2 Notations

Let  $F$ be a nonarchimedean local field of residual characteristic  $p$ and residual cardinality  $q$ . Throughout,  $R$ will denote one of the fields  $\mathbb{C}$ ,  $\overline{\mathbb{Q}_{\ell }}$ , and  $\overline{\mathbb{F}_{\ell }}$ and we assume that $\ell \neq p$ . For  $E$ any extension of  $F$ , we denote by  $\mathfrak{o}_{E}$ the ring of integers of  $E$ ; by  $\unicode[STIX]{x1D71B}_{E}$ a uniformizer of  $E$ ; by  $\mathfrak{p}_{E}=(\unicode[STIX]{x1D71B}_{E})$ the unique maximal ideal of  $\mathfrak{o}_{E}$ ; by  $q_{E}$ the residual cardinality of  $E$ ; and let  $||_{E,R}:E^{\times }\rightarrow R^{\times }$ denote the unramified character defined by  $|\unicode[STIX]{x1D71B}_{E}|_{E,R}=q_{E}^{-1}$ , thus  $||_{E,\mathbb{C}}$ is the restriction of the absolute value to  $E^{\times }$ normalized in the usual way. When the field  $R$ considered is clear we remove the index  $R$ from  $||_{E,R}$ , and when  $E=F$ we remove the index  $F$ as well.

Let  $G_{n}=\text{GL}_{n}(F)$ ,  $K_{n}=\text{GL}_{n}(\mathfrak{o}_{F})$ ,  $K_{n}^{1}=1+\text{Mat}_{n,n}(\mathfrak{p}_{F})$ , and let  $Z_{n}$ be the center of  $G_{n}$ . For  $g$ in  $G_{n}$ , by abuse of notation, we denote by  $|g|$ the quantity  $|\!\det (g)|$ . Put  $\unicode[STIX]{x1D702}_{n}=(0~\cdots ~0~1)\in \text{Mat}_{1,n}(F)$ , and let  $P_{n}$ be the standard mirabolic subgroup of  $G_{n}$ , that is, the set of all matrices  $g$ in  $G_{n}$ such that  $\unicode[STIX]{x1D702}_{n}g=\unicode[STIX]{x1D702}_{n}$ . Let  $N_{n}$ be the unipotent radical of the standard Borel subgroup of upper triangular matrices in  $G_{n}$ . For  $k\in \mathbb{Z}$ , let  $G_{n}^{(k)}=\{g\in G_{n}:~|g|_{F}=q^{-k}\}$ . For any subset  $X$ of  $G_{n}$ , let  $X^{(k)}=X\cap G_{n}^{(k)}$ , and let  $\mathbf{1}_{X}$ denote the characteristic function of  $X$ .

Let  $\overline{\mathbb{Q}_{\ell }}$ denote an algebraic closure of the  $\ell$ -adic numbers,  $\overline{\mathbb{Z}_{\ell }}$ denote its ring of integers, and  $\overline{\mathbb{F}_{\ell }}$ denote its residue field which is an algebraic closure of the finite field of  $\ell$ -elements.

3 Representations with coefficients in  $R$

We only consider smooth  $R$ -representations, that is smooth representations with coefficients in  $R$ , and we use  $\vee$ as an exponent to denote the contragredient. We call a representation on a  $\overline{\mathbb{Q}_{\ell }}$ -vector space an  $\ell$ -adic representation, and a representation on an  $\overline{\mathbb{F}_{\ell }}$ -vector space an  $\ell$ -modular representation. Let  $(\unicode[STIX]{x1D70B},{\mathcal{V}})$ be an irreducible  $\ell$ -adic representation of  $G_{n}$ . We call  $\unicode[STIX]{x1D70B}$ integral if  ${\mathcal{V}}$ contains a  $G_{n}$ -stable  $\overline{\mathbb{Z}_{\ell }}$ -lattice. Notice that for an  $\ell$ -adic character  $\unicode[STIX]{x1D708}:G_{n}\rightarrow \overline{\mathbb{Q}_{\ell }}^{\times }$ this just means that  $\unicode[STIX]{x1D708}$ takes values in  $\overline{\mathbb{Z}_{\ell }}^{\times }$ .

An  $R$ -representation is called cuspidal if it is irreducible and never appears as a quotient of a properly parabolically induced representation. By [Reference Vignéras13, II 4.12], a cuspidal  $\ell$ -adic representation is integral if and only if its central character is integral, hence the contragredient of a cuspidal  $\ell$ -adic representation  $\unicode[STIX]{x1D70B}$ is integral if and only if  $\unicode[STIX]{x1D70B}$ is integral. Let  $\unicode[STIX]{x1D70B}$ be an integral cuspidal  $\ell$ -adic representation and  $\mathfrak{L}$ be a  $G_{n}$ -stable  $\overline{\mathbb{Z}_{\ell }}$ -lattice in the space of  $\unicode[STIX]{x1D70B}$ . Let  $r_{\mathfrak{L}}(\unicode[STIX]{x1D70B})$ be the  $\ell$ -modular representation induced on the space  $\mathfrak{L}\otimes _{\overline{\mathbb{Z}_{\ell }}}\overline{\mathbb{F}_{\ell }}$ . This  $\ell$ -modular representation is also cuspidal (and irreducible) by [Reference Vignéras13, III 5.10], and hence independent of the choice of the lattice  $\mathfrak{L}$ by the Brauer–Nesbitt principle [Reference Vignéras14, Theorem 1], we thus write  $r_{\ell }(\unicode[STIX]{x1D70B})$ for  $r_{\mathfrak{L}}(\unicode[STIX]{x1D70B})$ and call  $r_{\ell }(\unicode[STIX]{x1D70B})$ the reduction modulo  $\ell$ of  $\unicode[STIX]{x1D70B}$ . We also say that  $\unicode[STIX]{x1D70B}$ lifts  $r_{\ell }(\unicode[STIX]{x1D70B})$ , and it follows from [Reference Vignéras13, III 5.10] that all cuspidal  $\ell$ -modular representations lift to cuspidal  $\ell$ -adic representations. Following [Reference Mínguez and Sécherre11, Remark 8.15], we call a cuspidal  $\ell$ -modular representation  $\unicode[STIX]{x1D70F}$ banal if  $\unicode[STIX]{x1D70F}\not \simeq \unicode[STIX]{x1D70F}\otimes ||_{F}$ (notice that the definition in [Reference Mínguez and Sécherre11, Remark 8.15] refers to a condition given in Proposition 8.9 of this reference, which in the cuspidal case reduces to the condition we give here). For  $H$ a closed subgroup of  $G$ , we write  $\text{Ind}_{H}^{G}$ for the functor of smooth induction taking representations of  $H$ to representations of  $G$ , and write  $\text{ind}_{H}^{G}$ for the functor of smooth induction with compact support.

4 Normalization of Haar measures

We now discuss our normalization of Haar measures. The basic reference for  $R$ -Haar measures is [Reference Vignéras14, I 2], but we also refer the reader to [Reference Kurinczuk and Matringe8, Section 2.2] for more details on the splitting of Haar measures with respect to standard decompositions. Let  $dg$ be the Haar measure on  $G_{n}$ normalized to give  $K_{n}^{1}$ volume  $1$ .

We normalize the right Haar measure on  $P_{n}$ so that  $dp(P_{n}\cap K_{n}^{1})=1$ , on $N_{n}$ so that  $dn(N_{n}\cap K_{n}^{1})=1$ , and on  $Z_{n}$ so that  $dz(Z_{n}\cap K_{n}^{1})=1$ . For the remainder of this section, let  $G$ denote a closed subgroup of  $G_{n}$ with Haar measure  $d_{G}g$ . For any open subgroup  $U$ of  $G$ , we define the Haar measure  $d_{U}g$ on  $U$ as the restriction of  $d_{G}g$ , in particular  $d_{U}g$ is normalized as soon as  $d_{G}g$ is.

If  $H$ is a closed subgroup of  $G$ with right Haar measure  $d_{H}h$ , and such that the modulus character of  $G$ restricts to  $H$ as the modulus character of  $H$ , we descend  $d_{G}g$ to a right-invariant measure  $d_{H\backslash G}g$ on  $H\backslash G$ as explained in [Reference Vignéras14, I 2.8]. For  $f$ a smooth map from  $G$ to  $R$ with compact support, denoting by  $f^{H}$ the map on  $H\backslash G$ defined by

$$\begin{eqnarray}f^{H}(g)=\int _{H}f(hg)\,d_{H}h,\end{eqnarray}$$

the usual relation is satisfied:

$$\begin{eqnarray}\int _{H\backslash G}f^{H}(g)\,d_{H\backslash G}g=\int _{G}f(g)\,d_{G}g.\end{eqnarray}$$

This implies that  $d_{H\backslash G}g$ is normalized as soon as  $d_{G}g$ and  $d_{H}g$ are.

Indeed, if  $K$ is a compact subgroup of  $G$ , applying the equality above to  $f=\mathbf{1}_{K}$ , so that

$$\begin{eqnarray}f^{H}=d_{H}(K\cap H)\mathbf{1}_{H\backslash HK}\end{eqnarray}$$

gives the relation

(1) $$\begin{eqnarray}d_{G}(K)=d_{H\backslash G}(H\backslash HK)d_{H}(K\cap H).\end{eqnarray}$$

This gives for example the normalization

$$\begin{eqnarray}d_{H\backslash G}(H\backslash HK_{n}^{1})=d_{H}(H\cap K_{n}^{1})\backslash d_{G}(G\cap K_{n}^{1}).\end{eqnarray}$$

With these normalizations, we have the splitting

$$\begin{eqnarray}dg=|p|_{F}^{-1}\,dp\,dz\,dk.\end{eqnarray}$$

This splitting descends on  $N_{n}\backslash G_{n}$ , in which case  $dg$ denotes the normalized right-invariant measure on  $N_{n}\backslash G_{n}$ and  $dp$ the right-invariant measure on $N_{n}\backslash P_{n}$ . Notice that with such normalizations, the volume of all pro- $p$ subgroups of  $G_{n}$ , of  $P_{n}$ and of  $Z_{n}$ will be (positive or negative) powers of  $q$ . Moreover, for such choices, reduction modulo  $\ell$ commutes with integration (cf. [Reference Kurinczuk and Matringe8, Remark 2.1]), that is, if  $f\in {\mathcal{C}}_{c}^{\infty }(X,\overline{\mathbb{Z}_{\ell }})$ for  $X$ equal to  $G_{n}$ or any of the homogeneous spaces  $K\backslash L$ with  $L$ a subgroup of  $G_{n}$ considered above, then  $\int _{X}f(x)\,dx\in \overline{\mathbb{Z}_{\ell }}$ , and 

$$\begin{eqnarray}r_{\ell }\biggl(\int _{X}f(x)\,dx\biggr)=\int _{X}r_{\ell }(f(x))\,dx.\end{eqnarray}$$

For the rest of this section, we suppose that  $R$  has characteristic zero, and we recall some classical equalities, which all follow from Relation (1).

For a finite set  $A$ , we let  $|A|$ denote its cardinality in  $R$ . Suppose that $G=K$ compact, and  $U$ is an open subgroup of  $K$ , then

(2) $$\begin{eqnarray}d_{U\backslash K}(U\backslash K)=\frac{d_{K}(K)}{d_{K}(U)}=|U\backslash K|\in R.\end{eqnarray}$$

Finally, if  $V$ is a closed subgroup of  $K$ (using the fact that  $K$ is unimodular, hence that  $d_{K}(UV)=d_{K}(V^{-1}U^{-1})=d_{K}(VU)$ ), one obtains

(3) $$\begin{eqnarray}\displaystyle d_{V\backslash K}(V\backslash VU) & = & \displaystyle \frac{d_{K}(VU)}{d_{V}(V)}=\frac{d_{K}(UV)}{d_{K}(U)}\frac{d_{V}(V\cap U)}{d_{V}(V)}\frac{d_{K}(U)}{d_{V}(V\cap U)}\nonumber\\ \displaystyle & = & \displaystyle \frac{|U\backslash UV|}{|V\cap U\backslash V|}\frac{d_{K}(U)}{d_{V}(V\cap U)}=\frac{d_{K}(U)}{d_{V}(V\cap U)}=d_{V\cap U\backslash U}(V\cap U\backslash U).\end{eqnarray}$$

By convention, from now on, we use the same letter for the measure on  $G$ and its descent to  $H\backslash G$ (and when the context is clear for its restriction to an open subgroup as well).

5 Rankin–Selberg integrals and local factors

Let  $\unicode[STIX]{x1D713}$ be an additive character of  $F$ which is trivial on  $\mathfrak{p}_{F}$ , but nontrivial on  $\mathfrak{o}_{F}$ . By abuse of notation, also denote by  $\unicode[STIX]{x1D713}$ the nondegenerate character of  $N_{n}$ defined for  $x=(x_{i,j})\in N_{n}$ by

$$\begin{eqnarray}\unicode[STIX]{x1D713}(x)=\unicode[STIX]{x1D713}\biggl(\mathop{\sum }_{i=1}^{n-1}x_{i,i+1}\biggr),\end{eqnarray}$$

which is necessarily integral in the  $\ell$ -adic case because  $N_{n}$ is exhausted by its pro- $p$ subgroups. If  $\unicode[STIX]{x1D70B}$ is a cuspidal representation of  $G_{n}$ , then it is generic (cf. [Reference Bernstein and Zelevinsky1] in the complex or  $\ell$ -adic case, and [Reference Vignéras13, III 5.10] for  $R=\overline{\mathbb{F}_{\ell }}$ ), meaning  $\text{dim}(\text{Hom}_{N_{n}}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713}))=1$ , and hence it has a unique Whittaker model  $W(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})$ , equal to the image of  $\unicode[STIX]{x1D70B}$ in  $\text{Ind}_{N_{n}}^{G_{n}}(\unicode[STIX]{x1D713})$ . Suppose that  $\unicode[STIX]{x1D70B}$ is an integral cuspidal  $\ell$ -adic representation of  $G_{n}$ , then the  $\overline{\mathbb{Z}_{\ell }}$ -submodule  $W_{e}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})$ of  $W(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})$ consisting of all functions in  $W(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})$ which take values in  $\overline{\mathbb{Z}_{\ell }}$ is a  $G_{n}$ -stable lattice in  $\unicode[STIX]{x1D70B}$ (cf. [Reference Vignéras14, Theorem 2]). Then by definition  $r_{\ell }(\unicode[STIX]{x1D70B})\simeq W_{e}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})\otimes _{\overline{\mathbb{Z}_{\ell }}}\overline{\mathbb{F}_{\ell }}$ , which is irreducible and cuspidal (cf. Section 2.1 and the references given there). Thus  $W_{e}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})\otimes _{\overline{\mathbb{Z}_{\ell }}}\overline{\mathbb{F}_{\ell }}$ is a space of Whittaker functions for  $\unicode[STIX]{x1D70B}$ with values in  $\overline{\mathbb{F}_{\ell }}$ , hence equal to  $W(r_{\ell }(\unicode[STIX]{x1D70B}),r_{\ell }(\unicode[STIX]{x1D713}))$ . For  $W\in W_{e}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713})$ , we write  $r_{\ell }(W)$ for the image of  $W$ in  $W(r_{\ell }(\unicode[STIX]{x1D70B}),r_{\ell }(\unicode[STIX]{x1D713}))$ .

Finally, we recall the definition of the Rankin–Selberg local  $L$ -factors for a pair of cuspidal  $R$ -representations of  $G_{n}$ . The construction is originally due to Jacquet–Piatetski-Shapiro–Shalika [Reference Jacquet, Piatetski-Shapiro and Shalika5] for complex representations, and works equally well for  $\overline{\mathbb{Q}_{\ell }}$ -representations. This construction was extended to a construction for representations over any algebraically closed field of characteristic prime to  $p$ in [Reference Kurinczuk and Matringe8]. As we are ultimately interested in  $\mathbb{C},\overline{\mathbb{Q}_{\ell }}$ and  $\overline{\mathbb{F}_{\ell }}$ representations we give precise references to the construction in [Reference Kurinczuk and Matringe8].

Let  $\unicode[STIX]{x1D70B}_{1}$ and  $\unicode[STIX]{x1D70B}_{2}$ be cuspidal representations of  $G_{n}$ ,  $W_{1}\in W(\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D713})$ ,  $W_{2}\in W(\unicode[STIX]{x1D70B}_{2},\unicode[STIX]{x1D713}^{-1})$ , and  $\unicode[STIX]{x1D6F7}\in {\mathcal{C}}_{c}^{\infty }(F^{n})$ be a locally constant function from  $F^{n}$ to  $R$ with compact support. By [Reference Kurinczuk and Matringe8, Proposition 3.3], for  $k\in \mathbb{Z}$ , the coefficients

$$\begin{eqnarray}c_{k}(W_{1},W_{2},\unicode[STIX]{x1D6F7})=\int _{N_{n}\backslash G_{n}^{(k)}}W_{1}(g)W_{2}(g)\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D702}_{n}g)\,dg\end{eqnarray}$$

are well defined and vanish for  $k$ sufficiently negative. In fact, these coefficients vanish for  $k$ sufficiently negative because both  $W_{1}$ and  $W_{2}$ vanish on  $P_{n}^{(k)}$ for such  $k$ , as a consequence of [Reference Jacquet, Piatetski-Shapiro and Shalika5, Proposition 2.2]. Hence the local Rankin–Selberg integral

$$\begin{eqnarray}I(X,W_{1},W_{2},\unicode[STIX]{x1D6F7})=\mathop{\sum }_{k\in \mathbb{Z}}c_{k}(W_{1},W_{2},\unicode[STIX]{x1D6F7})X^{k}\end{eqnarray}$$

is a formal Laurent series with coefficients in  $R$ . In fact, by [Reference Kurinczuk and Matringe8, Theorem 3.5],  $I(X,W_{1},W_{2},\unicode[STIX]{x1D6F7})\in R(X)$ is a rational function, and as  $W_{1}$ varies in  $W(\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D713})$ ,  $W_{2}$ varies in  $W(\unicode[STIX]{x1D70B}_{2},\unicode[STIX]{x1D713}^{-1})$ , and  $\unicode[STIX]{x1D6F7}$ varies in  ${\mathcal{C}}_{c}^{\infty }(F^{n})$ , the  $R$ -submodule of  $R(X)$ spanned by  $I(X,W_{1},W_{2},\unicode[STIX]{x1D6F7})$ is a fractional ideal of $R[X^{\pm 1}]$ , and has a unique generator  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ which is an Euler factor. We call  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ the local Rankin–Selberg  $L$ -factor, and note that it does not depend on the choice of the character  $\unicode[STIX]{x1D713}$ . If  $R=\overline{\mathbb{Q}_{\ell }}$ , it is shown in [Reference Kurinczuk and Matringe8, Corollary 3.6] that the  $L$ -factor is the inverse of a polynomial in  $\overline{\mathbb{Z}_{\ell }}[X]$ , and thus it makes sense to talk of its reduction modulo  $\ell$ . Moreover, it follows from [Reference Kurinczuk and Matringe8, Theorem 3.13], that if  $\unicode[STIX]{x1D70B}_{1}$ and  $\unicode[STIX]{x1D70B}_{2}$ are two integral cuspidal  $\ell$ -adic representations of  $G_{n}$ , then one has

$$\begin{eqnarray}L(X,r_{\ell }(\unicode[STIX]{x1D70B}_{1}),r_{\ell }(\unicode[STIX]{x1D70B}_{2}))|r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})).\end{eqnarray}$$

Now by [Reference Jacquet, Piatetski-Shapiro and Shalika5, Proposition 8.1, (ii)], the  $L$ -factor  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is equal to  $1$ unless $\unicode[STIX]{x1D70B}_{2}\simeq \unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee }$ for some unramified character  $\unicode[STIX]{x1D712}$ of  $F^{\times }$ . Hence if  $\unicode[STIX]{x1D70B}_{2}\not \simeq \unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee }$  then  $L(X,r_{\ell }(\unicode[STIX]{x1D70B}_{1}),r_{\ell }(\unicode[STIX]{x1D70B}_{2}))=r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}))=1$ .

For our computations to come, we use a decomposition of the Rankin–Selberg integral in the special case where  $\unicode[STIX]{x1D70B}_{2}\simeq \unicode[STIX]{x1D70B}_{1}^{\vee }$ , in particular their central characters are inverse of each other. Thus we assume this is the case for the rest of this section. For  $k\in \mathbb{Z}$ , we set

$$\begin{eqnarray}b_{k}(W_{1},W_{2})=\int _{N_{n}\backslash P_{n}^{(k)}}W_{1}(p)W_{2}(p)\,dp,\end{eqnarray}$$

which, similarly to  $c_{k}$ , vanishes for  $k$ sufficiently negative, and we put

$$\begin{eqnarray}I_{(0)}(X,W_{1},W_{2})=\mathop{\sum }_{k\in \mathbb{Z}}b_{k}(W_{1},W_{2})q^{k}X^{k}.\end{eqnarray}$$

Let  $\unicode[STIX]{x1D6F7}\in {\mathcal{C}}_{c}^{\infty }(F^{n})$ be a  $K_{n}$ -invariant function, for  $i\in \mathbb{Z}$ , we set

$$\begin{eqnarray}a_{ni}(\unicode[STIX]{x1D6F7})=\int _{z\in G_{1}^{(ni)}}\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D702}_{n}z)\,dz,\end{eqnarray}$$

which vanishes for  $i$ sufficiently negative, and we put

$$\begin{eqnarray}Z(X,\unicode[STIX]{x1D6F7})=\mathop{\sum }_{i\in \mathbb{Z}}a_{ni}(\unicode[STIX]{x1D6F7})X^{ni}.\end{eqnarray}$$

As  $G_{n}^{(k)}=\coprod _{i\in \mathbb{Z}}P_{n}^{(k-ni)}Z_{n}^{(ni)}K_{n}$ , from the splitting of Section 4 we find

$$\begin{eqnarray}c_{k}(W_{1},W_{2},\unicode[STIX]{x1D6F7})=\mathop{\sum }_{i\in \mathbb{Z}}a_{ni}(\unicode[STIX]{x1D6F7})q^{k-ni}\int _{(K_{n}\cap P_{n})\backslash K_{n}}b_{k-ni}(\unicode[STIX]{x1D70C}(k)W_{1},\unicode[STIX]{x1D70C}(k)W_{2})\,dk,\end{eqnarray}$$

from which we deduce

$$\begin{eqnarray}I(X,W_{1},W_{2},\unicode[STIX]{x1D6F7})=Z(X,\unicode[STIX]{x1D6F7})\biggl(\int _{(K_{n}\cap P_{n})\backslash K_{n}}I_{(0)}(X,\unicode[STIX]{x1D70C}(k)W_{1},\unicode[STIX]{x1D70C}(k)W_{2})\,dk\biggr).\end{eqnarray}$$

Taking  $\unicode[STIX]{x1D6F7}$ equal to the characteristic function  $\mathbf{1}_{\mathfrak{o}_{F}^{n}}$ , we obtain the formula

(4) $$\begin{eqnarray}I(X,W_{1},W_{2},\mathbf{1}_{\mathfrak{o}_{F}^{n}})=\frac{q-1}{1-X^{n}}\int _{(K_{n}\cap P_{n})\backslash K_{n}}I_{(0)}(X,\unicode[STIX]{x1D70C}(k)W_{1},\unicode[STIX]{x1D70C}(k)W_{2})\,dk.\end{eqnarray}$$

The equality  $Z(X,\mathbf{1}_{\mathfrak{o}_{F}^{n}})=(q-1/1-X^{n})$ is standard (cf. [Reference Mínguez10, Theorem 3.1]) except that in our setting, we get the extra constant  $q-1$ from our choice of normalization on  $Z_{n}$ , as we set  $dz(Z_{n}\cap K_{n}^{1})=1$ instead of the usual  $dz(Z_{n}\cap K_{n})=1$ .

6 Simple types and reduction modulo  $\ell$

For the beginning of this section we assume that  $R=\mathbb{C}$  or  $\overline{\mathbb{Q}_{\ell }}$ . Let  $V$ be an  $n$ -dimensional  $F$ -vector space, let  $\text{End}_{F}(V)$ denote the  $F$ -algebra $\text{End}_{F}(V)$ of  $F$ -endomorphisms of  $V$ and let  $G$ denote the group  $\text{Aut}_{F}(V)$ of  $F$ -automorphisms of  $V$ . Hence  $G$ identifies with  $G_{n}$ as soon as we choose a basis of  $V$ . In [Reference Bushnell and Kutzko2], every cuspidal  $R$ -representation of  $G$ is constructed explicitly as  $\text{ind}_{\mathbf{J}}^{G}(\unicode[STIX]{x1D6EC})$ , where  $\mathbf{J}$ is an open and compact-mod-center subgroup of  $G$ , and  $\unicode[STIX]{x1D6EC}$ is an irreducible representation of  $\mathbf{J}$ of finite dimension. The pairs  $(\mathbf{J},\unicode[STIX]{x1D6EC})$ are called extended maximal simple types, and for any such pair  $\text{ind}_{\mathbf{J}}^{G}(\unicode[STIX]{x1D6EC})$ is (irreducible and) cuspidal by [Reference Bushnell and Kutzko2, Chapter 6]. We briefly explain the construction of the group  $\mathbf{J}$ , focusing on the properties which we shall use.

An  $\mathfrak{o}_{F}$ -lattice chain  ${\mathcal{L}}$ in  $V$ is a nonempty set of  $\mathfrak{o}_{F}$ -lattices  $\{L_{i}:i\in \mathbb{Z}\}$ such that, for all  $i\in \mathbb{Z}$ ,  $L_{i+1}\subsetneq L_{i}$ and there exists  $e({\mathcal{L}})\in \mathbb{Z}$ such that  $L_{i+e({\mathcal{L}})}=\unicode[STIX]{x1D71B}_{F}L_{i}$ . The construction of [Reference Bushnell and Kutzko2], starts with data  $(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ called maximal simple strata consisting of

1. (1) an element  $\unicode[STIX]{x1D6FD}\in \text{End}_{F}(V)$ which generates a simple field extension $E=F[\unicode[STIX]{x1D6FD}]$ ;

2. (2) an $\mathfrak{o}_{F}$ -lattice chain  ${\mathcal{L}}$ in  $V$ such that  $E^{\times }{\mathcal{L}}\subset {\mathcal{L}}$ (i.e., for any $x\in E^{\times }$ and $L\in {\mathcal{L}}$ we have $xL\in {\mathcal{L}}$ ); in particular  ${\mathcal{L}}$ is an $\mathfrak{o}_{E}$ -lattice chain, and it is required (as  $(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ is maximal) that  $L_{i+1}=\unicode[STIX]{x1D71B}_{E}L_{i}$ ;

which satisfy a technical condition (cf. [Reference Bushnell and Kutzko2, 1.5.5] where the simple strata we consider are among those denoted $[\mathfrak{A},-,0,\unicode[STIX]{x1D6FD}]$ ).

Let  $(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ be a maximal simple strata. We denote by  $\mathfrak{A}=\mathfrak{A}({\mathcal{L}})$ the  $\mathfrak{o}_{F}$ -order in  $\text{End}_{F}(V)$ and  $\mathfrak{B}=\mathfrak{B}(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ the  $\mathfrak{o}_{E}$ -order in  $\text{End}_{E}(V)$ defined by  ${\mathcal{L}}$ ,

$$\begin{eqnarray}\mathfrak{A}=\text{End}_{\mathfrak{o}_{F}}({\mathcal{L}})=\mathop{\bigcap }_{k}\text{End}_{\mathfrak{o}_{F}}(L_{k}),\qquad \mathfrak{B}=\mathfrak{B}(\unicode[STIX]{x1D6FD},{\mathcal{L}})=\text{End}_{\mathfrak{o}_{E}}({\mathcal{L}})=\text{End}_{\mathfrak{o}_{E}}(L_{0}).\end{eqnarray}$$

In [Reference Bushnell and Kutzko2, 3.1] Bushnell–Kutzko define compact open subgroups of  $G$ denoted by  $H^{1}=H^{1}(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ ,  $J^{1}=J^{1}(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ , and  $J=J(\unicode[STIX]{x1D6FD},{\mathcal{L}})$ . The properties we need are:

1. (1) the groups  $H^{1}\leqslant J^{1}$ are pro- $p$ (by definition), are normalized by  $E^{\times }$ and are normal subgroups of  $J$ by [Reference Bushnell and Kutzko2, 3.1.15], moreover $J\subset \text{Aut}_{\mathfrak{o}_{F}}(L_{0})$ (by definition).

2. (2) Put  $m=n/[E:F]$ , by [Reference Bushnell and Kutzko2, 3.1.15] we have

   $$\begin{eqnarray}\displaystyle J & = & \displaystyle \mathfrak{B}^{\times }J^{1},\qquad \mathfrak{B}^{\times }\cap J^{1}=1+\unicode[STIX]{x1D71B}_{E}\mathfrak{B}\qquad \text{and}\qquad \nonumber\\ \displaystyle & & \displaystyle J/J^{1}\simeq \mathfrak{B}^{\times }/(1+\unicode[STIX]{x1D71B}_{E}\mathfrak{B})\simeq G_{m}(k_{E}).\nonumber\end{eqnarray}$$

We then set  $\mathbf{J}=\mathbf{J}(\unicode[STIX]{x1D6FD},{\mathcal{L}})=E^{\times }J$ , in particular  $\mathbf{J}$ is compact mod  $E^{\times }$ and hence compact mod  $F^{\times }$ . Notice that if  $\unicode[STIX]{x1D70B}\simeq \text{ind}_{\mathbf{J}}^{G}\unicode[STIX]{x1D6EC}$ , the center  $F^{\times }$ of  $G$ acts by the central character  $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}$ of  $\unicode[STIX]{x1D70B}$ through  $\unicode[STIX]{x1D6EC}$ . Finally, we note that the construction of  $\unicode[STIX]{x1D6EC}$ depends on our fixed additive character  $\unicode[STIX]{x1D713}$ (cf. [Reference Bushnell and Kutzko2, 3.2]).

The definitions above do not include the groups of the maximal simple types for level zero cuspidal representations (see [Reference Bushnell and Kutzko2, 5.5.10(b)]), although these can be considered formally as part of the construction described above for the maximal zero strata  $(0,{\mathcal{L}})$ with  $\unicode[STIX]{x1D6FD}=0$ and  $e({\mathcal{L}})=1$ . In this case, we put  $J=\mathfrak{A}^{\times }$ ,  $\mathbf{J}=F^{\times }J$ ,  $H^{1}=J^{1}=1+\unicode[STIX]{x1D71B}_{F}\mathfrak{A}$ , and  $J/J^{1}=\mathfrak{A}^{\times }/(1+\unicode[STIX]{x1D71B}_{F}\mathfrak{A})\simeq G_{n}(k_{F})$ .

Now we consider  $\overline{\mathbb{F}_{\ell }}$ -representations. It follows from [Reference Vignéras13, Chapitre IV] that the Bushnell–Kutzko classification of cuspidal  $\overline{\mathbb{Q}_{\ell }}$ -representations adapts well to  $\overline{\mathbb{F}_{\ell }}$ -representations. We only need to know the following facts:

Let  $\unicode[STIX]{x1D70F}$ be a cuspidal  $\ell$ -modular representation of  $G$ . As we recalled in Section 3, there exists an integral cuspidal  $\ell$ -adic representation  $\unicode[STIX]{x1D70B}$ such that  $\unicode[STIX]{x1D70F}=r_{\ell }(\unicode[STIX]{x1D70B})$ . Choose an extended maximal simple type  $(\mathbf{J},\unicode[STIX]{x1D6EC})$ such that  $\unicode[STIX]{x1D70B}\simeq \text{ind}_{\mathbf{J}}^{G}(\unicode[STIX]{x1D6EC})$ , as in the beginning of this section. A cuspidal  $\ell$ -adic representation is integral if and only if its central character  $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}$ is integral, by [Reference Vignéras13, II 4.13] (the direction integral implies integral central character being clear). We recall why this is true. First as  $\mathbf{J}$ is compact mod  $F^{\times }$ , we claim that the irreducible representation  $\unicode[STIX]{x1D6EC}$ is integral if and only if  $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}$ is integral. Again, one direction is clear. For the other, suppose that  $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}$ is integral and choose a random not necessarily  $\mathbf{J}$ -stable lattice  $\mathfrak{L}_{0}$ in the space  $V_{\unicode[STIX]{x1D6EC}}$ of  $\unicode[STIX]{x1D6EC}$ . It is stabilized by a compact open subgroup  $U$ of  $\mathbf{J}$ , and choosing representatives  $c_{1},\ldots ,c_{r}$ of $\mathbf{J}/F^{\times }U$ , one has  $\unicode[STIX]{x1D6EC}(\mathbf{J})(\mathfrak{L}_{0})=\sum _{i=1}^{r}\unicode[STIX]{x1D6EC}(c_{i})(\mathfrak{L}_{0})$ , hence  $\mathfrak{L}_{\unicode[STIX]{x1D6EC}}=\unicode[STIX]{x1D6EC}(\mathbf{J})(\mathfrak{L}_{0})$ is a  $\mathbf{J}$ -stable lattice in  $V_{\unicode[STIX]{x1D6EC}}$ by [Reference Vignéras13, 9.3]. The induced  $\overline{\mathbb{Z}_{\ell }}$ -representation  $\text{ind}_{\mathbf{J}}^{G}(\mathfrak{L}_{\unicode[STIX]{x1D6EC}})$ is then a lattice in  $\unicode[STIX]{x1D70B}$ by [Reference Vignéras13, 9.3]. Moreover $\unicode[STIX]{x1D70F}=r_{\ell }(\unicode[STIX]{x1D70B})\simeq \text{ind}_{\mathbf{J}}^{G}(r_{\ell }(\unicode[STIX]{x1D6EC}))$ , and  $r_{\ell }(\unicode[STIX]{x1D6EC})$ is an irreducible representation of  $\mathbf{J}$ by irreducibility of  $\unicode[STIX]{x1D70F}$ .

Finally, we give another characterization of banal cuspidal representations: recall, from Section 3, by definition  $\unicode[STIX]{x1D70F}$ is banal if and only if the cardinality of the cuspidal line $\mathbb{Z}_{\unicode[STIX]{x1D70F}}=\{||^{k}\unicode[STIX]{x1D70F},~k\in \mathbb{Z}\}$ is greater than  $1$ . By [Reference Mínguez and Sécherre11, Lemme 5.3], this cardinality is the same as the integer  $o(\unicode[STIX]{x1D70F})$ introduced in [Reference Mínguez and Sécherre11, Section 5.2, (5.4)]. From [Reference Mínguez and Sécherre11, Section 5.2, (5.4)],  $o(\unicode[STIX]{x1D70F})$ is the order of  $q^{n/e}$ in  $\overline{\mathbb{F}_{\ell }}^{\times }$ , where  $e=e(E/F)$ is the ramification index attached to  $(\mathbf{J},\unicode[STIX]{x1D6EC})$ which in particular does not depend on the choice of extended maximal simple type. Hence  $\unicode[STIX]{x1D70F}$ is banal if and only if  $q^{n/e}-1\neq 0$ in  $\overline{\mathbb{F}_{\ell }}$ .

7 The modified Paskunas–Stevens basis

For this section  $R=\mathbb{C}$  or  $\overline{\mathbb{Q}_{\ell }}$ . Let  $\unicode[STIX]{x1D70B}$ be a cuspidal  $R$ -representation of  $G$ and  $(\mathbf{J}=\mathbf{J}(\unicode[STIX]{x1D6FD},{\mathcal{L}}),\unicode[STIX]{x1D6EC})$ be an extended maximal simple type in  $\unicode[STIX]{x1D70B}$ . According to [Reference Paskunas and Stevens12, Corollaries 3.4 and 4.13], there exists an  $F$ -basis  ${\mathcal{B}}=(v_{1},\ldots ,v_{n})$ of  $V$ particularly suited to relating the Whittaker model of  $\unicode[STIX]{x1D70B}$ and the model  $\text{ind}_{\mathbf{J}}^{G}(\unicode[STIX]{x1D6EC})$ defined via type theory. In particular,  ${\mathcal{B}}$ splits  ${\mathcal{L}}$ , that is,  $L_{k}=\bigoplus _{i=1}^{n}\mathfrak{p}_{F}^{a_{i}(k)}v_{i}$ with  $a_{i}(k)\in \mathbb{Z}$ for all  $k\in \mathbb{Z}$ , and is such that if  $N$ is the maximal unipotent subgroup of  $G$ attached to the maximal flag defined by  ${\mathcal{B}}$ , and if  $\unicode[STIX]{x1D713}$ , by abuse of notation, denotes the nondegenerate character of  $N$ defined for  $x\in N$ by

$$\begin{eqnarray}\unicode[STIX]{x1D713}(x)=\unicode[STIX]{x1D713}\biggl(\mathop{\sum }_{i=1}^{n-1}\text{Mat}_{{\mathcal{B}}}(x)_{i,i+1}\biggr),\end{eqnarray}$$

where  $\text{Mat}_{{\mathcal{B}}}(x)$ denotes the matrix of  $x$ with respect to the basis  ${\mathcal{B}}$ , then the triple  $(J,\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D713})$ satisfies

$$\begin{eqnarray}\text{Hom}_{N\cap J}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6EC})\neq 0.\end{eqnarray}$$

Let  $P$ be the mirabolic subgroup defined by

$$\begin{eqnarray}P=\{g\in G,(g-\text{Id})V\subset \text{Vect}_{F}(v_{1},\ldots ,v_{n-1})\}.\end{eqnarray}$$

We put  ${\mathcal{M}}=(P\cap J)J^{1}$ , which is a group as  $J^{1}$ is normal in  $J$ . It follows from [Reference Paskunas and Stevens12] that the image of  ${\mathcal{M}}$ in  $J/J^{1}\simeq G_{m}(k_{E})$ is isomorphic to  $P_{m}(k_{E})$ . We now explain how to extract this from [Reference Paskunas and Stevens12]: in the notation of [Reference Paskunas and Stevens12], our group  $P$ is denoted  ${\mathcal{M}}_{F}$ and [Reference Paskunas and Stevens12, Corollary 4.8] shows that

(5) $$\begin{eqnarray}{\mathcal{M}}=(P\cap \mathfrak{B}^{\times })J^{1}.\end{eqnarray}$$

In [Reference Paskunas and Stevens12, Section 4.1], Paskunas–Stevens introduce another mirabolic group they denote by  ${\mathcal{M}}_{E}$ which satisfies  $P\cap \mathfrak{B}^{\times }={\mathcal{M}}_{E}\cap \mathfrak{B}^{\times }$ by the equality just before [Reference Paskunas and Stevens12, Corollary 4.7], and they also denote by  ${\mathcal{M}}_{\mathfrak{B}}$ the group  $({\mathcal{M}}_{E}\cap \mathfrak{B}^{\times })(1+\unicode[STIX]{x1D71B}_{E}\mathfrak{B})$ . Hence Equation (5) gives  ${\mathcal{M}}={\mathcal{M}}_{\mathfrak{B}}J^{1}$ as  $(1+\unicode[STIX]{x1D71B}_{E}\mathfrak{B})=\mathfrak{B}^{\times }\cap J^{1}$ . Finally, from the discussion after the proof of [Reference Paskunas and Stevens12, Lemma 4.10], the image of  ${\mathcal{M}}_{\mathfrak{B}}$ in  $\mathfrak{B}^{\times }/1+\unicode[STIX]{x1D71B}_{E}\mathfrak{B}\simeq G_{m}(k_{E})$ is isomorphic to  $P_{m}(k_{E})$ , hence the same is true for the image of  ${\mathcal{M}}$ in  $J/J^{1}\simeq \mathfrak{B}^{\times }/1+\unicode[STIX]{x1D71B}_{E}\mathfrak{B}\simeq G_{m}(k_{E})$ . In particular, the following index will appear in our computation:

$$\begin{eqnarray}|J/{\mathcal{M}}|=|G_{m}(k_{E})/P_{m}(k_{E})|=q_{E}^{m}-1=q^{n/e}-1.\end{eqnarray}$$

For  $i\in \{1,\ldots ,n\}$ , the functions  $a_{i}:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfy the relation  $a_{i+e}(k)=a_{i}(k)+1$ . In particular, this holds for  $i=n$ , and the map  $k\mapsto a_{n}(k)$ is increasing with values in  $\mathbb{Z}$ , so there is  $k_{0}$ between  $1$ and  $e$ such that  $a_{n}(k_{0})=a_{n}(k_{0}-1)+1$ , and then  $a_{n}(k_{0}+i)=a_{n}(k_{0})$ for  $i\in \{0,\ldots ,e-1\}$ . Hence by reindexing the lattice chain  ${\mathcal{L}}$ if necessary, by a translation,  $k\mapsto k-k_{0}$ , we can suppose that

$$\begin{eqnarray}a_{n}(0)=a_{n}(-1)+1=0,\qquad \text{and}\qquad a_{n}(1)=\cdots =a_{n}(e-1)=0.\end{eqnarray}$$

We recall that  $L_{0}=\bigoplus _{i=1}^{n}\mathfrak{p}_{F}^{a_{i}(0)}v_{i}$ , and we set  ${\mathcal{B}}^{\prime }=(\unicode[STIX]{x1D71B}_{F}^{a_{1}(0)}v_{1},\ldots ,\unicode[STIX]{x1D71B}_{F}^{a_{n}(0)}v_{n})$ , which we write as  ${\mathcal{B}}^{\prime }=(w_{1},\ldots ,w_{n})$ .

We use this basis to identify  $G$ with  $G_{n}$ . With this choice, one has  $J\subset K_{n}$ because  $J\subset \text{Aut}_{\mathfrak{o}_{F}}(L_{0})$ . The group  $P$ identifies with  $P_{n}$ , the group  $N$ identifies with  $N_{n}$ , and the character  $\unicode[STIX]{x1D713}$ of $N_{n}$ identifies with

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{t}:n\mapsto \unicode[STIX]{x1D713}\biggl(\mathop{\sum }_{i=1}^{n-1}t_{i}n_{i,i+1}\biggr),\end{eqnarray}$$

where  $t_{i}=\unicode[STIX]{x1D71B}_{F}^{a_{i}(0)-a_{i+1}(0)}$ .

For our computation to come, it will be useful to notice the following property of  ${\mathcal{B}}^{\prime }$ : one has

$$\begin{eqnarray}L_{0}=\bigoplus _{i=1}^{n}\mathfrak{o}_{F}w_{i},\qquad L_{k}=\bigoplus _{i=1}^{n-1}\mathfrak{p}_{F}^{a_{i}(k)-a_{i}(0)}w_{i}\oplus \mathfrak{o}_{F}w_{n},\end{eqnarray}$$

for  $k\in \{1,\ldots ,e-1\}.$ As  $\unicode[STIX]{x1D71B}_{E}L_{k}=L_{k+1}$ for any  $k\in \mathbb{Z}$ , the properties above and the fact that  $L_{k+e}=\unicode[STIX]{x1D71B}_{F}L_{k}$ , imply that the last row of  $\unicode[STIX]{x1D71B}_{E}^{i}\in G_{n}$ belongs to  $(\mathfrak{o}_{F})^{n}-(\mathfrak{p}_{F})^{n}$ for  $i=0,\ldots ,e-1$ , and more generally that it belongs to  $(\mathfrak{p}_{F}^{l})^{n}-(\mathfrak{p}_{F}^{l+1})^{n}$ if  $i=le+r$ , with  $r\in \{0,\ldots ,e-1\}$ . As an immediate consequence, if we write an Iwasawa decomposition of  $\unicode[STIX]{x1D71B}_{E}^{i}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D71B}_{E}^{i}=p_{i}z_{i}k_{i},\qquad p_{i}\in P_{n},~z_{i}\in Z_{n},~k_{i}\in K_{n},\end{eqnarray}$$

we can choose  $z_{i}=I_{n}$ for  $i=0,\ldots ,e-1$ , and more generally  $z_{i}=\unicode[STIX]{x1D71B}_{F}^{l}I_{n}$ for  $i=le+r$ , with  $r\in \{0,\ldots ,e-1\}$ . In particular  $|p_{i}|=q^{-in/e}$ , for $i=0,\ldots ,e-1$ .

For clarity, we list the properties of the data  $(J,\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D713}_{t})$ that we use.

For the remainder, we consider the  $k_{i}\in K_{n}$ and  $p_{i}\in P_{n}$ chosen in Proposition 7.1 Statement (4) as fixed.

As  $P_{n}\cap J^{1}$ is a pro- $p$ subgroup of  $P_{n}$ , and  $J^{1}$ is a pro- $p$ subgroup of  $G_{n}$ , the volume

$$\begin{eqnarray}dk(P_{n}\cap J^{1}\backslash J^{1})=\frac{dk(J^{1})}{dp(P_{n}\cap J^{1})}\end{eqnarray}$$

is a power of  $q$ thanks to our normalization of measures, and we write

$$\begin{eqnarray}dk(P_{n}\cap J^{1}\backslash J^{1})=q^{r_{1}}.\end{eqnarray}$$

A certain volume will appear in our later computation, we compute it in the next lemma.

Lemma 7.2. For any  $i\in \{0,\ldots ,e-1\}$ , we have

$$\begin{eqnarray}dk((P_{n}\cap K_{n})\backslash (P_{n}\cap K_{n})k_{i}J)=q^{r_{1}}(q^{n/e}-1)q^{-in/e}.\end{eqnarray}$$

Proof. We have

$$\begin{eqnarray}\displaystyle dk((P_{n}\cap K_{n})\backslash (P_{n}\cap K_{n})k_{i}J) & = & \displaystyle dk((P_{n}\cap K_{n})\backslash (P_{n}\cap K_{n})k_{i}Jk_{i}^{-1})\nonumber\\ \displaystyle & = & \displaystyle dk((P_{n}\cap k_{i}Jk_{i}^{-1})\backslash k_{i}Jk_{i}^{-1}),\nonumber\end{eqnarray}$$

the last equality thanks to Relation (3). Now,  $dk(k_{i}Jk_{i}^{-1})=dk(J)$ . We also notice that

$$\begin{eqnarray}p_{i}(P_{n}\cap k_{i}Jk_{i}^{-1})p_{i}^{-1}=P_{n}\cap \unicode[STIX]{x1D71B}_{E}^{i}J\unicode[STIX]{x1D71B}_{E}^{-i}=P_{n}\cap J,\end{eqnarray}$$

hence

$$\begin{eqnarray}P_{n}\cap k_{i}Jk_{i}^{-1}=p_{i}^{-1}(P_{n}\cap J)p_{i}.\end{eqnarray}$$

As for any compact open subset  $A$ of  $P_{n}$ , one has  $dp(pAp^{-1})=|p|dp(A)$ , as is easily seen by writing  $dp=dgdu$ , with  $dg$ on  $G_{n-1}$ and  $du$ on  $U_{n}$ , we obtain the relation

$$\begin{eqnarray}dp(P_{n}\cap k_{i}Jk_{i}^{-1})=|p_{i}|^{-1}dp(P_{n}\cap J)=q^{in/e}dp(P_{n}\cap J).\end{eqnarray}$$

We then obtain from Relations (1) and (2):

$$\begin{eqnarray}\displaystyle dk((P_{n}\cap k_{i}Jk_{i}^{-1})\backslash k_{i}Jk_{i}^{-1}) & = & \displaystyle \frac{dk(k_{i}Jk_{i}^{-1})}{dp(P_{n}\cap k_{i}Jk_{i}^{-1})}\nonumber\\ \displaystyle & = & \displaystyle q^{-in/e}\frac{dk(J)}{dp(P_{n}\cap J)}\nonumber\\ \displaystyle & = & \displaystyle q^{-in/e}dk((P_{n}\cap J)\backslash J).\nonumber\end{eqnarray}$$

Now by Relations (1) and (2) again, one has

$$\begin{eqnarray}\displaystyle dk((P_{n}\cap J)\backslash J)) & = & \displaystyle \frac{dk(J)}{dp(P_{n}\cap J)}=\frac{dk(J)}{dk({\mathcal{M}})}\frac{dk({\mathcal{M}})}{dp(P_{n}\cap J)}\nonumber\\ \displaystyle & = & \displaystyle |J\backslash {\mathcal{M}}|dk(P_{n}\cap J\backslash {\mathcal{M}}).\nonumber\end{eqnarray}$$

Finally, because  ${\mathcal{M}}=(P_{n}\cap J)J^{1}$ , applying Relation (3) gives:

$$\begin{eqnarray}dk((P_{n}\cap J)\backslash J))=|J\backslash {\mathcal{M}}|dk(P_{n}\cap J^{1}\backslash J^{1})=q^{r_{1}}(q^{n/e}-1)\end{eqnarray}$$

by Proposition 7.1(3) and our definition of  $r_{1}$ . This concludes the proof.◻

8 Explicit Whittaker functions of Paskunas–Stevens

In this section we continue to assume that  $R=\mathbb{C}$ or  $\overline{\mathbb{Q}_{\ell }}$ . We now recall the definition and some properties of the explicit Whittaker functions of [Reference Paskunas and Stevens12]. We set

$$\begin{eqnarray}{\mathcal{U}}=(N_{n}\cap J)H^{1}.\end{eqnarray}$$

We extend  $\unicode[STIX]{x1D713}_{t}$ to the group  ${\mathcal{U}}$ as in [Reference Paskunas and Stevens12, Definition 4.2], and, by abuse of notation, denote this extension by  $\unicode[STIX]{x1D713}_{t}$ . We fix a normal compact open subgroup  ${\mathcal{N}}$ of  ${\mathcal{U}}$ contained in  $\ker (\unicode[STIX]{x1D713}_{t})$ . We also denote by  $\unicode[STIX]{x1D70C}$ the trace character of  $\unicode[STIX]{x1D6EC}$ and  $\unicode[STIX]{x1D70C}^{\vee }$ that of  $\unicode[STIX]{x1D6EC}^{\vee }$ .

Definition 8.1. (Bessel functions)

For  $j\in \mathbf{J}$ , we define

$$\begin{eqnarray}\displaystyle {\mathcal{J}}(j) & = & \displaystyle |{\mathcal{N}}\backslash {\mathcal{U}}|^{-1}\mathop{\sum }_{{\mathcal{N}}\backslash {\mathcal{U}}}\unicode[STIX]{x1D713}_{t}(u)^{-1}\unicode[STIX]{x1D70C}(ju),\qquad \text{and}\nonumber\\ \displaystyle {\mathcal{J}}^{\vee }(j) & = & \displaystyle |{\mathcal{N}}\backslash {\mathcal{U}}|^{-1}\mathop{\sum }_{{\mathcal{N}}\backslash {\mathcal{U}}}\unicode[STIX]{x1D713}_{t}(u)\unicode[STIX]{x1D70C}^{\vee }(ju).\nonumber\end{eqnarray}$$

The Bessel functions enjoy the following properties:

We can now define the explicit Whittaker functions  $W$ and  $W^{\vee }$ of Paskunas–Stevens following [Reference Paskunas and Stevens12, Section 5.2] and recall a first property.

Definition 8.3. Both  $W$ and  $W^{\vee }$ are supported on  $N_{n}\mathbf{J}$ , and

$$\begin{eqnarray}W(nj)=\unicode[STIX]{x1D713}_{t}(n){\mathcal{J}}(j)\end{eqnarray}$$

for  $n\in N_{n}$ and  $j\in \mathbf{J}$ , whereas

$$\begin{eqnarray}W^{\vee }(nj)=\unicode[STIX]{x1D713}_{t}^{-1}(n){\mathcal{J}}^{\vee }(j)=\unicode[STIX]{x1D713}_{t}^{-1}(n){\mathcal{J}}(j^{-1})\end{eqnarray}$$

for  $n\in N_{n}$ and  $j\in \mathbf{J}$ . Moreover,  $W$ belongs to  $W(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D713}_{t})$ and  $W^{\vee }$ belongs to  $W(\unicode[STIX]{x1D70B}^{\vee },\unicode[STIX]{x1D713}_{t}^{-1})$ .

We now prove further properties of  $W$ and  $W^{\vee }$ .

9 Test vectors

Again, we assume that  $R=\mathbb{C}$ , or  $\overline{\mathbb{Q}_{\ell }}$ , and  $\unicode[STIX]{x1D70B}_{1}$ and  $\unicode[STIX]{x1D70B}_{2}$ are cuspidal  $R$ -representations of  $G_{n}$ . We denote by  $(\mathbf{J},\unicode[STIX]{x1D6EC})$ the extended maximal simple type of  $\unicode[STIX]{x1D70B}_{1}$ , by  $e=e(E/F)$ the ramification index of the field extension associated to  $(\mathbf{J},\unicode[STIX]{x1D6EC})$ , and by  $W,W^{\vee }$ the explicit Whittaker functions associated to  $\unicode[STIX]{x1D70B}_{1}$ (see Definition 8.3). This section is dedicated to proving our main result on test vectors.

Theorem 9.1. Suppose that  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is nontrivial, so that  $\unicode[STIX]{x1D70B}_{2}\simeq \unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee }$ for some unramified character  $\unicode[STIX]{x1D712}$ of  $F^{\times }$ . Then there is an integer  $r$ such that

$$\begin{eqnarray}I(X,W,\unicode[STIX]{x1D712}W^{\vee },1_{\mathfrak{ o}_{F}^{n}})=\frac{q^{r}(q-1)(q^{n/e}-1)}{1-(\unicode[STIX]{x1D712}(\unicode[STIX]{x1D71B}_{F})X)^{n/e}}=q^{r}(q-1)(q^{n/e}-1)L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}).\end{eqnarray}$$

We are now ready to prove the following crucial proposition. We recall that for all integers  $l\geqslant 0$ , the restriction  $W_{l}$ has been defined in Proposition 8.4.

Proposition 9.2. Let  $F_{l}:(K_{n}\cap P_{n})\backslash K_{n}/J^{1}\rightarrow R$ be defined by

$$\begin{eqnarray}F_{l}(k)=\int _{j\in J^{1}}\int _{N_{n}\backslash P_{n}}W_{l}(pkj)W_{l}^{\vee }(pkj)\,dp\,dj.\end{eqnarray}$$

Then  $F_{l}$ is nonzero if an only if  $l=in/e$ and  $i\in \{0,\ldots ,e-1\}$ , and in this case, it is supported on  $(K_{n}\cap P_{n})k_{i}J$ . Moreover, for  $i\in \{0,\ldots ,e-1\}$ , and for  $k\in (K_{n}\cap P_{n})k_{i}J$ , there is an integer  $r_{2}$ independent of  $i$ such that

$$\begin{eqnarray}F_{in/e}(k)=q^{r_{2}}.\end{eqnarray}$$

Proof. If  $F_{l}(k)$ is nonzero, then  $W_{l}(pkj)$ is nonzero at least for some  $p\in P_{n}$ and  $j\in J$ , but then according to Statements (1) and (2) of Proposition 8.4, this implies that  $l$ is of the form  $l=in/e$ with  $i\in \{0,\ldots ,e-1\}$ , and  $k\in (K_{n}\cap P_{n})k_{i}J$ . Moreover, from Statement (2) of the same proposition, we can write  $k=p_{0}\unicode[STIX]{x1D71B}_{E}^{i}j_{0}$ for  $p_{0}\in P_{n}$ and  $j_{0}\in J$ . But now notice that for such a  $k$ , we have

$$\begin{eqnarray}\displaystyle F_{l}(k) & = & \displaystyle \int _{j\in J^{1}}\int _{N_{n}\backslash P_{n}}W_{l}(pp_{0}\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)W_{l}^{\vee }(pp_{0}\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)\,dp\,dj\nonumber\\ \displaystyle & = & \displaystyle \int _{j\in J^{1}}\int _{N_{n}\backslash P_{n}}W_{l}(p\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)W_{l}^{\vee }(p\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)\,dp\,dj.\nonumber\end{eqnarray}$$

Hence by Statement (3) of Proposition 8.4

$$\begin{eqnarray}\displaystyle F_{l}(k) & = & \displaystyle \int _{j\in J^{1}}\int _{N_{n}\backslash N_{n}(P_{n}\cap J)}W_{l}(p\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)W_{l}^{\vee }(p\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)\,dp\,dj\nonumber\\ \displaystyle & = & \displaystyle \int _{j\in J^{1}}\int _{N_{n}\cap J\backslash P_{n}\cap J}W_{l}(m\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)W_{l}^{\vee }(m\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j)\,dm\,dj\nonumber\\ \displaystyle & = & \displaystyle \int _{j\in J^{1}}\int _{N_{n}\cap J\backslash P_{n}\cap J}{\mathcal{J}}(m\unicode[STIX]{x1D71B}_{E}^{i}j_{0}j){\mathcal{J}}(j^{-1}j_{0}^{-1}\unicode[STIX]{x1D71B}_{E}^{-i}m^{-1})\,dm\,dj,\nonumber\end{eqnarray}$$

the last equality according to Proposition 8.2(3). Now, as  $\mathbf{J}$ normalizes  $J^{1}$ , and as for any  $t\in G_{n}$ normalizing  $J^{1}$ , the automorphism  $j\mapsto tjt^{-1}$ of  $J^{1}$ has modulus character equal to  $\mathbf{1}$ , because  $J^{1}$ is an open subgroup of the unimodular group  $G_{n}$ , we have

$$\begin{eqnarray}\displaystyle F_{l}(k) & = & \displaystyle \int _{j\in J^{1}}\int _{N_{n}\cap J\backslash P_{n}\cap J}{\mathcal{J}}(mj\unicode[STIX]{x1D71B}_{E}^{i}j_{0}){\mathcal{J}}(j_{0}^{-1}\unicode[STIX]{x1D71B}_{E}^{-i}(mj)^{-1})\,dm\,dj\nonumber\\ \displaystyle & = & \displaystyle \int _{N_{n}\cap J\backslash {\mathcal{M}}}{\mathcal{J}}(m\unicode[STIX]{x1D71B}_{E}^{i}j_{0}){\mathcal{J}}(j_{0}^{-1}\unicode[STIX]{x1D71B}_{E}^{-i}m^{-1})\,dm.\nonumber\end{eqnarray}$$

We write

$$\begin{eqnarray}dm(N_{n}\cap J\backslash (N_{n}\cap J)H^{1})=dm(N_{n}\cap H^{1}\backslash H^{1})=q^{r_{2}},\end{eqnarray}$$

which is indeed a power of  $q$ as  $H^{1}$ is pro- $p$ . Moreover, as  $H^{1}$ is normal in  $J$ , and as the integrand is invariant under  ${\mathcal{U}}$ thanks to Property (2) in Proposition 8.2

$$\begin{eqnarray}F_{l}(k)=q^{r_{2}}\int _{{\mathcal{U}}\backslash {\mathcal{M}}}{\mathcal{J}}(m\unicode[STIX]{x1D71B}_{E}^{i}j){\mathcal{J}}(j^{-1}\unicode[STIX]{x1D71B}_{E}^{-i}m^{-1})\,dm=q^{r_{2}},\end{eqnarray}$$

the last equality thanks to Statement (4) of Proposition 8.2. ◻

Proposition 9.3. The coefficient

$$\begin{eqnarray}b_{l}=\int _{P_{n}\cap K_{n}\backslash K_{n}}\int _{N_{n}\backslash P_{n}}W_{l}(pk)W_{l}^{\vee }(pk)\,dp\,dk\end{eqnarray}$$

is zero unless  $l=in/e$ for some  $i\in \{0,\ldots ,e-1\}$ , in which case there is an integer  $r$ such that

$$\begin{eqnarray}b_{l}=q^{r}(q^{n/e}-1)q^{-in/e}.\end{eqnarray}$$

Proof. By definition,  $b_{l}$ is equal to

$$\begin{eqnarray}\int _{P_{n}\cap K_{n}\backslash K_{n}/J^{1}}F_{l}(k)\,dk=q^{r_{3}}\int _{P_{n}\cap K_{n}\backslash K_{n}}F_{l}(k)\,dk\end{eqnarray}$$

with  $dk(J^{1})=q^{r_{3}}$ ( $J^{1}$ is pro- $p$ ). So according to Proposition 9.2, this is zero if  $l\neq in/e$ for  $i\in \{0,\ldots ,e-1\}$ , and if  $l=in/e$ for  $i\in \{0,\ldots ,e-1\}$ , it is equal to

$$\begin{eqnarray}\displaystyle q^{r_{3}}\int _{P_{n}\cap K_{n}\backslash (P_{n}\cap K_{n})k_{i}J}F_{l}(k)\,dk & = & \displaystyle q^{r_{2}+r_{3}}dk(P_{n}\cap K_{n}\backslash (P_{n}\cap K_{n})k_{i}J)\nonumber\\ \displaystyle & = & \displaystyle q^{r}(q^{n/e}-1)q^{-in/e},\nonumber\end{eqnarray}$$

where we write  $r=r_{1}+r_{2}+r_{3}$ , from Lemma 7.2.◻

If  $\unicode[STIX]{x1D70B}$ is a cuspidal  $R$ -representation of  $G_{n}$ of ramification index  $e$ , we denote by  $R(\unicode[STIX]{x1D70B})$ its ramification group, that is the group of unramified characters  $\unicode[STIX]{x1D708}$ of  $F^{\times }$ which satisfy  $\unicode[STIX]{x1D708}\unicode[STIX]{x1D70B}\simeq \unicode[STIX]{x1D70B}$ . It follows from [Reference Bushnell and Kutzko2, 6.2.5], that  $R(\unicode[STIX]{x1D70B})$ is isomorphic to the group of  $n/e$ th roots of unity in  $R^{\times }$ , via  $\unicode[STIX]{x1D708}\mapsto \unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{F})$ .

Proof of Theorem 9.1.

We first suppose that  $\unicode[STIX]{x1D70B}_{2}\simeq \unicode[STIX]{x1D70B}_{1}^{\vee }$ . By Equation (4), the integral  $I(X,W,W^{\vee },\mathbf{1}_{\mathfrak{o}_{F}^{n}})$ is equal to

$$\begin{eqnarray}\frac{q-1}{1-X^{n}}\int _{(K_{n}\cap P_{n})\backslash K_{n}}I_{(0)}(X,\unicode[STIX]{x1D70C}(k)W,\unicode[STIX]{x1D70C}(k)W^{\vee })\,dk.\end{eqnarray}$$

Now, as  $WW^{\vee }=\sum _{l\in \mathbb{Z}}W_{l}W_{l}^{\vee }$ , by Statement (1) of Propositions 8.4 and 9.3, we have

$$\begin{eqnarray}\displaystyle \int _{(K_{n}\cap P_{n})\backslash K_{n}}I_{(0)}(X,\unicode[STIX]{x1D70C}(k)W,\unicode[STIX]{x1D70C}(k)W^{\vee })\,dk & = & \displaystyle \mathop{\sum }_{i=0}^{e-1}b_{in/e}q^{in/e}X^{in/e}\nonumber\\ \displaystyle =q^{r}(q^{n/e}-1)\mathop{\sum }_{i=0}^{e-1}X^{in/e} & = & \displaystyle q^{r}(q^{n/e}-1)\frac{1-X^{n}}{1-X^{n/e}}.\nonumber\end{eqnarray}$$

This gives the equality

$$\begin{eqnarray}I(X,W,W^{\vee },\mathbf{1}_{\mathfrak{ o}_{F}^{n}})=(q-1)(q^{n/e}-1)\frac{q^{r}}{1-X^{n/e}}.\end{eqnarray}$$

On the other hand, and by [Reference Jacquet, Piatetski-Shapiro and Shalika5, Proposition 8.1], the factor  $L(X,\unicode[STIX]{x1D70B},\unicode[STIX]{x1D70B}^{\vee })$ is equal to

$$\begin{eqnarray}L(X,\unicode[STIX]{x1D70B},\unicode[STIX]{x1D70B}^{\vee })=\mathop{\prod }_{\unicode[STIX]{x1D708}\in R(\unicode[STIX]{x1D70B})}\frac{1}{1-\unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{f})X}=\frac{1}{1-X^{n/e}}.\end{eqnarray}$$

Now in general, as we supposed that  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is not equal to  $1$ , we have  $\unicode[STIX]{x1D70B}_{2}\simeq \unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee }$ for some unramified character  $\unicode[STIX]{x1D712}$ of  $F^{\times }$ . However, we have

$$\begin{eqnarray}L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})=L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee })=L(\unicode[STIX]{x1D712}(\unicode[STIX]{x1D71B}_{F})X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{1}^{\vee }).\end{eqnarray}$$

On the other hand, we have

$$\begin{eqnarray}\displaystyle I(X,W,\unicode[STIX]{x1D712}W^{\vee },\mathbf{1}_{\mathfrak{o}_{F}^{n}}) & = & \displaystyle I(\unicode[STIX]{x1D712}(\unicode[STIX]{x1D71B}_{F})X,W,W^{\vee },\mathbf{1}_{\mathfrak{o}_{F}^{n}})\nonumber\\ \displaystyle & = & \displaystyle (q-1)(q^{n/e}-1)\frac{q^{r}}{1-(\unicode[STIX]{x1D712}(\unicode[STIX]{x1D71B}_{F})X)^{n/e}}.\nonumber\end{eqnarray}$$

However,

$$\begin{eqnarray}\displaystyle L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})=L(\unicode[STIX]{x1D712}(\unicode[STIX]{x1D71B}_{F})X,\unicode[STIX]{x1D70B},\unicode[STIX]{x1D70B}^{\vee })=\frac{1}{1-(\unicode[STIX]{x1D712}(\unicode[STIX]{x1D71B}_{F})X)^{n/e}}, & & \displaystyle \nonumber\end{eqnarray}$$

and we are done.◻

10 $L$ -factors of banal cuspidal  $\ell$ -modular representations

In this section, we consider the cases  $R=\overline{\mathbb{F}_{\ell }}$ , and  $R=\overline{\mathbb{Q}_{\ell }}$ . In the  $\overline{\mathbb{Q}_{\ell }}$ setting, we continue with the notations of the last section, and note that as  $\unicode[STIX]{x1D713}$ is integral, so are  $\unicode[STIX]{x1D713}_{t}$ and  $\unicode[STIX]{x1D713}_{t}^{-1}$ . Our main theorem has the following interesting corollary.

Corollary 10.1. Let  $\unicode[STIX]{x1D70F}_{1}$ and  $\unicode[STIX]{x1D70F}_{2}$ be two banal cuspidal  $\ell$ -modular representations of  $G_{n}$ , and  $\unicode[STIX]{x1D70B}_{1}$ and  $\unicode[STIX]{x1D70B}_{2}$ be any cuspidal  $\ell$ -adic lifts, then

$$\begin{eqnarray}L(X,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})).\end{eqnarray}$$

Proof. We already noticed in Section 5 that if  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is equal to  $1$ , then

$$\begin{eqnarray}L(X,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}))=1,\end{eqnarray}$$

whether  $\unicode[STIX]{x1D70F}_{1}$ and  $\unicode[STIX]{x1D70F}_{2}$ are banal or not. Hence we only need to focus on the case when  $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is not equal to  $1$ , that is  $\unicode[STIX]{x1D70B}_{2}\simeq \unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee }$ for some unramified character  $\unicode[STIX]{x1D712}$ . Let  $W$ be the Stevens–Paskunas explicit Whittaker function associated to an extended maximal simple type of  $\unicode[STIX]{x1D70B}_{1}$ as in the statement of Theorem 9.1.

Lemma 10.2. The explicit Whittaker functions  $W$ and  $\unicode[STIX]{x1D712}W^{\vee }$ lie in the  $\overline{\mathbb{Z}_{\ell }}$ -submodules  $W_{e}(\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D713}_{t})$ and  $W_{e}(\unicode[STIX]{x1D70B}_{2},\unicode[STIX]{x1D713}_{t}^{-1})$ respectively.

Proof. As in the proof of Theorem 9.1, the representation  $\unicode[STIX]{x1D70B}_{1}$ contains an extended maximal simple type  $(\mathbf{J}_{1},\unicode[STIX]{x1D6EC}_{1})$ and  $W$ is chosen to be the Paskunas–Stevens Whittaker function of Definition 8.3 relative to this data. As  $\unicode[STIX]{x1D70B}_{1}$ is integral,  $\unicode[STIX]{x1D6EC}_{1}$ is integral by the end of Section 6. This implies that the trace character  $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6EC}_{1}}$ of  $\unicode[STIX]{x1D6EC}_{1}$ has values in  $\overline{\mathbb{Z}_{\ell }}$ . In particular the Bessel function  ${\mathcal{J}}_{1}$ (see Definition 8.1) associated to the pair $(\mathbf{J}_{1},\unicode[STIX]{x1D6EC}_{1})$ takes values in  $\overline{\mathbb{Z}_{\ell }}$ . Hence, as  $\unicode[STIX]{x1D713}_{t}$ is integral,  $W\in W_{e}(\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D713}_{t})$ (see Definition 8.3). Now,  $\unicode[STIX]{x1D70B}_{2}$ is of the form  $\unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}_{1}^{\vee }$ with  $\unicode[STIX]{x1D712}$ an unramified character of  $F^{\times }$ (which is integral as  $\unicode[STIX]{x1D712}$ is unramified), so Proposition 8.2(3) implies that the Bessel function $\unicode[STIX]{x1D712}{\mathcal{J}}_{1}^{\vee }$ is integral. We conclude that  $\unicode[STIX]{x1D712}W^{\vee }$ belongs to  $W_{e}(\unicode[STIX]{x1D70B}_{2},\unicode[STIX]{x1D713}_{t}^{-1})$ (see Definition 8.3 again).◻

Granted  $W\in W_{e}(\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D713}_{t})$ and  $\unicode[STIX]{x1D712}W^{\vee }\in W_{e}(\unicode[STIX]{x1D70B}_{2},\unicode[STIX]{x1D713}_{t})$ , we have

$$\begin{eqnarray}\displaystyle r_{\ell }(q^{r}(q^{n/e}-1))r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})) & = & \displaystyle r_{\ell }(I(X,W,\unicode[STIX]{x1D712}W^{\vee },\mathbf{1}_{\mathfrak{o}_{F}^{n}}))\nonumber\\ \displaystyle & = & \displaystyle I(X,r_{\ell }(W),r_{\ell }(\unicode[STIX]{x1D712}W^{\vee }),r_{\ell }(\mathbf{1}_{\mathfrak{o}_{F}^{n}})).\nonumber\end{eqnarray}$$

Notice that  $r_{\ell }(q^{r}(q-1)(q^{n/e}-1))$ is nonzero if and only if  $\unicode[STIX]{x1D70B}_{1}$ (hence  $\unicode[STIX]{x1D70B}_{2}$ ) is banal by the end of Section 6. As the integral  $I(X,r_{\ell }(W),r_{\ell }(\unicode[STIX]{x1D712}W^{\vee }),r_{\ell }(\mathbf{1}_{\mathfrak{o}_{F}^{n}}))$ belongs to the fractional ideal  $(L(X,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}))$ of  $\overline{\mathbb{F}_{\ell }}[X^{\pm 1}]$ , we deduce that $r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}))$ divides  $L(X,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})$ . As in any case, thanks to [Reference Kurinczuk and Matringe8, Theorem 3.13], the  $L$ -factor  $L(X,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})$ divides  $r_{\ell }(L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}))$ , we deduce the desired equality.◻