{equation} \kappa_j \circ \tau_\beta^{\circ}\circ \iota_\beta^{*} : \operatorname{H}^{q}_c({S^{G}_K}, \mathcal{V}_{\mathcal{O}}^{\mu}) \longrightarrow \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}, \mathcal{V}^{(j,{{\sf w}}-j)}_{\mathcal{O}}). \end{equation}

2.2.4 Trivializations

Given any $\delta \in H(\mathbb {A}_f)$ the map

\[ \operatorname{triv}_\delta:H(\mathbb{Q})\delta L_\beta H_\infty^{\circ}\times V^{(j,{{\sf w}}-j)}_{\mathcal{O}}\to H(\mathbb{Q})\delta L_\beta H_\infty^{\circ}\times V^{(j,{{\sf w}}-j)}_{\mathcal{O}}\, , \, (\gamma\delta l h_\infty, v)\mapsto (\gamma\delta l h_\infty, l_p^{-1}\cdot v), \]

is well-defined because $H(\mathbb {Q})\cap L_\beta H_\infty ^{\circ }\subset \ker (\operatorname {N}_{F/\mathbb {Q}}\circ \det )$ acts trivially on $V^{(j,{{\sf w}}-j)}_{\mathcal {O}}$. An easy check shows that $\operatorname {triv}_\delta$ induces a homomorphism of local systems

\[ \tilde{S}^{H}_{L_\beta}[\delta] \times V^{(j,{{\sf w}}-j)}_{\mathcal{O}}\to \big(\mathcal{V}^{(j,{{\sf w}}-j)}_{\mathcal{O}}\big)|_{\tilde{S}^{H}_{L_\beta}[\delta]}, \]

where $[\delta ]$ denotes the image of $\delta$ in $\pi _0(\tilde {S}^{H}_{L_\beta })$, hence yields a homomorphism:

\[ \operatorname{triv}_\delta^{*}: \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}[\delta], \mathcal{V}^{(j,{{\sf w}}-j)}_{\mathcal{O}}) \to \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}[\delta], \mathbb{Z})\otimes V^{(j,{{\sf w}}-j)}_{\mathcal{O}}. \]

We now render the trivializations independent of the choice of $\delta \in [\delta ]\in \pi _0(\tilde {S}^{H}_{L_\beta })$. By definition, for any $\delta '\in H(\mathbb {Q}) \delta l H_\infty$ one has

(31)\begin{equation} \operatorname{triv}_{\delta'}^{*}= (\operatorname{id}\otimes l_p^{-1})\cdot \operatorname{triv}_{\delta}^{*}=\operatorname{N}_{F_p/\mathbb{Q}_p}^{-1} \big(\det(l_{1,p})^{j}\det(l_{2,p})^{{{\sf w}}-j}\big)\operatorname{triv}_{\delta}^{*}. \end{equation}

The $p$-adic cyclotomic character $\varepsilon$ seen as idele class character $F^{\times } \backslash \mathbb {A}_F^{\times } \to \mathbb {Z}_p^{\times }$ sends $y$ to $\operatorname {N}_{F_p/\mathbb {Q}_p}(y_p)|y_f|_F\prod _{\sigma \in \Sigma _\infty } {\rm sgn}(y_\sigma )$, is trivial on $F_\infty ^{\times \circ }$ and given by $\operatorname {N}_{F_p/\mathbb {Q}_p}$ on $(\mathcal {O}_{F}\otimes \mathbb {Z}_p)^{\times }$. Hence,

(32)\begin{equation} \operatorname{triv}_{[\delta]}^{*}= \varepsilon \big(\det(\delta_{1}^{j}\delta_{2}^{{{\sf w}}-j})\big) \operatorname{triv}_\delta^{*} : \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}[\delta], \mathcal{V}^{(j,{{\sf w}}-j)}_{\mathcal{O}}) \to \operatorname{H}^{q}_c(\tilde{S}^{H}_{L_\beta}[\delta], \mathbb{Z})\otimes V^{(j,{{\sf w}}-j)}_{\mathcal{O}}, \end{equation}

is independent of the particular choice of $\delta \in [\delta ]\in \pi _0(\tilde {S}^{H}_{L_\beta })$.

2.2.5 Connected components and fundamental classes

Recall that for each $[\delta ]\in \pi _0(\tilde {S}^{H}_{L_\beta })$, $\tilde {S}^{H}_{L_\beta }[\delta ]$ is a $t$-dimensional connected orientable real manifold and that choosing an orientation amounts to choosing a fundamental class, that is, a basis $\theta _{[\delta ]}$ of its Borel–Moore homology $H_{t}^{\mathrm {BM}}(\tilde {S}^{H}_{L_\beta }[\delta ])\simeq \mathbb {Z}$. We choose such orientations in a consistent manner when $\beta$ and $[\delta ]$ vary as follows. First, we fix, once and for all, an ordered basis on the tangent space of the symmetric space $H_\infty ^{\circ }/L_\infty ^{\circ }$ yielding fundamental classes $\theta _{\beta }$ of the connected components of identity $\tilde {S}^{H}_{L_\beta }[1]$, when $\beta$ varies. Then for each $[\delta ]\in \pi _0(\tilde {S}^{H}_{L_\beta })$, we consider the isomorphism $\tilde {S}^{H}_{L_\beta }[1]\underset {\sim }{\overset {\cdot \delta }{\longrightarrow }} \tilde {S}^{H}_{L_\beta }[\delta ]$ and define $\theta _{[\delta ]}=\delta _*\theta _{\beta }$, which is clearly independent of the particular choice of $\delta \in [\delta ]$. Capping with $\theta _{[\delta ]}$ and fixing a basis of $V^{(j,{{\sf w}}-j)}_{\mathcal {O}}$ (later in (40) we fix a particular basis in order to compare evaluations at different $j$) yields an isomorphism:

\[ \operatorname{H}^{t}_c(\tilde{S}^{H}_{L_\beta}[\delta], \mathbb{Z})\otimes V^{(j,{{\sf w}}-j)}_{\mathcal{O}} \overset{\sim}{\longrightarrow} V^{(j,{{\sf w}}-j)}_{\mathcal{O}}\overset{\sim}{\longrightarrow} \mathcal{O}. \]

Combining this with (30) and (32) gives homomorphisms:

(33)\begin{align} \begin{aligned} \mathcal{E}^{j,{{\sf w}}}_{\beta,\delta}&=(\!\cap \theta_{[\delta]}) \circ\operatorname{triv}_\delta^{*}\circ\kappa_j \circ \tau_\beta^{\circ}\circ \iota_\beta^{*}: \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\mathcal{O}}^{\mu})\to \mathcal{O},\\ \mathcal{E}^{j,{{\sf w}}}_{\beta, [\delta]}&= \varepsilon \big(\det(\delta_{1}^{j}\delta_{2}^{{{\sf w}}-j})\big) \cdot \mathcal{E}^{j,{{\sf w}}}_{\beta,\delta}=(\!\cap \theta_{[\delta]}) \circ \operatorname{triv}_{[\delta]}^{*} \circ\kappa_j \circ \tau_\beta^{\circ}\circ \iota_\beta^{*} : \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\mathcal{O}}^{\mu})\to \mathcal{O}.\end{aligned} \end{align}

2.2.6 Summing over the second component

Consider a finite order $\mathcal {O}$-valued idele class character $\eta _0$ of $F$ that is trivial on $I(\mathfrak {m})$, in particular unramified at all places above $p$. The character $\eta =\eta _0|\cdot |_F^{-{{\sf w}}}$ later plays a role when we discuss Shalika models for automorphic representations of $G$. The following map provides a section of (22):

(34)\begin{equation} \delta(x,y) = (\operatorname{diag}(xy, 1,\ldots, 1), \operatorname{diag}(y, 1,\ldots ,1)) \in H. \end{equation}

When $(x,y)\in (\mathbb {A}_F^{\times })^{2}$ runs over a set of representatives of $\mathscr {C}\!\ell _F^{+}(p^{\beta }\mathfrak {m})\times \mathscr {C}\!\ell _F^{+}(\mathfrak {m})$, $\tilde {S}^{H}_{L_\beta }[\delta (x,y)]$ runs over the set of connected components of $\tilde {S}^{H}_{L_\beta }.$ Define the level $\beta$ evaluation:

(35)\begin{equation} \mathcal{E}^{j,\eta}_{\beta}= \sum_{[\bar x] \in \mathscr{C}\!\ell_F^{+}(p^{\beta})} \mathcal{E}^{j,\eta}_{\beta,[\bar x]}[\bar x],\quad \text{where } \mathcal{E}^{j,\eta}_{\beta,[\bar x]}=\sum_{[y] \in \mathscr{C}\!\ell_F^{+}(\mathfrak{m})} \sum_{[x]} \eta_0([y]) \mathcal{E}^{j,{{\sf w}}}_{\beta,[\delta( x,y)]}, \end{equation}

where the last sum runs over all $[x] \in \mathscr {C}\!\ell _F^{+}(p^{\beta }\mathfrak {m})$ mapping to $[\bar x]$ under the natural projection.

The following diagram recapitulates the steps in the construction of $\mathcal {E}^{j,\eta }_{\beta }$.

(36)

2.3.1 The distributive property

Fix a $\beta =(\beta _{\mathfrak {p}})_{\mathfrak {p}\mid p}$ with $\beta _{\mathfrak {p}}\in \mathbb {Z}_{>0}$ for all $\mathfrak {p}\mid p$.

Theorem 2.2 Given a prime $\mathfrak {p}\mid p$ we let $p^{\beta '}=p^{\beta }\mathfrak {p}$ and consider the canonical projection $\operatorname {pr}_{\beta ',\beta } : \mathscr {C}\!\ell _F^{+}(p^{\beta '}) \to \mathscr {C}\!\ell _F^{+}(p^{\beta })$. For all $[x] \in \mathscr {C}\!\ell _F^{+}(p^{\beta })$ we have $\mathcal {E}^{j,\eta }_{\beta }\circ U_\mathfrak {p}^{\circ }=\operatorname {pr}_{\beta ',\beta }\circ \mathcal {E}^{j,\eta }_{\beta '},$ that is,

\[ \mathcal{E}^{j,\eta}_{\beta,[x]}\circ U_\mathfrak{p}^{\circ} = \sum_{[x'] \in \operatorname{pr}_{\beta',\beta}^{-1}([x])} \mathcal{E}^{j,\eta}_{\beta',[x']}. \]

Proof. Using (33), (34), and (35) one has to show that for all $[x] \in \mathscr {C}\!\ell _F^{+}(p^{\beta }\mathfrak {m}), [y]\in \mathscr {C}\!\ell _F^{+}(\mathfrak {m})$:

\[ \mathcal{E}^{j,{{\sf w}}}_{\beta,\delta(x,y)}\circ U_\mathfrak{p}^{\circ} = \sum_{[x'] \in \operatorname{pr}_{\beta',\beta}^{-1}([x])} \operatorname{N}_{F_p/\mathbb{Q}_p}^{j}(u_{x'})\cdot \mathcal{E}^{j,{{\sf w}}}_{\beta',\delta(x',y)}, \]

where $u_{x'}\in I(p^{\beta })$ is such that $x'\in F^{\times } x u_{x'} F_\infty ^{\times \circ }$. We proceed as in the proof of [Reference Barrera, Dimitrov and JorzaBDJ, Proposition 3.4]. Pulling back the definition of the Hecke operator $U_\mathfrak {p}^{\circ }$ (see § 1.4) by the automorphic symbols (see § 2.2.1) and the twisting operators (see § 2.2.2) yields the following commutative diagram (we use implicitly that $p_{K^{0}(\mathfrak {p}),K}$ and $\operatorname {pr}_{\beta ',\beta }$ have the same degree as $L_{\beta }/L_{\beta '}\simeq M_n(\mathcal {O}/\mathfrak {p})$)

where the upper $[t_\mathfrak {p}]$ is induced by the morphism $(g,v)\mapsto (g\cdot t_\mathfrak {p}^{-1}, t_\mathfrak {p}\bullet v)$ of local systems, whereas the lower $[t_\mathfrak {p}]$ is induced by the morphism $(h,v)\mapsto (h, t_\mathfrak {p}\bullet v)$. Then

is another commutative diagram by (31), hence the claim.

2.3.2 Distributions for finite slope eigenvectors

Let $\phi \in \operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{\mathcal {O}}^{\mu })$ be an eigenvector for $U_{\mathfrak {p}}^{\circ }$ with eigenvalue $\alpha _\mathfrak {p}^{\circ }$ for all $\mathfrak {p}\mid p$. Then, for all $\beta =(\beta _\mathfrak {p})_{\mathfrak {p}\mid p}$, it is an eigenvector for $U_{p^{\beta }}^{\circ }$ with eigenvalue $\alpha _{p^{\beta }}^{\circ }=\prod _{\mathfrak {p}\mid p}(\alpha _\mathfrak {p}^{\circ })^{\beta _\mathfrak {p}}$. We say that $\phi$ is of finite slope if $\alpha _{p}^{\circ } \neq 0$ and in which case we define its slope as $v_p(\alpha _p^{\circ })$. A eigenvector $\phi$ of slope $0$ is called $Q$-ordinary. Being $Q$-ordinary is equivalent to saying that the $U_{\mathfrak {p}}$-eigenvalue $\alpha _\mathfrak {p}$ satisfies $|\alpha _\mathfrak {p}|_p = |\mu ^{\vee }(t_\mathfrak {p})|_p^{-1}$ for all $\mathfrak {p}\mid p$ (see § 4.2 for more details).

Given any $U_p$-eigenvector $\phi$ of finite slope and any $j\in \operatorname {Crit}(\mu )$ by Theorem 2.2 one has a well-defined element

(37)\begin{equation} \boldsymbol{\mu}^{j,\eta}_\phi=((\alpha_{p^{\beta}}^{\circ})^{-1} \mathcal{E}^{j,\eta}_{\beta}(\phi))_\beta, \end{equation}

which is thought of as an $E$-valued distribution on $\mathscr {C}\!\ell _F^{+}(p^{\infty })$.

We write $\operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{\mathcal {O}}^{\mu })^{Q-\mathrm {ord}}$ for the maximal $\mathcal {O}$-submodule of $\operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{\mathcal {O}}^{\mu })$ on which the operators $U_{\mathfrak {p}}^{\circ }$ are invertible for all $\mathfrak {p}\mid p$ (it is a direct $\mathcal {O}$-factor). Given any (not necessarily $U_p^{\circ }$-eigen) non-torsion element $\phi \in \operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{\mathcal {O}}^{\mu })^{Q-\mathrm {ord}}$ one defines

(38)\begin{equation} \boldsymbol{\mu}^{j,\eta}_\phi=(\mathcal{E}^{j,\eta}_{\beta}((U_{p^{\beta}}^{\circ})^{-1}(\phi)))_\beta \in \mathcal{O}[[\mathscr{C}\!\ell_F^{+}(p^{\infty})]]=\varprojlim_{\beta}\mathcal{O}[\mathscr{C}\!\ell_F^{+}(p^{\beta})], \end{equation}

which can be reinterpreted as a measure (i.e. a bounded distribution) on $\mathscr {C}\!\ell _F^{+}(p^{\infty })$.

2.4.1 Lie theoretic considerations

By the distributive property (see Theorem 2.2) we may reduce to strict $p^{\beta }$-power level with integral exponents $\beta \in \mathbb {Z}_{>0},$ ignoring the finer components $\mathfrak {p}\mid p$ for simplicity of notation. Recall that ${{\mathfrak {b}}}={{\mathfrak {t}}}\oplus {{\mathfrak {n}}}$ and ${{\mathfrak {q}}}={{\mathfrak {h}}}\oplus {{\mathfrak {u}}}$. With the notation $t_p=\iota (p\textbf {1}_n,\textbf {1}_n)$, we observe for any $\beta \geqslant 0$ the relations

\[ t_p^{\beta}{{\mathfrak{n}}}_\mathcal{O} t_p^{-\beta}\subseteq {{\mathfrak{n}}}_\mathcal{O}, \quad t_p^{\beta}{{\mathfrak{u}}}_\mathcal{O} t_p^{-\beta}=p^{\beta}{{\mathfrak{u}}}_\mathcal{O}. \]

Recall the matrix $\xi = \big (\begin {smallmatrix} \textbf {1}_n & w_n\\ \textbf {0}_n & w_n \end {smallmatrix}\big ).$ A superscript ${}^{\xi }(-)$ denotes left conjugation action by $\xi$.

Proof. (i) As $\xi \in G(\mathcal {O})$, it suffices to verify it over $E$, where it amounts to show that $\dim _E({{\mathfrak {h}}}_E\cap {}^{\xi }\,{{\mathfrak {b}}}_E^{-}) = n$. To this end, let $l_1,l_2$ be lower triangular matrices in $M_n(E)$ and $u\in M_n(E)$. Then

\[ \xi\cdot \begin{pmatrix} l_1 & \\ u & l_2 \end{pmatrix}\cdot \xi^{-1} = \begin{pmatrix} l_1+w_nu & w_n l_2 w_n-l_1-w_nu\\ w_nu & w_n l_2 w_n-w_nu \end{pmatrix}, \]

lies in ${{\mathfrak {h}}}_E$ if and only if $u = 0$ and $l_1 = w_n l_2 w_n$. Therefore, $l_1$ and $l_2$ are diagonal matrices determining each other uniquely.

(ii) Conjugation by $\xi ^{-1}$ reduces the claim to the problem of solving

\[ \begin{pmatrix} n_1 & \\ & n_2 \end{pmatrix}= \begin{pmatrix} h_1 & (h_1-h_2)w_n\\ & w_n h_2 w_n \end{pmatrix} \;+\; \begin{pmatrix} \bar{n}_1 & \\ & \bar{n}_2 \end{pmatrix}, \]

for given $\iota (n_1,n_2)\in {{\mathfrak {h}}}\cap {{\mathfrak {n}}}_\mathcal {O}$ and unknowns $\iota (h_1,h_2)\in [{{\mathfrak {h}}}_\mathcal {O},{{\mathfrak {h}}}_\mathcal {O}]$ and $\iota (\bar {n}_1,\bar {n}_2)\in {{\mathfrak {n}}}^{-}$. The choice

\[ h_1=h_2=n_1+w_n n_2 w_n,\quad \bar{n}_1=-w_n n_2 w_n,\quad \bar{n}_2=-w_n n_1 w_n, \]

is a solution with the desired properties.

Proof. (i) This is a consequence of Proposition 2.4(i) and the Poincaré–Birkhoff–Witt theorem.

(ii) The decomposition ${{\mathfrak {n}}}_\mathcal {O} = ({{\mathfrak {h}}}_\mathcal {O}\cap {{\mathfrak {n}}}_\mathcal {O})\oplus {{\mathfrak {u}}}_\mathcal {O},$ gives $t_p^{\beta }{{\mathfrak {n}}}_\mathcal {O} t_p^{-\beta } = ({{\mathfrak {h}}}_\mathcal {O}\cap {{\mathfrak {n}}}_\mathcal {O})\oplus p^{\beta }{{\mathfrak {u}}}_\mathcal {O}$. Conjugating by $\xi$, we obtain

\[ \xi t_p^{\beta}{{\mathfrak{n}}}_\mathcal{O} t_p^{-\beta}\xi^{-1}= {}^{\xi} \big({{\mathfrak{h}}}_\mathcal{O} \cap {{\mathfrak{n}}}_\mathcal{O} \big)\oplus p^{\beta} \, {}^{\xi}{{\mathfrak{u}}}_\mathcal{O}. \]

Applying Proposition 2.4(ii) to the first summand and Proposition 2.4(i) to the second we obtain

\[ \xi t_p^{\beta}{{\mathfrak{n}}}_\mathcal{O} t_p^{-\beta}\xi^{-1}\subseteq \big([{{\mathfrak{h}}},{{\mathfrak{h}}}]_\mathcal{O}+p^{\beta}{{\mathfrak{h}}}_\mathcal{O}\big)+ \big({}^{\xi}{{\mathfrak{n}}}_\mathcal{O}^{-} + p^{\beta} \, {}^{\xi} {{\mathfrak{b}}}_\mathcal{O}^{-} \big). \]

One concludes again by the Poincaré–Birkhoff–Witt theorem, because the sums within the parentheses on the right-hand side are Lie $\mathcal {O}$-algebras.

2.4.2 Lattices and the projection formula

Recall from (5) the lowest weight vector $v_0 \in V^{\mu }_{E}$ and the $G(\mathcal {O})$-lattice $V^{\mu }_{\mathcal {O}} = \mathcal {U}({{\mathfrak {g}}}_\mathcal {O})\cdot v_0 = \mathcal {U}({{\mathfrak {n}}}_\mathcal {O})\cdot v_0$. Recall also the $\bullet$-action of the semi-group $\Lambda _p$ on $V^{\mu }_{\mathcal {O}}$ as in (12).

Given $j \in \operatorname {Crit}(\mu )$ recall from § 2.2.3 the map $\kappa _j: V^{\mu }_{\mathcal {O}} \to V^{(j,{{\sf w}}-j)}_{\mathcal {O}}$. By Corollary 2.5(i)

\[ V^{\mu}_{\mathcal{O}}=\mathcal{U}({{\mathfrak{h}}}_\mathcal{O})\cdot \xi v_0, \]

which implies that $\kappa _j(\xi v_0)$ is an $\mathcal {O}$-basis of $V^{(j,{{\sf w}}-j)}_{\mathcal {O}}$ yielding a surjective $\mathcal {O}$-linear map

(40)\begin{equation} \kappa_j^{\circ} : V^{\mu}_{\mathcal{O}} \to \mathcal{O}, \quad \mbox{defined by } \kappa_j(v) = \kappa_j^{\circ}(v) \kappa_j(\xi v_0). \end{equation}

It is independent from the choice of $\kappa _j$ because of (28), and $\kappa _j^{\circ }(\xi v_0)=1$. We now come to the main technical result that is at the heart of our proof of the Manin relations.

Proposition 2.6 For any $\beta \geqslant 0$, $v\in (\xi t_p^{\beta })\bullet V^{\mu }_{\mathcal {O}} \subset V^{\mu }_{\mathcal {O}}$, and for all $j,j'\in \operatorname {Crit}(\mu )$, we have

(41)\begin{equation} \kappa^{\circ}_{j}(v) \equiv \kappa^{\circ}_{j'}(v)\pmod{p^{\beta}}. \end{equation}

Proof. By (12) for $v\in (\xi t_p^{\beta })\bullet V^{\mu }_{\mathcal {O}}$ there exists $m\in \mathcal {U}({{\mathfrak {n}}}_\mathcal {O})$ with

\[ v = \xi\cdot t_p^{\beta} \bullet (m v_0) = {}^{\xi t_p^{\beta}} m \cdot\xi (t_p^{\beta} \bullet v_0) = {}^{\xi t_p^{\beta}} m \cdot\xi v_0 \in \mathcal{U}({}^{\xi t_p^{\beta}}{{\mathfrak{n}}}_{\mathcal{O}}) \cdot\xi v_0. \]

By Corollary 2.5(ii) we can write

\[ {}^{\xi t_p^{\beta}} m = xy, \text{ with } x \in \mathcal{U}([{{\mathfrak{h}}},{{\mathfrak{h}}}]_\mathcal{O}+p^{\beta}{{\mathfrak{h}}}) \text{ and } y \in \mathcal{U}({}^{\xi}{{\mathfrak{n}}}_\mathcal{O}^{-}+p^{\beta}{}^{\xi}{{\mathfrak{b}}}_\mathcal{O}^{-}). \]

Let $x_0, y_0 \in \mathcal {O}$ be the degree zero terms of $x$ and $y$, respectively, and let

\begin{gather*} x_1 = x - x_0 \in ([{{\mathfrak{h}}},{{\mathfrak{h}}}]_\mathcal{O}+p^{\beta}{{\mathfrak{h}}})\cdot \mathcal{U}([{{\mathfrak{h}}},{{\mathfrak{h}}}]_\mathcal{O}+p^{\beta}{{\mathfrak{h}}}),\\ y_1 = y - y_0 \in ({}^{\xi}{{\mathfrak{n}}}_\mathcal{O}^{-}+p^{\beta}{}^{\xi}{{\mathfrak{b}}}_\mathcal{O}^{-})\cdot \mathcal{U}({}^{\xi}{{\mathfrak{n}}}_\mathcal{O}^{-}+p^{\beta}{}^{\xi}{{\mathfrak{b}}}_\mathcal{O}^{-}), \end{gather*}

be the higher degree terms in their respective enveloping algebras. Then

\begin{align*} \begin{aligned} \kappa_j^{\circ}(v)&= \kappa_j^{\circ}(xy \cdot \xi v_0) = x \cdot \kappa_j^{\circ}(y \cdot \xi v_0)\quad \mbox{(because $\kappa_j^{\circ}$ is $H$-equivariant)}\\ &\equiv x \cdot \kappa_j^{\circ}(y_0 \cdot \xi v_0)\pmod{p^{\beta}} \qquad\mbox{(because ${}^{\xi}{{\mathfrak{n}}}_\mathcal{O}^{-}$ acts trivially on $\xi v_0$)}\\ &\equiv x_0 \cdot\kappa_j^{\circ}(y_0\cdot\xi v_0)\pmod{p^{\beta}} \quad\,\,\,\mbox{(because $[{{\mathfrak{h}}},{{\mathfrak{h}}}]_\mathcal{O}$ acts trivially on a line)}\\ &= x_0y_0\cdot\kappa_j^{\circ}(\xi v_0) = x_0y_0,\end{aligned}\end{align*}

which does not depend on $j$ as claimed.

2.4.3 Proof of Theorem 2.3

As $\mathcal {O}[[\mathscr {C}\!\ell _F^{+}(p^{\infty })]]= \varprojlim _{\beta }(\mathcal {O}/p^{\beta }\mathcal {O})[\mathscr {C}\!\ell _F^{+}(p^{\beta })]$ and because $\varepsilon \pmod {p^{\beta }}$ factors through $\mathscr {C}\!\ell _F^{+}(p^{\beta })$, it is enough to check that given $\beta \geqslant 1$ and $[x]\in \mathscr {C}\!\ell _F^{+}(p^{\beta })$ one has

\[ \varepsilon ([x]) \mathcal{E}^{j,\eta}_{\beta,[x]}(\phi) \equiv \mathcal{E}^{j+1,\eta}_{\beta,[x]}(\phi) \pmod{p^{\beta}}. \]

As, by (32), one has $\operatorname {triv}_{[\delta (x,y)]}^{*}= \varepsilon (x^{j} y^{{{\sf w}}})\operatorname {triv}_{\delta (x,y)}^{*}$, it suffices to show that (see (36))

\[ \mathcal{E}^{j,\eta}_{\beta,\delta}(\phi)=(\!\cap \theta_{[\delta]}) \circ\operatorname{triv}_\delta^{*}\circ\kappa_j \circ \tau_\beta^{\circ}\circ \iota_\beta^{*}(\phi)\equiv (\!\cap \theta_{[\delta]}) \circ \operatorname{triv}_\delta^{*}\circ\kappa_{j+1}^{*} \circ \tau_\beta^{\circ}\circ \iota_\beta^{*}(\phi)\pmod{p^{\beta}}. \]

Now, by definition, the homomorphism of sheaves $\tau _\beta ^{\circ }$ defined in § 2.2.2 factors as

\[ \iota_\beta^{*}\mathcal{V}_{\mathcal{O}}^{\mu} \longrightarrow\iota^{*}\big((\xi t_p^{\beta})\bullet\mathcal{V}_{\mathcal{O}}^{\mu}\big) \longrightarrow \iota^{*}\mathcal{V}_{\mathcal{O}}^{\mu}. \]

Hence, Proposition 2.6 translates to the statement that (for the choice of basis of $V^{(j,{{\sf w}}-j)}_{\mathcal {O}}$ as in (40)) $(\!\cap \theta _{[\delta ]}) \circ \operatorname {triv}_\delta ^{*}\circ \kappa _j \circ \tau _\beta ^{\circ }\circ \iota _\beta ^{*}(\phi )\pmod {p^{\beta }}$ is independent of $j\in \operatorname {Crit}(\mu )$.

4.1.1 Global Shalika models

Let $\eta : F^{\times } \backslash \mathbb {A}_F^{\times } \to \mathbb {C}^{\times }$ be a Hecke character such that $\eta ^{n}$ equals the central character $\omega _\Pi$ of $\Pi$. We obtain an automorphic character:

\[ \eta \otimes \psi: S(F) \backslash S(\mathbb{A}_F)\to \mathbb{C}^{\times}, \quad \begin{pmatrix} h & hX\\ 0 & h \end{pmatrix}\mapsto \eta(\det(h))\psi(Tr(X)). \]

For a cusp form $\varphi \in \Pi$ and $g\in G(\mathbb {A})$ consider the integral

(54)\begin{equation} W^{\eta}_{\varphi}(g)=\int_{Z(\mathbb{A})S(F)\backslash S(\mathbb{A}_F)} \varphi(sg)(\eta \otimes \psi)^{-1}(s) \, d s, \end{equation}

where Haar measures are normalized as in [Reference Grobner and RaghuramGR14, § 2.8]. It is well-defined by the cuspidality of the function $\varphi$ (see [Reference Jacquet and ShalikaJS90, § 8.1]) and, hence, yields a function $W^{\eta }_{\varphi }: G(\mathbb {A})\rightarrow \mathbb {C}$ such that

\[ W^{\eta}_{\varphi}(sg) = (\eta\otimes\psi)(s)\cdot W^{\eta}_{\varphi}(g), \]

for all $g\in G(\mathbb {A})$ and $s\in S(\mathbb {A})$. In particular, we obtain an intertwining of $G(\mathbb {A})$-modules

(55)\begin{equation} \mathcal{S}^{\eta}_\psi: \Pi\rightarrow {\rm Ind}_{S(\mathbb{A})}^{G(\mathbb{A})}(\eta\otimes\psi), \quad \varphi\mapsto W^{\eta}_{\varphi}. \end{equation}

The following theorem, due to Jacquet and Shalika, gives a necessary and sufficient conditions for the existence of a non-zero intertwining as in (55).

This is proved in [Reference Jacquet and ShalikaJS90] for unitary representations and its extension to the non-unitary case is easy. If $\Pi$ satisfies any one, and hence all, of the equivalent conditions of Theorem 4.1, then we say that $\Pi$ has an $(\eta ,\psi )$-Shalika model, and we call the isomorphic image $\mathcal {S}^{\eta }_\psi (\Pi )$ of $\Pi$ under (55) a global $(\eta ,\psi )$-Shalika model of $\Pi$. Then clearly $\Pi \otimes \chi$ has an $(\eta \chi ^{2},\psi )$-Shalika model for any Hecke character $\chi$, by keeping the same model and only twisting the action.

The following proposition (see [Reference Asgari and ShahidiAS14]) gives another equivalent condition for $\Pi$ to have a global Shalika model.

4.1.2 Period integrals and $L$-functions

The following proposition, due to Friedberg and Jacquet, is crucial for much that follows. It relates the period integral over $H$ of a cusp form $\varphi$ of $G$ to a certain zeta integral of the function $W^{\eta }_{\varphi }$ in the Shalika model corresponding to $\varphi$ over one copy of $\operatorname {GL}_n$.

Proposition 4.3 [FJ93, Proposition 2.3] Assume that $\Pi$ has an $(\eta ,\psi )$-Shalika model. For $\varphi \in \Pi$,

\[ \Psi(s,\varphi,\chi,\eta)= \int_{Z(\mathbb{A})H(\mathbb{Q})\backslash H(\mathbb{A})} \varphi\bigg(\begin{pmatrix} h_1 & 0\\ 0 & h_2 \end{pmatrix}\bigg)(\chi | \cdot|^{s-{1}/{2}})\bigg(\frac{\det(h_1)}{\det(h_2)}\bigg)\eta^{-1}(\det(h_2)) \, d h_1 \,d h_2 \]

converges absolutely for all $s\in \mathbb {C}$. For $\Re (s)\gg 0$ it is equal to

\[ \zeta(s;W^{\eta}_{\varphi},\chi)=\int_{\operatorname{GL}_n(\mathbb{A}_F)} W^{\eta}_{\varphi}\bigg(\begin{pmatrix}h & 0\\0 & 1 \end{pmatrix}\bigg) \chi(\det(h)) |\!\det(h)|^{s-{1}/{2}} \, dh, \]

thus providing an analytic continuation of $\zeta (s;W^{\eta }_{\varphi },\chi )$ to all of $\mathbb {C}$.

Suppose the representation $\Pi$ of $G(\mathbb {A}) = \operatorname {GL}_{2n}(\mathbb {A}_F)$ decomposes as $\Pi = \otimes '_v \Pi _v$, where $\Pi _v$ is an irreducible admissible representation of $\operatorname {GL}_{2n}(F_v)$.

If $\Pi$ has a global Shalika model, then $\mathcal {S}^{\eta }_\psi$ defines local Shalika models at every place (see Definition 3.2). The corresponding local intertwining operators are denoted by $\mathcal {S}^{\eta _v}_{\psi _v}$ and their images by $\mathcal {S}^{\eta _v}_{\psi _v}(\Pi _v)$, whence $\mathcal {S}^{\eta }_\psi (\Pi )=\otimes '_v \mathcal {S}^{\eta _v}_{\psi _v}(\Pi _v)$. We can now consider cusp forms $\varphi$ such that the function $W_{\varphi }\in \mathcal {S}^{\eta }_\psi (\Pi )$ is factorizable as $W_\varphi =\otimes '_v W_{\varphi _v},$ where

\[ W_{\varphi_v}\in \mathcal{S}^{\eta_v}_{\psi_v}(\Pi_v)\subset {\rm Ind}_{S(F_v)}^{\operatorname{GL}_{2n}(F_v)}(\eta_v\otimes\psi_v). \]

Then the following factorization holds for $\Re (s) \gg 0$:

(56)\begin{equation} \zeta(s;W_{\varphi},\chi)=\prod_{v } \zeta_v(s;W_{\varphi_v},\chi_v), \end{equation}

where the non-Archimedean local zeta integrals $\zeta _v(s;W_{\varphi _v},\chi _v)$ are related to $L$-functions in Proposition 3.3.

Proposition 4.3 relates this Shalika zeta integral to a period integral over $H$, and the main thrust of [Reference Ash and GinzburgAG94], refined and generalized in [Reference Grobner and RaghuramGR14], is that the period integral over $H$ admits a cohomological interpretation, provided that $\Pi$ is of cohomological type.

4.1.3 Shalika models and cuspidal cohomology

In this section, we recall some well-known facts from Clozel [Reference ClozelClo90, § 3] (see also [Reference Grobner and RaghuramGR14, § 3.4]). Assume from now on that the cuspidal automorphic representation $\Pi$ is cohomological with respect to a dominant integral weight $\mu \in X_+^{*}(T)$ (see (3)), that is,

\[ \operatorname{H}^{q}({{\mathfrak{g}}}_\infty,K_\infty^{\circ} ;\Pi\otimes V^{\mu}_{\mathbb{C}}) = \operatorname{H}^{q}({{\mathfrak{g}}}_\infty,K_\infty^{\circ} ;\Pi_\infty\otimes V^{\mu}_{\mathbb{C}})\otimes\Pi_f \neq 0 \]

for some degree $q$. A necessary condition for the non-vanishing of this cohomology group is that the weight $\mu$ is pure, that is, $\mu \in X_0^{*}(T)$. For each archimedean place $\sigma \in \Sigma _{\infty }$, $\Pi _\sigma$ can be described explicitly as follows. For any integer $\ell \geqslant 1$ consider the unitary discrete series representation $D(\ell )$ of $\operatorname {GL}_2(\mathbb {R})$ of lowest non-negative ${\rm SO}_2$-type $\ell +1$ and central character $\mathrm {sgn}^{\ell +1}$. Let $P$ be the parabolic subgroup of $\operatorname {GL}_{2n}$ with Levi factor $\prod _{i=1}^{n} \operatorname {GL}_2$. Then

\[ \Pi_\sigma\simeq{\rm Ind}^{\operatorname{GL}_{2n}(\mathbb{R})}_{P(\mathbb{R})} \bigg(\bigotimes_{i=1}^{n} D(2(\mu_{\sigma,i}+n-i)+ 1-{{\sf w}})\otimes |\!\det\!|^{-{{\sf w}}/2}\bigg), \]

in particular $\omega _{\Pi _\sigma }=|\cdot |^{-n {{\sf w}}}$. The highest degree supporting cuspidal cohomology of $G$ is $t=|\Sigma _\infty |(n^{2}+n-1)$. For any character $\epsilon$ of $K_{\infty }/K_\infty ^{\circ }$ the $\epsilon$-eigenspace of

(57)\begin{equation} \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_\infty^{\circ} ; \Pi_\infty \otimes V^{\mu}_{\mathbb{C}}) = \operatorname{Hom}_{K_\infty^{\circ} } \big(\wedge^{t}({{\mathfrak{g}}}_\infty/{{\mathfrak{k}}}_\infty), \Pi_\infty \otimes V^{\mu}_{\mathbb{C}}\big), \end{equation}

is a line. If, in addition, $\Pi$ admits an $(\eta ,\psi )$-Shalika model, then $\eta$ is forced to be algebraic of the form $\eta = \eta _0 | \cdot |_F^{-{{\sf w}}}$ with $\eta _0$ of finite order, ${{\sf w}}$ is the purity weight of $\mu$ (see [Reference Gan and RaghuramGR13, Theorem 5.3]). Using the multiplicity one theorem for local Shalika models [Reference NienNie09] (see also [Reference Chen and SunCS20]), one deduces that, for any character $\epsilon$ of $K_{\infty }/K_\infty ^{\circ }$,

\[ \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_{\infty}^{\circ} ;\mathcal{S}^{\eta_\infty}_{\psi_\infty}(\Pi_\infty)\otimes V^{\mu}_{\mathbb{C}})[\epsilon], \]

is a line, a basis $\Xi _\infty ^{\epsilon }$ of which we fix in way compatible with twisting (see [Reference Grobner and RaghuramGR14, Lemma 5.1.1]). The relative Lie algebra cohomology of $\Pi$ as previously is a summand of the cuspidal cohomology, which, in turn, injects into the cohomology with compact supports (see [Reference Gan and RaghuramGR13, § 2])

(58)\begin{equation} \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_{\infty}^{\circ} ;\Pi^{K}\otimes V^{\mu}_{\mathbb{C}}) \ \hookrightarrow \ \operatorname{H}^{t}_{\rm cusp}({S^{G}_K}, \mathcal{V}_{\mathbb{C}}^{\mu}) \ \hookrightarrow \ \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\mathbb{C}}^{\mu}). \end{equation}

We define an isomorphism $\Theta ^{\epsilon }$ of $G(\mathbb {A}_f)$-modules as the composition

(59)\begin{align} \mathcal{S}^{\eta_f}_{\psi_f}(\Pi_f) &\xrightarrow{\sim} \mathcal{S}^{\eta_f}_{\psi_f}(\Pi_f) \otimes \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_{\infty}^{\circ} ; \mathcal{S}^{\eta_{\infty}}_{\psi_{\infty}}(\Pi_\infty) \otimes V^{\mu}_{\mathbb{C}})[\epsilon] \nonumber\\ &\xrightarrow{\sim} \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_{\infty}^{\circ} ; \mathcal{S}^{\eta}_{\psi}(\Pi) \otimes V^{\mu}_{\mathbb{C}})[\epsilon] \xrightarrow{\sim} \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_{\infty}^{\circ} ; \Pi \otimes V^{\mu}_{\mathbb{C}})[\epsilon], \end{align}

where the first map is $W_f \mapsto W_f \otimes \Xi _\infty ^{\epsilon }$, the second map is the natural one and the third map is the map induced in cohomology by $(\mathcal {S}^{\eta }_\psi )^{-1}$ from (55). Taking $K$-invariants in (59) and composing with (58) yields a Hecke equivariant embedding

(60)\begin{equation} \Theta^{\epsilon}_K: \mathcal{S}^{\eta_f}_{\psi_f}(\Pi_f)^{K} \hookrightarrow \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\mathbb{C}}^{\mu})[\epsilon]. \end{equation}

The reader should appreciate that the analytic condition on $\Pi$ of admitting a Shalika model and the algebraic condition of contributing to the cuspidal cohomology of $G$ are of an entirely different nature. One may construct examples of representations satisfying only one of these conditions and not the other (see [Reference Grobner and RaghuramGR14, § 3.5]).

4.3.1 Choice of local Shalika vectors

For $v\nmid p\infty$ we recall the vector $W_{\Pi _v}\in \mathcal {S}^{\eta _v}_{\psi _v}(\Pi _v)$ from Proposition 3.3.

For $\mathfrak {p} \mid p$, $\Pi _\mathfrak {p}$ is spherical and $\tilde \Pi _\mathfrak {p}=(\Pi _\mathfrak {p}, \{n+1,\ldots , 2n\})$ is $Q$-regular, in particular, $\alpha _\mathfrak {p}=\prod _{n+1\leqslant i \leqslant 2n}\alpha _{\mathfrak {p},i}$ is a simple eigenvalue for the Hecke operator $U_\mathfrak {p}$ acting on $\Pi _\mathfrak {p}^{J_\mathfrak {p}}$. By Lemma 3.6 there exists a unique $W_{\tilde \Pi _\mathfrak {p}}$ on the line $\mathcal {S}^{\eta _\mathfrak {p}}_{\psi _\mathfrak {p}}(\Pi _\mathfrak {p})^{J_\mathfrak {p}}[U_\mathfrak {p}-\alpha _\mathfrak {p}]$ normalized so that $W_{\tilde \Pi _\mathfrak {p}}(t_\mathfrak {p}^{-\delta _\mathfrak {p}})=1$. Let

(69)\begin{equation} W_{\tilde\Pi_f} = \otimes_{\mathfrak{p}\mid p}W_{\tilde\Pi_\mathfrak{p}} \bigotimes \otimes_{v\nmid p\infty}'W_{\Pi_v} \in\mathcal{S}^{\eta_f}_{\psi_f}(\Pi_f). \end{equation}

In addition to conditions (K1) and (K2) on $K$ (see § 2), we henceforth assume that:

1. (K3) $K$ fixes $W_{\tilde \Pi _f}$ and $\eta$ is trivial on $I(\mathfrak {m})$, hence can be seen as a character of $\mathscr {C}\!\ell _F^{+}(\mathfrak {m})$.

For the local vectors at infinity, given any character $\epsilon$ of $K_{\infty }/K_\infty ^{\circ }$ we recall the basis $\Xi _\infty ^{\epsilon }$ of the line $\operatorname {H}^{t}({{\mathfrak {g}}}_\infty ,K_{\infty }^{\circ }; \mathcal {S}^{\eta _\infty }_{\psi _\infty }(\Pi _\infty )\otimes V^{\mu }_{\mathbb {C}})[\epsilon ]$ from § 4.1.3. As in [Reference Grobner and RaghuramGR14, § 4.1], we define the following.

Definition 4.5 Fix a basis $\{e_\alpha \}$ of $V^{\mu }_{\mathbb {C}}$ and a basis $\{\omega _i\}$ of $\big ({{\mathfrak {g}}}_\infty /{{\mathfrak {k}}}_\infty \big )^{\vee }$. For $\underline i= (i_1,\ldots ,i_{t}),$ with $1 \leqslant i_1 < \cdots < i_{t} \leqslant \dim \big ({{\mathfrak {g}}}_\infty /{{\mathfrak {k}}}_\infty \big )$, we let $\omega _{\underline i}= \omega _{i_1}\wedge \cdots \wedge \omega _{i_{t}}\in \bigwedge ^{t} \big ({{\mathfrak {g}}}_\infty /{{\mathfrak {k}}}_\infty \big )^{\vee }.$

Using (57), $\Xi _\infty ^{\epsilon }$ can be written as a $K_\infty ^{\circ }$-invariant element

(70)\begin{equation} \Xi_\infty^{\epsilon} = \sum_{\underline i,\alpha} \omega_{\underline i}\otimes W_{\infty,\underline i, \alpha}^{\epsilon} \otimes e_\alpha \in \wedge^{t}({{\mathfrak{g}}}_\infty/{{\mathfrak{k}}}_\infty)^{\vee} \otimes \mathcal{S}^{\eta_\infty}_{\psi_\infty}(\Pi_\infty)\otimes V^{\mu}_{\mathbb{C}}, \end{equation}

for a unique choice of $W_{\infty , \underline i, \alpha }^{\epsilon } \in \mathcal {S}^{\eta _\infty }_{\psi _\infty }(\Pi _\infty )$ called ‘cohomological vectors at infinity’.

For $\underline i$ and $\alpha$ as previously, let $\varphi _{\underline i, \alpha }^{\epsilon } \in \Pi$ be the unique vector whose image under (55) equals

\[ W^{\eta}_{\varphi_{\underline i, \alpha}^{\epsilon}}=W_{\tilde\Pi_f} \otimes W_{\infty,\underline i, \alpha}^{\epsilon} \in\mathcal{S}^{\eta}_{\psi}(\Pi). \]

For each character $\epsilon$ of $K_{\infty }/K_{\infty }^{\circ }$ the embedding $\Theta ^{\epsilon }_K$ defined in (60) yields

\[ \mathcal{S}^{\eta_f}_{\psi_f}(\Pi_f) \xrightarrow{\Theta^{\epsilon}_K} \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\mathbb{C}}^{\mu})[\epsilon] \xrightarrow{i_p^{*}} \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\bar{\mathbb{Q}}_p}^{\mu})[\epsilon] \xrightarrow{(g,v)\mapsto (g, g^{-1}\cdot v)} \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\bar{\mathbb{Q}}_p}^{\mu})[\epsilon]. \]

The image $\phi _{\tilde \Pi }^{\epsilon }$ of $W_{\tilde \Pi _f}$ under the composition of these three maps belongs to $\operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{E}^{\mu })[\epsilon ]$ for some finite extension $E$ of $\mathbb {Q}_p$, and after possibly rescaling the maps $\Theta ^{\epsilon }$, that is, rescaling the basis elements $\Xi _\infty ^{\epsilon }$, one can render the cohomology class $\mathcal {O}$-integral:

(71)\begin{equation} \phi_{\tilde\Pi}^{\epsilon}\in \operatorname{H}^{t}_c({S^{G}_K}, \mathcal{V}_{\mathcal{O}}^{\mu})[\epsilon]. \end{equation}

Recall that the Hecke operator $U_{p^{\beta }}^{\circ }$ defines an endomorphism of $\operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{\mathcal {O}}^{\mu })[\epsilon ]$. As $W_{\tilde \Pi _f}$ is an $U_{p^{\beta }}$-eigenvector with eigenvalue $\alpha _{p^{\beta }}=\prod _{\mathfrak {p}\mid p}\alpha _\mathfrak {p}^{\beta _\mathfrak {p}}$, it follows (after possibly additionally rescaling by a power of $p$ killing the torsion in $\operatorname {H}^{t}_c({S^{G}_K}, \mathcal {V}_{\mathcal {O}}^{\mu })$) that one can assume $\phi _{\tilde \Pi }^{\epsilon }$ is an $U_{p^{\beta }}^{\circ }$-eigenvector with eigenvalue $\alpha _{p^{\beta }}^{\circ }=\mu ^{\vee }(t_p^{\beta }) \alpha _{p^{\beta }}$.

4.3.2 Interpolation formula at critical points

In this section, we relate the image of $\phi _{\tilde \Pi }^{\epsilon }$ defined in (71) by the evaluation map $\mathcal {E}^{j,\eta }_{\beta ,[\delta ]}$ from (33), to the Friedberg–Jacquet integral from Proposition 4.3.

Proposition 4.6 For any character $\epsilon$ of $K_{\infty }/K_\infty ^{\circ }$ and any $[\delta ]\in \mathscr {C}\!\ell _F^{+}(p^{\beta }\mathfrak {m})\times \mathscr {C}\!\ell _F^{+}(\mathfrak {m})$ we have

\[ i_p^{-1}\big( \mu^{\vee}(t_p^{-\beta})\cdot \mathcal{E}^{j,{{\sf w}}}_{\beta,[\delta]}(\phi_{\tilde\Pi}^{\epsilon})\big)= \int_{\tilde{S}^{H}_{L_\beta}[\delta]} \varphi_{\tilde\Pi,j}^{\epsilon}(h \xi t_p^{\beta}) |\!\det(h_1^{j} h_2^{{{\sf w}}-j}) |_F \,d h, \]

where $\varphi _{\tilde \Pi ,j}^{\epsilon } = \sum _{\underline i} \sum _\alpha a_{\underline i, \alpha , j}^{\epsilon } \cdot \varphi _{\underline i, \alpha }^{\epsilon }$ for suitable $a_{\underline i, \alpha , j}^{\epsilon } \in \mathbb {C}$.

In this proposition, and henceforth, $\sum _{\underline i}$ denotes summing over all ${\underline i} = (i_1,\ldots ,i_{t})$ and $\sum _\alpha$ denotes summing over all $1 \leqslant \alpha \leqslant \dim (V^{\mu }),$ as in Definition 4.5. A more careful choice of the bases $\{e_\alpha \}$ and $\{\omega _i\}$ as in [Reference JanuszewskiJan16, § 7] yields algebraic coefficients $a_{\underline i,\alpha ,j}^{\epsilon }\in \bar {\mathbb {Q}}$.

Proof. We follow closely the proof of [Reference Barrera, Dimitrov and JorzaBDJ, Proposition 4.1]. Consider the commutative diagram

where $\tau _\beta =\mu ^{\vee }(t_p^{-\beta })\tau _\beta ^{\circ }$ is defined in (27), the horizontal maps are induced from the morphisms of local systems written above them, the map $\mathcal {T}_{\beta }$ is induced from the morphisms of local systems $(h,v)\mapsto (h\xi t_p^{\beta }, v)$, and $\operatorname {triv}'_\delta$ is induced from the morphisms of local systems:

\[ H(\mathbb{Q})\delta L_\beta H_\infty^{\circ} \times V^{(j,{{\sf w}}-j)}_{\mathbb{Q}(\tilde\Pi)} \to \mathcal{V}^{(j,{{\sf w}}-j)}_{\mathbb{Q}(\tilde\Pi)| \tilde{S}^{H}_{L_\beta}[\delta]},\quad (\gamma\delta\ell h_\infty,v)\mapsto(\gamma\delta\ell h_\infty, \gamma^{-1}\cdot v). \]

As $\mathcal {E}^{j,{{\sf w}}}_{\beta , [\delta ]}= \varepsilon (\det (\delta _{1}^{j}\delta _{2}^{{{\sf w}}-j})) \mathcal {E}^{j,{{\sf w}}}_{\beta ,\delta }$, with $\varepsilon _f = |\cdot |_{F,f}\operatorname {N}_{F_p/\mathbb {Q}_p}$, the previous diagram shows that the proposition is equivalent to

(72)\begin{equation} |\!\det(\delta_{1,f}^{j}\delta_{2,f}^{{{\sf w}}-j})|_F \big(\!\cap \theta_{[\delta]})\circ\operatorname{triv}_\delta^{*}\circ\kappa_j\circ \mathcal{T}_{\beta}\big)(\phi_{\tilde\Pi}^{\epsilon}) = \int_{\tilde{S}^{H}_{L_\beta}[\delta]} \varphi_{\tilde\Pi,j}^{\epsilon}(h\xi t_p^{\beta}) |\!\det(h_1^{j} h_2^{{{\sf w}}-j}) |_F \,d h, \end{equation}

the left-hand side being considered over $\mathbb {C}$ via $i_p^{-1}:\bar{\mathbb {Q}}_p \xrightarrow {\sim } \mathbb {C}$. By Definition 4.5

\[ \phi_{\tilde\Pi}^{\epsilon}= (\mathcal{S}^{\eta}_\psi)^{-1}( W_{\tilde\Pi_f} \otimes \Xi_\infty^{\epsilon}) = \sum_{\underline i} \sum_{\alpha} \omega_{\underline i}\otimes \varphi_{\underline i, \alpha}^{\epsilon} \otimes e_\alpha \in \big(\wedge^{t}({{\mathfrak{g}}}_\infty/{{\mathfrak{k}}}_\infty)^{\vee}\otimes \Pi \otimes V^{\mu}_{\mathbb{C}}\big)^{K_\infty^{\circ}}, \]

yielding a $V^{\mu }_{\mathbb {C}}$-valued differential $t$-form on $G_\infty ^{\circ }/K_\infty ^{\circ }$. Now, recall the basis $\kappa _j$ of the one-dimensional $\operatorname {Hom}_H(V^{\mu }, V^{(j,{{\sf w}}-j)})$ from (28) and consider the map

\[ \kappa_j\circ \iota^{*}: \operatorname{H}^{t}({{\mathfrak{g}}}_\infty,K_\infty^{\circ} ;\Pi_\infty \otimes V^{\mu}_{\mathbb{C}})\to \operatorname{H}^{t}({{\mathfrak{h}}}_\infty,L^{\circ}_\infty;\Pi_\infty \otimes V^{(j,{{\sf w}}-j)}_{\mathbb{C}}). \]

We obtain

\[ (\kappa_j\circ \mathcal{T}_{\beta})(\phi_{\tilde\Pi}^{\epsilon}) = \sum_{\underline i} \sum_\alpha \iota^{*} \omega_{\underline i} \otimes \varphi_{\underline i, \alpha}^{\epsilon}(-\cdot \xi t_p^{\beta}) \otimes \kappa_j(e_\alpha) \in \big(\wedge^{t}({{\mathfrak{h}}}_\infty/{{\mathfrak{l}}}_\infty)^{\vee}\otimes \Pi \otimes V^{(j,{{\sf w}}-j)}_{\mathbb{C}}\big)^{L_\infty^{\circ}}. \]

First, let $\kappa _j^{\circ }: V^{(j,{{\sf w}}-j)}_{\mathbb {C}}\xrightarrow {\sim } \mathbb {C}$ be the scalar extension of (40), and so $\kappa _j(e_\alpha )$ corresponds to a complex number. Next, after the discussion in the paragraph following (23), we can fix a basis for the top exterior $\wedge ^{t}({{\mathfrak {h}}}_\infty /{{\mathfrak {l}}}_\infty )^{\vee }$ corresponding to the Haar measure $dh_\infty$; hence $\iota ^{*} \omega _{\underline i}$ is a scalar multiple of $dh_\infty$. Putting both together, the restriction to $\tilde {S}^{H}_{L_\beta }[\delta ]$ of $\kappa _j( \mathcal {T}_{\beta }( \phi _{\tilde \Pi }^{\epsilon }))$ can be seen as $V^{(j,{{\sf w}}-j)}_{\mathbb {C}}$-valued top-degree differential form on $H_\infty ^{\circ }/L_\infty ^{\circ }$ given by

\[ \sum_{\underline i} \sum_\alpha a_{\underline i, \alpha, j}^{\epsilon} \cdot \varphi_{\underline i, \alpha}^{\epsilon}(\delta h_\infty \xi t_p^{\beta}) \det(h_{1,\infty}^{ j} h_{2,\infty}^{ {{\sf w}}-j}) \, d h_\infty \]

for suitable $a_{\underline i, \alpha , j}^{\epsilon }\in \mathbb {C}$. Writing $h=\gamma \delta l h'_\infty \in H(\mathbb {Q})\delta L_\beta H_\infty ^{\circ }\subset H(\mathbb {A})$, and using $\operatorname {triv}'_\delta (\gamma \delta \ell h'_\infty ,v)=(\gamma \delta \ell h'_\infty , \det (\gamma _{1,\infty }^{-j}\gamma _{2,\infty }^{j-{{\sf w}}}) v)$ one obtains (72) and the proposition from

\begin{align*} |\!\det(\delta_{1,f}^{j}\delta_{2,f}^{{{\sf w}}-j})|_F \det(\gamma_{1,\infty}^{-j}\gamma_{2,\infty}^{j-{{\sf w}}})\det(h_{1,\infty}^{ j} h_{2,\infty}^{ {{\sf w}}-j}) & = |\!\det(\delta_{1,f}^{j}\delta_{2,f}^{{{\sf w}}-j})|_F\det(h_{1,\infty}^{\prime j} h_{2,\infty}^{\prime {{\sf w}}-j})\\ & = |\!\det(h_1^{j} h_2^{{{\sf w}}-j})|_F. \end{align*}

4.3.3 $p$-adic distributions attached to $\tilde \Pi$

Recall that $\Pi$ is a cuspidal automorphic representation of $G(\mathbb {A})$ admitting a global $(\psi ,\eta )$-Shalika model, which is cohomological with respect to a pure dominant integral weight $\mu$. Recall also that $\Pi _\mathfrak {p}$ is spherical for all $\mathfrak {p}\mid p$ and that $\tilde \Pi _\mathfrak {p}=(\Pi _\mathfrak {p}, \{n+1,\ldots , 2n\})$ is $Q$-regular (see Definition 3.5), which by Lemma 4.4 is automatically fulfilled if $\Pi _\mathfrak {p}$ is $Q$-ordinary. In all cases $\Pi _\mathfrak {p}^{J_\mathfrak {p}}$ contains a unique line on which $U_\mathfrak {p}$ acts by $\alpha _{\mathfrak {p}}$. Finally recall the $U_\mathfrak {p}^{\circ }$-eigenvectors $\phi _{\tilde \Pi }^{\epsilon }$ constructed in (71). Then

(73)\begin{equation} \phi_{\tilde\Pi}=\sum_{\epsilon: F_{\infty}^{\times}/F_{\infty}^{\times\circ} \to \{\pm 1\}}\phi_{\tilde\Pi}^{\epsilon} \end{equation}

is an $U_\mathfrak {p}^{\circ }$-eigenvector with same eigenvalue. When $\tilde \Pi _p$ is $Q$-ordinary, consider the element

(74)\begin{equation} \boldsymbol{\mu}_{\tilde\Pi}^{\eta}=\boldsymbol{\mu}_{\phi_{\tilde\Pi}}^{\eta}= \varepsilon_{\operatorname{cyc}}^{-j}(\boldsymbol{\mu}^{j,\eta}_{\phi_{\tilde\Pi}})\in \mathcal{O}[[\mathscr{C}\!\ell_F^{+}(p^{\infty})]], \end{equation}

constructed in (37) and (39), which defines a measure $d\boldsymbol {\mu }_{\tilde \Pi }^{\eta }$ on $\mathscr {C}\!\ell _F^{+}(p^{\infty })$.

4.3.4 Main theorem on $p$-adic interpolation

Fix any character $\epsilon$ of $K_{\infty }/K_\infty ^{\circ }$ and any $j\in \operatorname {Crit}(\mu )$. Consider the following cohomological test vector:

(75)\begin{equation} W_{\Pi_\infty,j}^{\epsilon} = \sum_{\underline i,\alpha} a_{\underline i, \alpha, j}^{\epsilon} W_{\infty, \underline i, \alpha}^{\epsilon}. \end{equation}

A crucial result of Sun [Reference SunSun19, Theorem 5.5] asserts the following non-vanishing:

(76)\begin{equation} \zeta_\infty\big(j+\tfrac{1}{2};W_{\Pi_\infty,j}^{\epsilon}\big) \in \mathbb{C}^{\times}. \end{equation}

Note that because $\Pi _\infty \otimes {\rm sgn} = \Pi _\infty$ we have suppressed $\chi _\infty$ from the notation.

In fact, using Künneth's theorem, it is easy to see that $\zeta _\infty (j+\frac {1}{2};W_{\Pi _\infty ,j}^{\epsilon })$ is a product of similar quantities parsed over the archimedean places. As we have multiplicity one for cuspidal cohomology in top-degree (see (57)), we can only change the class $\Xi _\infty ^{\epsilon }$ by a non-zero scalar, which correspondingly scales the cohomological test vector and so also the zeta integral. The variation of the complex period $\zeta _\infty (j+\frac {1}{2};W_{\Pi _\infty ,j}^{\epsilon })$ in $j$ is studied in [Reference JanuszewskiJan16]. The reader is also referred to the discussion around [Reference Grobner and RaghuramGR14, Theorem 6.6.2].

Recall the auxiliary ideal $\mathfrak {m}$ from (L1) in § 2.1 and, for brevity, let us define

(77)\begin{equation} \gamma= \#\mathscr{C}\!\ell_F^{+}(\mathfrak{m})\cdot \# \operatorname{GL}_n(\mathcal{O}_F/\mathfrak{m})\cdot \# \operatorname{PGL}_n(\mathcal{O}_F/\mathfrak{m})\cdot \prod_{\mathfrak{p}\mid p} \big(q_\mathfrak{p}^{-n^{2}} \cdot \# \operatorname{GL}_n(\mathcal{O}_F/\mathfrak{p})\big) \in\mathbb{Q}^{\times}. \end{equation}

Theorem 4.7 Let $\Pi$ be a cuspidal automorphic representation of $\operatorname {GL}_{2n}/F$ admitting a $(\psi ,\eta )$-Shalika model and such that $\Pi _\infty$ is cohomological of weight $\mu$. Assume that for all $\mathfrak {p}\mid p$, $\Pi _\mathfrak {p}$ is spherical and admits a $Q$-regular refinement $\tilde \Pi _{\mathfrak {p}}$. Then for any $j\in \operatorname {Crit}(\mu )$ and for any finite order character $\chi$ of $\mathscr {C}\!\ell _F^{+}(p^{\infty })$ of conductor $\beta _\mathfrak {p} \geqslant 1$ at $\mathfrak {p}\mid p$:

\begin{align*} i_p^{-1}\bigg( \int_{\mathscr{C}\!\ell_F^{+}(p^{\infty})} \chi(x) \, d \boldsymbol{\mu}_{\phi_{\tilde\Pi}}^{\eta,j}(x) \bigg) &= \gamma \cdot \operatorname{N}^{jn}_{F/\mathbb{Q}}(\mathfrak{d}) \cdot \prod_{\mathfrak{p}\mid p}\big(\alpha_{\mathfrak{p}}^{-1} q_\mathfrak{p}^{n(j+1)}\big)^{\beta_\mathfrak{p}} \\ &\quad \cdot {\mathcal{G}(\chi_f)^{n} \cdot L\bigg(j+\frac{1}{2}, \Pi_f \otimes \chi_f\bigg) } \zeta_\infty\bigg(j+\frac{1}{2};W_{\Pi_\infty,j}^{(\varepsilon^{j}\chi\eta)_\infty}\bigg). \end{align*}

Proof. Using (37) and (35) we find that $\int _{\mathscr {C}\!\ell _F^{+}(p^{\infty })} \chi (x) \, d \boldsymbol {\mu }_{\phi _{\tilde \Pi }}^{\eta ,j}(x)$ equals

\[ (\alpha_{p^{\beta}}^{\circ})^{-1} \sum_{[x]\in \mathscr{C}\!\ell_F^{+}(p^{\beta}\mathfrak{m})}\chi([x])\mathcal{E}^{j,\eta}_{\beta,[x]}(\phi_{\tilde\Pi}) \nonumber = \alpha_{p^{\beta}}^{-1} \cdot \mu^{\vee}(t_p^{-\beta}) \sum_{\substack{[x]\in \mathscr{C}\!\ell_F^{+}(p^{\beta}\mathfrak{m}) \\ [y]\in \mathscr{C}\!\ell_F^{+}(\mathfrak{m})}} \chi([x]) \eta_0([y]) \mathcal{E}^{j,{{\sf w}}}_{\beta,[\delta(x,y)]}(\phi_{\tilde\Pi}). \]

As $\pi _0(\tilde {S}^{H}_{L_\beta })\simeq \mathscr {C}\!\ell _F^{+}(p^{\beta } \mathfrak {m})\times \mathscr {C}\!\ell _F^{+}(\mathfrak {m})$ by (73) and Proposition 4.6 the integral equals

\[ \alpha_{p^{\beta}}^{-1} \cdot \sum_{\epsilon: \{\pm 1\}^{\Sigma_\infty}\to \{\pm 1\}} \int_{\tilde{S}^{H}_{L_\beta}} \varphi_{\tilde\Pi,j}^{\epsilon}(h\xi t_p^{\beta}) \chi\bigg(\frac{\det(h_1)}{\det(h_2)}\bigg) \bigg| \frac{\det(h_1)}{\det(h_2)}\bigg|_F^{j} \eta^{-1}( \det(h_2)) \, d h. \]

Note that the integrand is $L_\infty ^{\circ } Z(\mathbb {A}_f)$-invariant and $L_\infty /L_\infty ^{\circ } Z_\infty$ acts on it by $\epsilon \varepsilon _\infty ^{j}\chi _\infty \eta _\infty$, hence the integral vanishes unless $\epsilon =(\varepsilon ^{j}\chi \eta )_\infty$. As $Z(\mathbb {A}_f)\cap L_\beta$ is independent of $\beta$, after some volume computation, one further finds

\[ i_p^{-1}\bigg( \int_{\mathscr{C}\!\ell_F^{+}(p^{\infty})}\varepsilon^{j}(x) \chi(x) \, d \boldsymbol{\mu}_{\tilde\Pi}^{\eta}(x)\bigg) =\gamma \cdot \Psi\bigg(j+\frac{1}{2},\varphi_{\tilde\Pi,j}^{(\varepsilon^{j}\chi\eta)_\infty}(-\cdot \xi t_p^{\beta}),\chi,\eta\bigg) \prod_{\mathfrak{p}\mid p} (\alpha_{\mathfrak{p}}^{-1} q_\mathfrak{p}^{n^{2}})^{\beta_\mathfrak{p}}. \]

By Proposition 4.3 the Friedberg–Jacquet integral has an Euler product for $\Re (s)\gg 0$:

\[ {\Psi(s,\varphi_{\tilde\Pi,j}^{\epsilon}(-\cdot \xi t_p^{\beta}),\chi,\eta)} = \prod_{v\nmid p\infty}\zeta_v(s;W_{\Pi_v},\chi_v)\cdot \prod_{\mathfrak{p}\mid p}\zeta_\mathfrak{p}(s;W_{\tilde\Pi_\mathfrak{p}}(-\cdot\xi t_\mathfrak{p}^{\beta_\mathfrak{p}}),\chi_\mathfrak{p})\cdot \zeta_\infty(s;W_{\Pi_\infty,j}^{\epsilon}, \chi_\infty). \]

As $L(s, \Pi \otimes \chi )$ has trivial Euler factors at all places $\mathfrak {p}\mid p$ (as $\Pi _\mathfrak {p}$ is spherical while $\chi _\mathfrak {p}$ is ramified), Proposition 3.3 implies that

\[ \prod_{v\nmid p\infty}\zeta_v\bigg(j+\frac{1}{2};W_{\Pi_v},\chi_v\bigg)= \operatorname{N}_{F/\mathbb{Q}}^{jn}(\mathfrak{d}^{(p)}) \chi(\mathfrak{d}^{-1})^{n} L\bigg(j+\frac{1}{2}, \Pi_f \otimes \chi_f\bigg). \]

The factor at $\mathfrak {p}\mid p$ is computed in Proposition 3.4, which together with Lemma 3.6 gives

\[ q_\mathfrak{p}^{\beta_\mathfrak{p} n^{2}} \zeta_\mathfrak{p}\big(j+\tfrac{1}{2};W_{\tilde\Pi_\mathfrak{p}},\chi_\mathfrak{p}\big) = \mathcal{G}(\chi_\mathfrak{p})^{n} \cdot q_\mathfrak{p}^{\beta_\mathfrak{p} n+ (\beta_\mathfrak{p}+\delta_\mathfrak{p})jn}W_{\tilde\Pi_\mathfrak{p}}(\textbf{1}_{2n})= \operatorname{N}_{F/\mathbb{Q}}^{jn}(\mathfrak{d}_p) \mathcal{G}(\chi_\mathfrak{p})^{n} \cdot q_\mathfrak{p}^{\beta_\mathfrak{p} n(j+1)}. \]

As $\mathcal {G}(\chi _f)=\chi (\mathfrak {d}^{-1}) \prod _{\mathfrak {p}\mid p} \mathcal {G}(\chi _\mathfrak {p})$ and $\mathfrak {d}=\mathfrak {d}_p\mathfrak {d}^{(p)}$ we obtain the desired formula.