2.1 $p$-adic automorphic forms

Let $E$ be an imaginary quadratic field over $\mathbb {Q}$, and $D$ a central simple $E$-algebra of rank $n^2$ which has an involution $x\mapsto x^*$ extending the nontrivial automorphism $\sigma$ of $E$ over $\mathbb {Q}$ (for example, $D$ could be ${GL}_n(E)$, in which case $x^*$ would be $\sigma (x)^T$). Let $G/\mathbb {Q}$ be the group whose $R$-points, for a $\mathbb {Q}$-algebra $R$, are

\[ G(R)=\{x\in D\otimes_\mathbb{Q} R\mid xx^*=1\}. \]

Note that $G(\mathbb {Q}_p)$ is isomorphic to ${GL}_n(\mathbb {Q}_p)$ if $p$ is split in $E$ (since then $E_p\cong \mathbb {Q}_p\oplus \mathbb {Q}_p$ with $\sigma$ switching factors) and to $U_n(\mathbb {Q}_p)$ if $p$ is inert in $E$; we assume that $p$ is split in $E$ and $G(\mathbb {Q}_p)\cong {GL}_n(\mathbb {Q}_p)$. In addition, $G(\mathbb {R})\cong U_{s,t}(\mathbb {R})$ for $(s,t)$ the signature of $Q(x)=xx^*$; we assume that $Q(x)$ has signature $(n,0)$ or $(0,n)$, so that $G(\mathbb {R})$ is compact.

As usual, we write $B$ and $\overline {B}$ for the upper and lower triangular Borel subgroups of ${GL}_n$, respectively, $T$ for the diagonal torus, and $N$ and $\overline {N}$ for the upper and lower unipotent subgroups of ${GL}_n$, respectively.

Write $\mathbb {A}=\mathbb {A}_\mathbb {Q}$, $\mathbb {A}_f$ for the finite adeles of $\mathbb {A}$, and $\mathbb {A}_f^p$ for the finite adeles trivial at $p$. Let $\mathscr {U}$ be a compact open subgroup of $G(\mathbb {A}_f)$ of the form $\mathscr {U}_p\times \mathscr {U}^p$, where $\mathscr {U}_p$ is a compact open subgroup of $G(\mathbb {Q}_p)$ (called the wild level structure) and $\mathscr {U}^p$ a compact open subgroup of $G(\mathbb {A}_f^p)$ (called the tame level structure). We can now define $V$-valued automorphic forms for any $\mathscr {U}_p$-module $V$.

In general, all our group and monoid actions will be left actions.

Definition 2.1.1 If $V$ is a $k[\mathscr {U}_p]$-module for any field $k$, write $V(G,\mathscr {U})$ for the $k$-vector space of maps

\[ f:G(\mathbb{Q})\backslash G(\mathbb{A}_f)\to V \]

such that $f(xu)=u_p^{-1}f(x)$ for all $x\in G(\mathbb {Q})\backslash G(\mathbb {A}_f)$ and $u\in \mathscr {U}$. Equivalently,

\[ V(G,\mathscr{U})=(\text{Hom}_{set}(G(\mathbb{Q})\backslash G(\mathbb{A}_f),k)\otimes_k V)^{\mathscr{U}}, \]

where the action of $\mathscr {U}$ on $\text {Hom}_{set}(G(\mathbb {Q})\backslash G(\mathbb {A}_f), k)$ is right translation and the action on $V$ is through $\mathscr {U}_p$. For any submonoid $\mathscr {U}'\supseteq \mathscr {U}$ of $G(\mathbb {A}_f)$ which has an action on $V$ that is trivial for $\mathscr {U}$, $V(G,\mathscr {U})$ is a $\mathscr {U}'$-module with action $(uf)(x)=u_pf(xu)$.

We frequently express examples using the following notation: if $B\subseteq H$ are groups, $R$ is a ring, and $s:B\to R^\times$ is a character, let

\[ \operatorname{Ind}_B^{H}s=\{f:H\to R\mid f(hb)=s(b)f(h)\text{ for all }h\in H,b\in B\}, \]

and if $P$ is a property of some functions $f\in \operatorname {Ind}_B^Hs$ which is invariant under left translation by $H$, let

\[ \operatorname{Ind}_B^{H,P}s=\{f\in\operatorname{Ind}_B^Hs\mid f\text{ has property }P\}. \]

Then $\operatorname {Ind}_B^{H,P}s$ is an $R$-module with a (left) action of $H$ given by $(hf)(x)=f(h^{-1}x)$ for all $h,x\in H$. Note that our left/right conventions for induction are unusual for convenience.

For example, if $k$ is a field, $t=(t_1,\dotsc,t_n)\in \mathbb {Z}^n$, and we write $\operatorname {diag}(d_1,\dotsc,d_n)$ for the diagonal matrix with entries $d_1,\dotsc,d_n$ along the diagonal, we can interpret $t$ as the character of the diagonal torus $T(k)$ of ${GL}_n(k)$ taking $\operatorname {diag}(d_1,\dotsc,d_n)$ to $\prod _{i=1}^n d_i^{t_i}$, and thus as the character of the upper triangular Borel $B(k)$ obtained by reducing to $T(k)$ and applying $t$. In the event that $t_1\ge \dotsb \ge t_n$, the $k$-vector space

\[ \operatorname{Ind}_{B(k)}^{{GL}_n(k),{\rm alg}}t, \]

where ‘alg’ stands for algebraic (i.e. $f:{GL}_n(k)\to k$ comes from an element of $k[{GL}_n]$), is the irreducible algebraic representation of ${GL}_n$ over $k$ of highest weight $t$ (see § 12.1.3 of [Reference Goodman and WallachGW09] and Proposition 2.2.1 of [Reference ChenevierChe04]). We call this representation $S_t(k)$. Then $S_t(k)(G,\mathscr {U})$ is the space of classical $p$-adic automorphic forms on $G$ of weight $t$ and level $\mathscr {U}$ with coefficients in $k$.

One way to picture $V(G,\mathscr {U})$ is as follows. By the generalized finiteness of class groups (see Theorem 5.1 of [Reference BorelBor63]), the set $G(\mathbb {Q})\backslash G(\mathbb {A}_f)/\mathscr {U}$ is finite. Fix double coset representatives $x_1,\dotsc,x_h\in G(\mathbb {A}_f)$. Then we have an isomorphism

\begin{align*} V(G,\mathscr{U}) & \xrightarrow{\sim} \bigoplus_{i=1}^h V^{x_i^{-1}G(\mathbb{Q})x_i\cap\mathscr{U}} \\ f & \mapsto (f(x_1),\dotsc,f(x_h)). \end{align*}

Because $G(\mathbb {R})\cong U_n(\mathbb {R})$ is compact, $G(\mathbb {Q})$ is discrete in $G(\mathbb {A}_f)$ (see, e.g., Proposition 1.4 of [Reference GrossGro99] or Proposition 3.1.2 of [Reference LoefflerLoe10]). Since, in addition, $\mathscr {U}$ is compact, the group $x_i^{-1}G(\mathbb {Q})x_i\cap \mathscr {U}$ is always finite, and it is trivial if $\mathscr {U}^p$ is sufficiently small. (For example, by Proposition 4.1.1 of [Reference ChenevierChe04], there is an integer $e_n$ depending only on $n$ such that $x_i^{-1}G(\mathbb {Q})x_i\cap \mathscr {U}$ is guaranteed to be trivial if the image of $\mathscr {U}^p$ in $G(\mathbb {Q}_l)$ is contained in $\Gamma (l)=\{g\in {GL}_n(\mathbb {Z}_p)\mid g\equiv 1\pmod {l}\}$ for some prime $l\nmid e_n$.) It is this fact that makes the construction of the eigenvariety for $G$ so sleek.

When convenient, we assume that $\mathscr {U}^p$ is sufficiently small (sometimes called ‘neat’ in the literature) and, thus, $V(G,\mathscr {U})\cong V^h$. As in Remark 2.14 of [Reference Liu, Wan and XiaoLWX17], this does not affect our results, because the eigenvariety for any $\mathscr {U}^p$ is a union of connected components of the eigenvariety for a sufficiently small subgroup of $\mathscr {U}^p$.

2.2 Weight space

A weight is a continuous character of $T(\mathbb {Z}_p)\cong (\mathbb {Z}_p^\times )^n$. Such a weight can be viewed as a character of $B(\mathbb {Z}_p)$ by reduction to $T(\mathbb {Z}_p)$. (In the introduction, we defined a weight instead to be a character of $(\mathbb {Z}_p^\times )^{n-1}$, that is, a character of $T(\mathbb {Z}_p)$ that is trivial on the last $\mathbb {Z}_p^\times$-factor. We will go back to restricting possible weights to the subset that is trivial on the last $\mathbb {Z}_p^\times$-factor whenever it is convenient, because any character of $T(\mathbb {Z}_p)$ can be twisted by a central character to one in this restricted subset, and central characters do not change spaces of automorphic forms in an interesting way.)

The weight space $\mathscr {W}^n$ is the rigid analytic space over $\mathbb {Q}_p$ such that for any affinoid $\mathbb {Q}_p$-algebra $A$, $\mathscr {W}^n(A)$ is the set of continuous characters $(\mathbb {Z}_p^\times )^n\to A^\times$. Let $\Delta ^n=((\mathbb {Z}/q\mathbb {Z})^\times )^n$. We have

\[ (\mathbb{Z}_p^\times)^n\cong \Delta^n\times(1+q\mathbb{Z}_p)^n, \]

so an $A$-point of $\mathscr {W}^n$ is determined by a character of $\Delta ^n$ and a character of $(1+q\mathbb {Z}_p)^n$. Furthermore, a character $s$ of $(1+q\mathbb {Z}_p)^n$ is determined by the values $T_i(s)=s(1,\dotsc,1,\exp (q),1,\dotsc,1)-1$ (where the $i$th entry is $\exp (q)$ and all the others are $1$), since $\exp (q)$ topologically generates $1+q\mathbb {Z}_p$. By Lemma 1 of [Reference BuzzardBuz04], the coordinates $(T_1,\dotsc,T_n)\in A^n$ come from an $A$-point of $\mathscr {W}^n$ precisely when they are topologically nilpotent. Thus $\mathscr {W}^n$ can be pictured as a finite disjoint union of $\varphi (q)$ open unit polydiscs with coordinates $(T_1,\dotsc,T_n)$, one for each tame character of $\Delta ^n$.

We use

\[ [\cdot]:(\mathbb{Z}_p^\times)^n\to\mathbb{Z}_p[\kern-1pt[ (\mathbb{Z}_p^\times)^n]\kern-1pt] \]

to denote the universal character of $(\mathbb {Z}_p^\times )^n$ and $\Lambda ^n$ to denote the Iwasawa algebra

\[ \Lambda^n=\mathbb{Z}_p[\kern-1pt[ (\mathbb{Z}_p^\times)^n]\kern-1pt]\cong\mathbb{Z}_p[\Delta^n]\otimes_{\mathbb{Z}_p}\mathbb{Z}_p[\kern-1pt[ (1+q\mathbb{Z}_p)^n]\kern-1pt]\cong\mathbb{Z}_p[\Delta^n]\otimes_{\mathbb{Z}_p}\mathbb{Z}_p[\kern-1pt[ T_1,\dotsc,T_n]\kern-1pt], \]

where $T_i=[(1,\dotsc,1,\exp (q),1,\dotsc,1)]-1$ with the $\exp (q)$ in the $i$th position; then continuous homomorphisms $\chi :\Lambda ^n\to A$ are in bijection with $A$-points of $\mathscr {W}^n$ via $\chi \mapsto \chi \circ [\cdot ]$.

Example 2.2.1 (Dominant algebraic weights)

If $t_1\ge \dotsb \ge t_n$ are integers, the algebraic character $(d_1,\dotsc,d_n)\mapsto \prod _{i=1}^n d_i^{t_i}$ is a $\mathbb {Q}_p$-point of $\mathscr {W}^n$ with $T$-coordinates

\[ (\exp(t_1q)-1,\dotsc,\exp(t_nq)-1), \]

such that the valuation of $T_i=\exp (t_iq)-1$ is

\[ v\biggl(\biggl(1+t_iq+\frac{(t_iq)^2}{2!}+\dotsb\biggr)-1\biggr)=v(t_iq). \]

We remark that the weight polydisc in which this character appears is determined by $(t_1,\dotsc,t_n)\pmod {\varphi (q)}$.

If $\chi :\mathbb {Z}_p^\times \to \mathbb {C}_p^\times$ is a finite-order character, we will borrow the following slightly nonstandard definition of the conductor $\operatorname {cond}(\chi )$ of $\chi$ from [Reference RocheRoc98, § 3]: it is the least positive integer $n$ such that $1+p^n\mathbb {Z}_p\subset \ker (\chi )$. Thus, the conductor of the trivial character is $1$ but the conductor of any other character is the same as with the usual definition.

Example 2.2.2 (Locally algebraic weights)

If $t_1\ge \dotsb \ge t_n$ are integers and $\chi _1,\dotsc,\chi _n$ are finite-order characters $\mathbb {Z}_p^\times \to \mathbb {C}_p^\times$, the ‘locally algebraic’ character

\[ (d_1,\dotsc,d_n)\mapsto\prod_{i=1}^n\chi_i(d_i)d_i^{t_i} \]

is a $\mathbb {C}_p$-point of $\mathscr {W}^n$. If $\chi _i$ is nontrivial with conductor $c_i$, we have

\[ v(T_i)=v(\chi_i(\exp(q))\exp(t_iq)-1)=\begin{cases} v(t_iq) & \text{ if }p>2 \text{ and }c_i=1, \\ \dfrac{q}{p^{c_i-1}(p-1)} & \text{ if } p>2 \text{ and } c_i\ge2, \\ v(t_iq) & \text{ if }p=2 \text{ and }c_i=3, \\ \dfrac{q}{p^{c_i-1}(p-1)}=\dfrac1{2^{c_i-3}} & \text{ if }p=2 \text{ and }c_i\ge4. \end{cases} \]

For completeness, we quickly prove the second case; the others are similar. The value $\chi _i(\exp (q))=\chi _i(\exp (p))$ is a primitive $p^{c_i}$th root of unity, say $\zeta _{p^{c_i}}$. Let

\[ f(X)=\frac{X^{p^{c_i}}-1}{X^{p^{c_i-1}}-1}=\prod_{a\in(\mathbb{Z}/p^{c_i}\mathbb{Z})^\times}(X-\zeta_{p^{c_i}}^a)=X^{p^{c_i}-p^{c_i-1}}+X^{p^{c_i}-2p^{c_i-1}}+\dotsb+X^{p^{c_i-1}}+1. \]

Then $f(1)=p=\prod _{a\in (\mathbb {Z}/p^{c_i}\mathbb {Z})^\times }(1-\zeta _{p^{c_i}}^a)$. Each term in the product has the same valuation, since they are Galois conjugate, and there are $p^{c_i-1}(p-1)$ such terms. Thus, $v(\chi _i(\exp (q))-1)=1/({p^{c_i-2}(p-1)})$. The factor of $\exp (t_iq)$ has no effect since it is $1\pmod {p}$.

In general, if $A$ is a Banach $\mathbb {Q}_p$-algebra, we say that a character $s:\mathbb {Z}_p^\times \to A^\times$ is $c$-locally analytic if its restriction to $1+p^c\mathbb {Z}_p$ is given by a convergent power series with coefficients in $A$. Every continuous character $s$ is $c$-locally analytic for some $c$: let $T=s(\exp (q))-1$ and choose $c$ such that $|T^{q^{-1}p^c}|< q^{-1}$. Then we have

\[ s(z)=s\bigl(\exp(q)^{(1/{q})\log z}\bigr)=[(1+T)^{q^{-1}p^c}]^{(1/{p^c})\log z}=[1+((1+T)^{q^{-1}p^c}-1)]^{(1/{p^c})\log z} \]

if this converges. But by our choice of $c$, we have $|(1+T)^{q^{-1}p^c}-1|< q^{-1}$, and if $z\in (1+p^c\mathbb {Z}_p)$, then $|(1/{p^c})\log z|\le 1$. By Lemma 3.6.1 of [Reference ChenevierChe04], this expression is a convergent power series in $z$.

Naturally, if $s:(\mathbb {Z}_p^\times )^n\to A^\times$ is a character, we say that it is $(c_1,\dotsc,c_n)$-locally analytic if it is $c_i$-locally analytic in the $i$th factor.

If $W$ is any open affinoid subset of $\mathscr {W}$, we use

\[ [\cdot]_W:(\mathbb{Z}_p^\times)^n\to\mathscr{O}(W)^\times \]

to denote the universal character of $(\mathbb {Z}_p^\times )^n$ with coefficients in $\mathscr {O}(W)$. Note that $[\cdot ]_W$ is $(c_1,\dotsc,c_n)$-locally analytic with $c_i$ depending on $\max _{s\in W(\mathbb {C}_p)}|T_i(s)|$.

2.3 Coordinates on spaces of functions on $\operatorname {Iw}_p$

If $A$ is an affinoid $\mathbb {Q}_p$-algebra and $s:(\mathbb {Z}_p^\times )^n\to A^\times$ is a weight, we can view any function $f\in \operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p}s$ as a function on $\mathbb {Z}_p^{n(n-1)/2}$ by restricting $f$ to the lower unipotent subgroup $\overline {N}$ and applying the map

\begin{align*} \mathbb{Z}_p^{n(n-1)/2} &\to \overline{N} \\ \underline{z}=(z_{ij})_{n\ge i>j\ge1} &\mapsto \overline{N}(\underline{z})= \begin{pmatrix} 1 & 0 & 0 & \dotsb & 0 \\ pz_{21} & 1 & 0 & \dotsb & 0 \\ pz_{31} & pz_{32} & 1 & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ pz_{n1} & pz_{n2} & pz_{n3} & \dotsb & 1 \end{pmatrix} \in \overline{N}. \end{align*}

We say that $f$ is continuous if it is continuous as a function on $\mathbb {Z}_p^{n(n-1)/2}$ via $\underline {z}\mapsto \overline {N}(\underline {z})$. Then $\mathscr {S}_s:=\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,{\rm cts}}(s)$, where ${\rm cts}$ denotes continuous, is an $A[\operatorname {Iw}_p]$-module. If $s^0:(\mathbb {Q}_p^\times )^n\to A^\times$ is the trivial extension of $s$ from $(\mathbb {Z}_p^\times )^n$ to $(\mathbb {Q}_p^\times )^n$ (that is, we set $s^0(d)=1$ for any $d\in (\mathbb {Q}_p^\times )^n$ whose entries are powers of $p$), $\mathscr {S}_s$ is isomorphic to $\operatorname {Ind}_{B(\mathbb {Q}_p)}^{B(\mathbb {Q}_p)\operatorname {Iw}_p,{\rm cts}}(s^0)$ by restriction of functions from $B(\mathbb {Q}_p)\operatorname {Iw}_p$ to $\operatorname {Iw}_p$. Consequently, it has an action by $B(\mathbb {Q}_p)\operatorname {Iw}_p$.

It will be useful to write out the natural action of $\operatorname {Iw}_p$ on $f\in \mathscr {S}_s$ more explicitly in terms of the coordinates $z_{ij}$. To do this, we interpret them as Plücker coordinates on $\overline {N}(\underline {z})$. Recall that for any $1\le j\le n$ and subset $\sigma$ of $\{1,\dotsc,n\}$ with $\#\sigma =j$, the Plücker coordinate $Z_{j,\sigma }$ associated to $(j,\sigma )$ is the algebraic function on ${GL}_n$ given by the determinant of the minor associated to the rows corresponding to $\sigma$ and the first $j$ columns.

Give $\mathbb {Q}_p^n$ the standard basis $e_1,\dotsc,e_n$ and interpret elements of $\mathbb {Q}_p^n$ as horizontal vectors. Give $\wedge ^j(\mathbb {Q}_p^n)$ the corresponding standard basis

\[ \{e_\sigma=e_{k_1}\wedge\dotsb\wedge e_{k_j}\mid \sigma=\{k_1<\dotsb< k_j\}\subset\{1,\dotsc,n\}\}, \]

ordered lexicographically, and again interpret elements of $\wedge ^j(\mathbb {Q}_p^n)$ as horizontal vectors. Let $1_j=\{1,\dotsc,j\}$. If ${GL}_n(\mathbb {Q}_p)$ acts on $\mathbb {Q}_p^n$ by right multiplication of horizontal vectors (the transpose of the standard action), and $\iota _j:{GL}_n(\mathbb {Q}_p)\hookrightarrow {GL}(\wedge ^j(\mathbb {Q}_p^n))$ gives the induced action of ${GL}_n(\mathbb {Q}_p)$ on $\wedge ^j(\mathbb {Q}_p^n)$ (where again ${GL}(\wedge ^j(\mathbb {Q}_p^n))$ acts on $\wedge ^j(\mathbb {Q}_p^n)$ by right multiplication of horizontal vectors), then for $x\in {GL}_n(\mathbb {Q}_p)$, $Z_{j,\sigma }(x)$ is the coefficient of $e_{1_j}$ in $e_\sigma \cdot \iota _j(x)$, or the entry of $\iota _j(x)$ in the $\sigma$th row and first column. If $b=(b_{ij})\in B(\mathbb {Q}_p)$, $\iota _j(b)$ is also upper triangular, so the coefficient of $e_{1_j}$ in $e_\sigma \cdot \iota _j(xb)=e_\sigma \cdot \iota _j(x)\cdot \iota _j(b)$ is $Z_{j,\sigma }(x)$ times the top left entry of $\iota _j(b)$, which is $b_{11}\dotsb b_{jj}=:t_j(b)$. That is, we have

\[ Z_{j,\sigma}(xb)=t_j(b)Z_{j,\sigma}(x). \]

Thus, $Z_{j,\sigma }$ is invariant under right multiplication by $N$, and $Z_{j,\sigma /1}:=Z_{j,\sigma }/Z_{j,1_j}$ is invariant under right multiplication by $B$.

If $u\in {GL}_n(\mathbb {Z}_p)$, we have $\iota _j(u^{-1}x)=\iota _j(u^{-1})\iota _j(x)$, so the entry of $\iota _j(u^{-1}x)$ in the $\sigma$th row and first column is a linear combination of all the entries of $\iota _j(x)$ in the first column. Hence, we can write

\[ Z_{j,\sigma}(u^{-1}x)=\sum_{\#\tau=j}a_{j,\sigma,\tau}(u)Z_{j,\tau}(x) \]

(note that $a_{j,\sigma,\tau }(u)\in \mathbb {Z}_p$, and if $u\in \operatorname {Iw}_p$, then $a_{j,1_j,1_j}(u)\in \mathbb {Z}_p^\times$), and

\[ Z_{j,\sigma/1}(u^{-1}x)=\frac{a_{j,\sigma,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,\sigma,\tau}(u)Z_{j,\tau/1}(x)}{a_{j,1_j,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,1_j,\tau}(u)Z_{j,\tau/1}(x)}. \]

For $i\ge j$, let $\sigma _{ij}=\{1,\dotsc,j-1,i\}$ (so $\sigma _{jj}=1_j$); then we can see that

\[ Z_{j,\sigma_{ij}}(\overline{N}(\underline{z}))=\begin{cases} Z_{j,1_j}(\overline{N}(\underline{z}))=1 & \text{ if } i=j, \\ pz_{ij} & \text{ if } i>j. \end{cases} \]

Thus, $z_{ij}$, or technically $pz_{ij}$, is indeed a Plücker coordinate for $\overline {N}(\underline {z})$ when $i>j$. Now using the Iwahori decomposition for $\operatorname {Iw}_p$, let

\[ u^{-1}\overline{N}(\underline{z})=\overline{N}(u,\underline{z})T(u,\underline{z})N(u,\underline{z}) \]

for some $\overline {N}(u,\underline {z})\in \overline {N}$, $T(u,\underline {z})\in T$, and $N(u,\underline {z})\in N$. Let $\underline {uz}=((uz)_{ij})_{n\ge i>j\ge 1}\in \mathbb {Z}_p^{n(n-1)/2}$ be the preimage of $\overline {N}(u,\underline {z})$ under $\mathbb {Z}_p^{n(n-1)/2}\to \overline {N}$, so that $\overline {N}(\underline {uz})=\overline {N}(u,\underline {z})$. Thus, if $f\in \mathscr {S}_s$, we have

\begin{align*} (uf)(\overline{N}(\underline{z}))=f(u^{-1}\overline{N}(\underline{z})) &=f(\overline{N}(\underline{uz})T(u,\underline{z})N(u,\underline{z})) \\ &=s(T(u,\underline{z}))f(\overline{N}(\underline{uz})). \end{align*}

We wish to write $\underline {uz}$ and $T(u,\underline {z})$ in terms of $u$ and $\underline {z}$. But we have

\begin{align*} Z_{j,\sigma_{ij}}(u^{-1}\overline{N}(\underline{z})) &=Z_{j,\sigma_{ij}}(\overline{N}(\underline{uz})T(u,\underline{z})N(u,\underline{z})) \\ &=Z_{j,\sigma_{ij}}(\overline{N}(\underline{uz}))t_j(T(u,\underline{z})). \end{align*}

Thus, in fact, setting $i=j$, we find

\[ t_j(T(u,\underline{z}))=Z_{j,\sigma_{jj}}(u^{-1}\overline{N}(\underline{z}))=\sum_{\#\tau=j}a_{j,1_j,\tau}(u)Z_{j,\tau}(\overline{N}(\underline{z})), \]

where $Z_{j,\tau }(\overline {N}(\underline {z}))$ is, by definition, a polynomial in the variables $\{z_{kl}\}_{l\le j,k>l}$ with coefficients in $p\mathbb {Z}_p$. Then when $i>j$, we have

\[ Z_{j,\sigma_{ij}}(u^{-1}\overline{N}(\underline{z}))=Z_{j,\sigma_{ij}}(\overline{N}(\underline{uz}))t_j(T(u,\underline{z}))=p(uz)_{ij}Z_{j,\sigma_{jj}}(u^{-1}\overline{N}(\underline{z})), \]

so

\[ p(uz)_{ij}=Z_{j,\sigma_{ij}/1}(u^{-1}\overline{N}(\underline{z}))=\frac{a_{j,\sigma,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,\sigma,\tau}(u)Z_{j,\tau/1}(\overline{N}(\underline{z}))}{a_{j,1_j,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,1_j,\tau}(u)Z_{j,\tau/1}(\overline{N}(\underline{z}))}, \]

where $Z_{j,\tau /1}(\overline {N}(\underline {z}))$ is again a polynomial in the variables $\{z_{kl}\}_{l\le j,k>l}$ with coefficients in $p\mathbb {Z}_p$.

2.4 Notation for subgroups of $\operatorname {Iw}_p$

Since we work with numerous subgroups of $\operatorname {Iw}_p$, we introduce some notation to identify them. If $\underline {c}=(c_{ij})\in \mathbb {Z}_{\ge 0}^{n\times n}$ is any $n\times n$ matrix of nonnegative integers, we write

\[ \Gamma(\underline{c})=\{(x_{ij})\in {GL}_n(\mathbb{Z}_p)\mid p^{c_{ij}}\mid (x_{ij}-\delta_{ij})\text{ for all }i,j\} \]

(where $\delta _{ij}$ is $1$ if $i=j$ and $0$ otherwise). By Lemma 3.2 of [Reference RocheRoc98], when $\underline {c}$ satisfies $c_{ij}\le c_{ik}+c_{kj}$ for all $i,j,k$ and $c_{ij}+c_{ji}\ge 1$ for all $i\neq j$, the set $\Gamma (\underline {c})$ is closed under multiplication and inverses, hence is a group. Note that this means that if $\Gamma (\underline {c})$ is a group, then so is $T(\mathbb {Z}_p)\Gamma (\underline {c})$. If we instead only have half a matrix of positive integers $\underline {c}=(c_{ij})_{n\ge i>j\ge 1}\in \mathbb {Z}_{>0}^{n(n-1)/2}$, we write

\begin{align*} \Gamma_1(\underline{c})&=\{(x_{ij})\in\operatorname{Iw}_p\mid v(x_{ij})\ge c_{ij}\forall i>j\text{ and }v(x_{ii}-1)\ge \min\{c_{ij}|j< i\}\cup\{c_{ji}|j>i\}\forall i\},\\ \Gamma_0(\underline{c})&=\{(x_{ij})\in\operatorname{Iw}_p\mid v(x_{ij})\ge c_{ij}\forall i>j\}=T(\mathbb{Z}_p)\Gamma_1(\underline{c})\subset\operatorname{Iw}_p. \end{align*}

Thus, $\Gamma _1(\underline {c})$ and $\Gamma _0(\underline {c})$ are subgroups whenever $\underline {c}$ is group-shaped.

Then we see that if $\chi =(\chi _1,\dotsc,\chi _n):T(\mathbb {Z}_p)\to \mathbb {C}^\times$ is a character of $T(\mathbb {Z}_p)$, and $\underline {c}\in \mathbb {Z}_p^{n(n-1)/2}$, $\chi$ extends to a well-defined character of $T(\mathbb {Z}_p)\Gamma _1(\underline {c})=\Gamma _0(\underline {c})$, trivial on $\Gamma _1(\underline {c})$, whenever $\underline {c}$ is compatible with $(\operatorname {cond}(\chi _1),\dotsc,\operatorname {cond}(\chi _n))$.

In the following calculations, whenever we write $\Gamma (\underline {c})$, $\Gamma _0(\underline {c})$, or $\Gamma _1(\underline {c})$ for a matrix or half-matrix of nonnegative integers $\underline {c}$, we implicitly assume that $\underline {c}$ has been chosen so that it is in fact a group.

Depending on convenience, we may also overload the notation in the following ways. First, if $\underline {r}=(r_{ij})\in [0,1]^{n\times n}$ is any $n\times n$ matrix of real numbers in $[0,1]$, we write

\[ \Gamma(\underline{r})=\{(x_{ij})\in {GL}_n(\mathbb{Z}_p)\mid |x_{ij}-\delta_{ij}|\le r_{ij}\text{ for all }i,j\}. \]

Then $\Gamma (\underline {r})$ is a group whenever $r_{ij}\ge r_{ik}r_{kj}$ for all $i,j,k$. We may define $\Gamma _1(\underline {r}),\Gamma _0(\underline {r})$ similarly. Second, if $c\in \mathbb {Z}_{>0}$ is a single integer, we write

\[ \Gamma(c)=\{(x_{ij})\in {GL}_n(\mathbb{Z}_p)\mid v(x_{ij}-\delta_{ij})\ge c\text{ for all }i,j\}. \]

This is always a group. We may define $\Gamma _1(c),\Gamma _0(c)$ similarly. Finally, if $r$ is a single real number in $[0,1]$, we write $\Gamma (r),\Gamma _1(r),\Gamma _0(r)$ for the obvious final abuse of the same notation.

2.5 The sheaf of $p$-adic automorphic forms on weight space

If $\underline {c}=(c_{ij})_{n\ge i>j\ge 1}\in \mathbb {Z}_{>0}^{n(n-1)/2}$, we say that $f\in \mathscr {S}_s$ is $\underline {c}$-locally analytic if, for any $\underline {a}=(a_{ij})\in \mathbb {Z}_p^{n(n-1)/2}$, the restriction of $f$ to

\[ B(\underline{a},\underline{c})=\{z=(z_{ij})_{n\ge i>j\ge1}\in\mathbb{Z}_p^{n(n-1)/2}\mid z_{ij}\in a_{ij}+p^{c_{ij}}\mathbb{Z}_p\forall i,j\} \]

is given by a convergent power series in the variables $z_{ij}$ with coefficients in $A$.

By Proposition 2.5.3, if $s$ is $\underline {c}$-locally analytic with $\underline {c}$ analytic-shaped, the space $\mathscr {S}_{s,\underline {c}}=\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,\underline {c}\text {-loc.an.}}(s)$, where $\underline {c}\text {-loc.an.}$ denotes $\underline {c}$-locally analytic, is well-defined and has an action by $\operatorname {Iw}_p$.

We let $\mathscr {S}=\mathscr {S}_{[\cdot ]}=\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,{\rm cts}}([\cdot ])$. If $\mathscr {U}_p=\operatorname {Iw}_p$, we call

\[ \mathscr{S}(G,\mathscr{U})=\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,{\rm cts}}([\cdot])(G,\mathscr{U}) \]

the space of integral $p$-adic automorphic forms for $G$ of level $\mathscr {U}$; it has an action by $B(\mathbb {Q}_p)\mathscr {U}$. This gives a sheaf on $\mathscr {W}$ whose fiber over $s$ is

\[ \mathscr{S}_s(G,\mathscr{U})=\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,{\rm cts}}(s)(G,\mathscr{U}). \]

Similarly, let $\mathscr {S}_{W,\underline {c}}=\mathscr {S}_{[\cdot ]_W,\underline {c}}=\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,\underline {c}\text {-loc.an.}}([\cdot ]_W)$ (for any $\underline {c}$ such that $[\cdot ]_W$ is $\underline {c}$-locally analytic). If $\mathscr {U}_p=\operatorname {Iw}_p$, we call

\[ \mathscr{S}_{W,\underline{c}}(G,\mathscr{U})=\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,\underline{c}\text{-loc.an.}}([\cdot]_W)(G,\mathscr{U}) \]

the space of $\underline {c}$-locally analytic $p$-adic automorphic forms for $G$ of level $\mathscr {U}$; we show in the next section that this does not have an action by $B(\mathbb {Q}_p)$, as some elements of $B(\mathbb {Q}_p)$ do not preserve the radius of local analyticity (Remark 2), but it does have an action by a certain submonoid.

2.6 The operators $U_p^a$

If $H$ is any locally compact, totally disconnected topological group, we write $\mathscr {H}(H)$ for the $k$-algebra of compactly supported, locally constant $k$-valued functions on $H$ with the convolution product

\[ (\varphi_1\star\varphi_2)(g)=\int_{h\in H}\varphi_1(h)\varphi_2(h^{-1}g)\,d\mu, \]

where $\mu$ is a Haar measure on $H$. This algebra usually has no identity, but many idempotents. If $K$ is a compact open subgroup of $H$, the idempotent $e_K={\mathbb {1}_K}/{\mu (K)}$ projects $\mathscr {H}(H)$ onto the subalgebra $\mathscr {H}(H// K)$ of functions that are both left- and right-invariant under $K$. If $V$ is a smooth $H$-module, it is an $\mathscr {H}(H)$-module via

\[ \varphi(v)=\int_H \varphi(h)(hv)\,dh \]

and, similarly, $V^K$ is an $\mathscr {H}(H// K)$-module.

In the particular case $H=B(\mathbb {Q}_p)\mathscr {U}$, $V=\mathscr {S}_s(G,\mathscr {U})$, $K=\mathscr {U}$, we can rephrase this as follows. We sometimes write $[\mathscr {U}\zeta \mathscr {U}]$ for the element $\mathbb {1}_{\mathscr {U}\zeta \mathscr {U}}$ of $\mathscr {H}(G(\mathbb {A}_f)// \mathscr {U})$. If $\zeta _1,\dotsc,\zeta _r$ are left $\mathscr {U}$-coset representatives of $\mathscr {U}\zeta \mathscr {U}$, so that

\[ \mathscr{U}\zeta\mathscr{U}=\coprod_{i=1}^r\zeta_i\mathscr{U}, \]

then for any $\varphi \in \mathscr {S}_s(G,\mathscr {U})$ and $x\in G(\mathbb {Q})\backslash G(\mathbb {A}_f)$, we have

\begin{align*} [\mathscr{U}\zeta\mathscr{U}](\varphi)(x) &=\int_{G(\mathbb{A}_f)}[\mathscr{U}\zeta\mathscr{U}](g)\cdot(g.\varphi)(x)\,dg \\ &=\int_{\mathscr{U}\zeta\mathscr{U}}g_p\varphi(xg)\,dg= \sum_{i=1}^r(\zeta_i)_p.\varphi(x\zeta_i). \end{align*}

The following is Lemma 4.5.2 of [Reference ChenevierChe04] or Proposition 3.3.3 of [Reference LoefflerLoe10].

Lemma 2.6.1 Fix coset representatives $x_1,\dotsc,x_h$ of $G(\mathbb {Q})\backslash G(\mathbb {A}_f)/\mathscr {U}$ and, thus, an isomorphism $\mathscr {S}_s(G,\mathscr {U})\cong \mathscr {S}_s^h$. Then we have

\[ [\mathscr{U}\zeta\mathscr{U}](\varphi)(x_j)=\sum_{k=1}^h\sum_{i\mid\zeta_i\in x_j^{-1}G(\mathbb{Q})x_k\mathscr{U}}(\zeta_iu_{ij}^{-1})_p.\varphi(x_k) \]

for some $u_{ij}\in \mathscr {U}$. That is, the action of $[\mathscr {U}\zeta \mathscr {U}]$ on $\mathscr {S}_s(G,\mathscr {U})$ is of the form $\sum T_j\circ \sigma _j$, where the $\sigma _j$ are compositions of permutation operators on the entries of vectors in $\mathscr {S}_s^h$ with projections onto one of the coordinates, and the $T_j$ are diagonal translations of $\mathscr {S}_s^h$ by elements of $\mathscr {U}\zeta \mathscr {U}$.

Proof. Write $x_j\zeta _i$ in the form $d_{ij}x_{k_{ij}}u_{ij}$ where $d_{ij}\in G(\mathbb {Q})$ and $u_{ij}\in \mathscr {U}$. Then

\begin{align*} [\mathscr{U}\zeta\mathscr{U}](\varphi)(x_j) &=\sum_{i=1}^r(\zeta_i)_p.\varphi(x_j\zeta_i) \\ &=\sum_{i=1}^r(\zeta_i)_p.\varphi(d_{ij}x_{k_{ij}}u_{ij})=\sum_{i=1}^r(\zeta_iu_{ij}^{-1})_p.\varphi(x_{k_{ij}}). \end{align*}

The values of $i$ for which $k_{ij}=k$ are those for which $\zeta _i=x_j^{-1}dx_ku$ for some $d\in G(\mathbb {Q})$ and $u\in \mathscr {U}$, that is, $\zeta _i\in x_j^{-1}G(\mathbb {Q})x_k\mathscr {U}$.

If $a=(a_1,\dotsc,a_n)\in \mathbb {Z}^n$, we write

\[ u^a=\operatorname{diag}(p^{a_1},\dotsc,p^{a_n}) \]

and define the subgroup

\[ \Sigma=\{u^a=\operatorname{diag}(p^{a_1},\dotsc,p^{a_n})\mid a=(a_1,\dotsc,a_n)\in\mathbb{Z}^n\}\subset {GL}_n(\mathbb{Q}_p) \]

and its submonoids

\begin{align*} \Sigma^-&=\{u^a=\operatorname{diag}(p^{a_1},\dotsc,p^{a_n})\mid a_1\ge a_2\ge\dotsb\ge a_n\}\subset \Sigma,\\ \Sigma^{--}&=\{u^a=\operatorname{diag}(p^{a_1},\dotsc,p^{a_n})\mid a_1> a_2>\dotsb> a_n\}\subset \Sigma^-. \end{align*}

We frequently choose $\zeta$ to be an element of $\Sigma ^-$. Let

\[ U_p^a=[\mathscr{U}\operatorname{diag}(p^{a_1},\dotsc,p^{a_n})\mathscr{U}]. \]

Proof. We have

\begin{align*} & f((u^a)^{-1}\overline{N}(z_{ij}))\\ & \quad =f\left( \begin{pmatrix} p^{-a_1} & \dotsb & 0 \\ 0 & \vdots & 0 \\ 0 & \dotsb & p^{-a_n} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & \dotsb & 0 \\ pz_{21} & 1 & 0 & \dotsb & 0 \\ pz_{31} & pz_{32} & 1 & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ pz_{n1} & pz_{n2} & pz_{n3} & \dotsb & 1 \end{pmatrix} \right)\\ & \quad =f\begin{pmatrix} p^{-a_1} & 0 & 0 & \dotsb & 0 \\ p^{-a_2+1}z_{21} & p^{-a_2} & 0 & \dotsb & 0 \\ p^{-a_3+1}z_{31} & p^{-a_3+1}z_{32} & p^{-a_3} & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ p^{-a_n+1}z_{n1} & p^{-a_n+1}z_{n2} & p^{-a_n+1}z_{n3} & \dotsb & p^{-a_n} \end{pmatrix}\\ & \quad =f\left(\begin{pmatrix} 1 & 0 & 0 & \dotsb & 0 \\ p^{a_1-a_2+1}z_{21} & 1 & 0 & \dotsb & 0 \\ p^{a_1-a_3+1}z_{31} & p^{a_2-a_3+1}z_{32} & 1 & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ p^{a_1-a_n+1}z_{n1} & p^{a_2-a_n+1}z_{n2} & p^{a_3-a_n+1}z_{n3} & \dotsb & 1 \end{pmatrix} \begin{pmatrix} p^{-a_1} & \dotsb & 0 \\ 0 & \vdots & 0 \\ 0 & \dotsb & p^{-a_n} \end{pmatrix} \right)\\ & \quad =f(\overline{N}(p^{a_i-a_j}z_{ij}))s^0(u^a)=f(\overline{N}(p^{a_i-a_j}z_{ij})). \end{align*}

Let $\underline {c}^0\in \mathbb {Z}_{>0}^{n(n-1)/2}$ be minimal such that $s$ is $\underline {c}^0$-locally analytic.

The space $\mathscr {S}_{s,\underline {c}}$ is an orthonormalizable $A$-module, for which we choose the following orthonormal basis: for each $\underline {a}\in \prod _{n\ge i>j\ge 1} \mathbb {Z}_p/p^{c_{ij}}\mathbb {Z}_p$, we choose the set of monomials $\prod _{n\ge i>j\ge 1}z_{ij}^{e_{ij}}$ as an orthonormal basis for the restriction of $\mathscr {S}_{s,\underline {c}}$ to $B(\underline {a},\underline {c})$; then for $\mathscr {S}_{s,\underline {c}}$, we may choose as orthonormal basis the set of monomials $\prod _{n\ge i>j\ge 1}(z_{ij}^{\underline {a}})^{e_{ij}}$, with one copy for each $\underline {a}\in \prod _{n\ge i>j\ge 1} \mathbb {Z}_p/p^{c_{ij}}\mathbb {Z}_p$, where $z_{ij}^{\underline {a}}$ is the function which equals $z_{ij}$ on $B(\underline {a},\underline {c})$ and $0$ elsewhere.

Since $U_p^a$ is completely continuous on $\mathscr {S}_{s,\underline {c}}(G,\mathscr {U})$, for any $k$, the matrix of the action of $U_p^a$ (in any basis) has a finite number of nonzero rows mod $p^k$. Suppose that this matrix has $r_k$ rows that are zero mod $p^k$ but nonzero mod $p^{k+1}$. Then for any $N\ge r_0+r_1+\dotsb +r_k$, the coefficient of $X^N$ in the characteristic power series

\[ P_{s,\underline{c}}^a(X)=\det(1-XU_p^a|\mathscr{S}_{s,\underline{c}}(G,\mathscr{U})) \]

of $U_p^a$ acting on $\mathscr {S}_{s,\underline {c}}(G,\mathscr {U})$, being a linear combination of minors of size $N\ge r_0+r_1+\dotsb +r_k$, is divisible by $r_1+2r_2+\dotsb +kr_k$. Since this lower bound grows faster than any linear function of $N$, $P_{s,\underline {c}}^a(X)$ is an entire function of $X$.

Let $U_p^\Sigma$ be the subring of $\mathscr {H}(G(\mathbb {A}_f)// \mathscr {U})$ generated by the elements $U_p^a$ for $a\in \Sigma ^-$ and their inverses (which exist, as discussed in [Reference Bellaïche and ChenevierBC09, § 6.4.1]). By Proposition 6.4.1 of [Reference Bellaïche and ChenevierBC09], the map from $k[\Sigma ]$ to $U_p^\Sigma$ sending $u^a$ to $U_p^b(U_p^c)^{-1}$ where $u^b,u^c$ are any elements of $\Sigma ^-$ such that $u^a=u^b(u^c)^{-1}$ is a well-defined isomorphism of rings. Thus, in particular, $U_p^\Sigma$ is abelian. Let $\mathscr {H}$ be a subalgebra of $\mathscr {H}(G(\mathbb {A}_f)// \mathscr {U})$ given by the product of $\mathbb {Z}[U_p^\Sigma ]$ at $p$ and some commutative subalgebra of $\mathscr {H}(G(\mathbb {A}_f^p)// \mathscr {U}^p)$ away from $p$.

We write $u_i$ for the image of $\operatorname {diag}(1,\dotsc,1,p,1,\dotsc,1)\in k[\Sigma ]$ in $U_p^\Sigma$. If $f$ is an element of an $\mathscr {H}$-module $S$ (such as $\mathscr {S}_{s,\underline {c}}(G,\mathscr {U})$) that is a generalized simultaneous eigenvector for $\mathscr {H}$, let $u_i(f)=\lambda _i f$. We call these the $\lambda$-values associated to $f$. We call the subspace generated by all the generalized simultaneous eigenvectors whose associated $\lambda$-values are nonzero the finite-slope subspace of $S$, and we denote it by $S^{{\rm fs}}$.

Unless otherwise specified, we will generally set $\mathscr {U}$ to be a compact open subgroup of $G(\mathbb {A}_f)$ given by the product of $\operatorname {Iw}_p$ at $p$ and a fixed tame level structure away from $p$ chosen so that $x^{-1}G(\mathbb {Q})x\cap \mathscr {U}=1$ for all $x$ (the condition of being ‘sufficiently small’ or ‘neat’ as described at the end of § 2.1). Call this subgroup $U_0(p)$. (Note that for the same reason as in Proposition 3.1.2, our choice of $\operatorname {Iw}_p$ as the wild level structure does not actually affect $P_s^a(X)$.)

2.7 The eigenvariety

Given our setup so far, the eigenvariety is easy to define. For a given $u^a\in \Sigma ^{--}$, let $\mathscr {Z}^a$ be the subvariety of $\mathscr {W}\times \mathbb {G}_m$ which, in any subset $W\times \mathbb {G}_m$ where $W\subset \mathscr {W}$ is open affinoid, is cut out by the characteristic power series $P_W^a(X)$ of $U_p^a$ acting on $\mathscr {S}_W(G,U_0(p))$. Let $w:\mathscr {Z}^a\to \mathscr {W}$ be the first projection (weight) map, and $a_p^a:\mathscr {Z}^a\to \mathbb {G}_m$ the inverse of the second projection ($U_p^a$-eigenvalue) map. Then for any point $z\in \mathscr {Z}^a$, $a_p^a(z)$ is a nonzero eigenvalue of $U_p^a$ acting on $\mathscr {S}_{w(z)}(G,U_0(p))$, and for any $w\in \mathscr {W}$, all nonzero eigenvalues of $U_p^a$ acting on $\mathscr {S}_w(G,U_0(p))$ can be found in the fiber of $\mathscr {Z}^a$ over $w$. We call $\mathscr {Z}^a$ the spectral variety associated to $U_p^a$.

It is convenient to fix a particular choice of $u^a\in \Sigma ^{--}$; we choose $a=(n-1,n-2,\dotsc,1,0)$. From now on, we write $U_p=U_p^{(n-1,n-2,\dotsc,1,0)}$ and $\mathscr {Z}=\mathscr {Z}^{(n-1,n-2,\dotsc,1,0)}$. We call an eigenform $f\in \mathscr {S}_w(G,U_0(p))$ finite-slope if $U_pf\neq 0$ (i.e. the valuation, or slope, of the $U_p$-eigenvalue is finite, and $f$ appears on the eigenvariety), and infinite-slope otherwise.

Since $\mathscr {H}$ is commutative, we can construct the space $\mathscr {D}$ whose points correspond to systems of eigenvalues of all Hecke operators in $\mathscr {H}$, including in particular all $U_p^a$ simultaneously, by simply taking $\mathscr {D}$ to be the finite cover of $\mathscr {Z}$ which, over an affinoid $W\subset \mathscr {W}$, is given by the MaxSpec of the image of $\mathscr {H}\otimes \Lambda ^n$ in the endomorphism ring of $\mathscr {S}_W(G,U_0(p))$. Then $\mathscr {D}$ inherits the weight map $w:\mathscr {Z}^a\to \mathscr {W}$ and each eigenvalue map $a_p^a:\mathscr {D}\to \mathbb {G}^m$. Because $\mathscr {D}\to \mathscr {Z}^a$ is a finite map, in general the bounds and geometric properties we get for $\mathscr {Z}^a$ should also apply to $\mathscr {D}$. For this paper, we focus on the properties of $\mathscr {Z}$ and/or $\mathscr {Z}^a$ for any fixed $a$.

For additional details on properties of $\mathscr {Z}^a$ and $\mathscr {D}$ and their proofs, see [Reference ChenevierChe04] or [Reference BuzzardBuz07].

3.1 $p$-adic automorphic forms of locally algebraic weights

In § 2.1, we defined classical forms of algebraic weights via the algebraic representation $S_t(k)$ of ${GL}_n(\mathbb {Q}_p)$. This construction may be generalized to locally algebraic weights as follows. Let $\chi =\chi _1\dotsb \chi _n$ be a finite character of $(\mathbb {Z}_p^\times )^n$. Then $t\chi$ is a locally algebraic character of $(\mathbb {Z}_p^\times )^n$, in the sense that it is algebraic upon restriction to $\prod _{i=1}^n(a_i+p^{c_i}\mathbb {Z}_p)$ for some choice of $c_i$ and any nonzero $a_i$. Similarly to earlier notation, for a positive integer $c$, let

\[ B(\underline{a},c)=\{z=(z_{ij})_{n\ge i>j\ge1}\in\mathbb{Z}_p^{n(n-1)/2}\mid z_{ij}\in a_{ij}+p^{c}\mathbb{Z}_p\forall i,j\}. \]

Then there are two equivalent definitions of the space

\[ S_{t\chi,c}:=\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,c\text{-loc.alg.}}(t\chi), \]

where $c\text {-loc.alg.}$ stands for $c$-locally algebraic. The first is through the usual induction operator, as follows. We say that $f\in \operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p}(t\chi )$ is $c$-locally algebraic if it has an algebraic extension to $B(\underline {a},c)$ for all $\underline {a}\in \mathbb {Z}_p^{n(n-1)/2}$ of degree bounded as follows: writing $f$ as a polynomial in the variables $Z_{i,k/1}$ as in § 2.3, we require that for each fixed $i$, the degree of $f$ as a polynomial in all the variables $Z_{i,k/1}$ should be at most $t_i-t_{i+1}=:m_i$. As in Proposition 2.5.3, one can see using the formulas in § 2.3 that assuming $\operatorname {cond}(\chi _i)\le c$ for all $i$, this condition is invariant under right translation by $\operatorname {Iw}_p$.

The second definition, coming from the perspective of Loeffler [Reference LoefflerLoe10, § 2.5], is

\[ \biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,{\rm alg}}t\biggr)\otimes\biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\Gamma(c)}\chi\biggr). \]

Note that $\Gamma (c)$ is normal in $\operatorname {Iw}_p$ because it is the kernel of the reduction map from $\operatorname {Iw}_p$ to the corresponding group with coefficients in $\mathbb {Z}_p/p^c\mathbb {Z}_p$.

Except for an annoying technical distinction which we discuss at the end of this subsection, the space $\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,{\rm alg}}t$ is the same (as an $\operatorname {Iw}_p$-representation) as the space $S_t(k)$ defined in § 2.1, since $\operatorname {Iw}_p$ is Zariski-dense in ${GL}_n$. Let $d_t=\dim \operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,{\rm alg}}t$. We now check that the two definitions just given are actually equivalent.

Proposition 3.1.1 The natural map

\begin{gather*} \biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,{\rm alg}}t\biggr)\otimes\biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\Gamma(c)}\chi\biggr)\to \operatorname{Ind}_B^{\operatorname{Iw}_p,c\text{-loc.alg.}}(t\chi)\\ f\otimes g\mapsto fg \end{gather*}

is an isomorphism.

Proof. To construct an inverse, let $\varphi \in \operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,c\text {-loc.alg.}}(t\chi )$. Let $\varphi _{{\rm alg}}:\operatorname {Iw}_p\to \mathbb {C}$ (the ‘algebraic part’ of $\phi$) be defined by

\[ \varphi_{{\rm alg}}(b\overline{n})=t(b)\varphi'(\overline{n}) \]

for all $b\in B,\overline {n}\in \overline {N}\cap \operatorname {Iw}_p$, where $\varphi '$ is the unique algebraic extension of $\varphi |_{\overline {N}\cap \Gamma (c)}$ to $\overline {N}\cap \operatorname {Iw}_p$. Let $\varphi _{{\rm sm}}:\operatorname {Iw}_p/\operatorname {Iw}_p\cap \Gamma (c)\to \mathbb {C}$ (the ‘smooth part’ of $\phi$) be defined by

\[ \varphi_{{\rm sm}}(\overline{b}\overline{\overline{n}})=\chi(b)(\varphi/\varphi')(\overline{n}), \]

where $b,\overline {n}$ are any lifts of $\overline {b}\in B/B\cap \Gamma (c)$, $\overline {\overline {n}}\in (\overline {N}\cap \operatorname {Iw}_p)/(\overline {N}\cap \Gamma (c))$. Then we have $\varphi _{{\rm alg}}\otimes \varphi _{{\rm sm}}\mapsto \varphi$, which suffices to prove surjectivity.

Injectivity follows from dimension counting: both sides have dimension $d_tp^{c\binom{n}{2}}$.

Remark 3 There is a simple isomorphism of $\operatorname {Iw}_p$-representations

\[ \operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\Gamma(c)}\chi\xrightarrow{\sim}\operatorname{Ind}_{\Gamma_0(c)}^{\operatorname{Iw}_p}\chi, \]

so we could just as easily have phrased this section in terms of $\operatorname {Ind}_{\Gamma _0(c)}^{\operatorname {Iw}_p}\chi$. For now, we have no particular reason to do this, but it may be more convenient for future work.

We call

\[ S_{t\chi,c}(G,\mathscr{U})=\operatorname{Ind}_B^{\operatorname{Iw}_p,c\text{-loc.alg.}}(t\chi)(G,\mathscr{U}) \]

the space of classical $p$-adic automorphic forms on $G$ of weight $t\chi$, radius $c$, and level $\mathscr {U}$. By the definitions, it embeds into $\mathscr {S}_{t\chi }(G,\mathscr {U})$, and we call its image a classical subspace of $\mathscr {S}_{t\chi }(G,\mathscr {U})$. The following proposition is a quick generalization of part 4 of Lemma 4 of [Reference BuzzardBuz04].

Proposition 3.1.2 For any positive integers $c$, $d$, and $e$ with $d\le e$ and $c+d-e\ge 1$, we have a natural vector space isomorphism

\[ S_{t\chi,c}(G,\mathscr{U}^p\Gamma_0(d))\cong S_{t\chi,c+d-e}(G,\mathscr{U}^p\Gamma_0(e)) \]

such that systems of $\mathscr {H}$-eigenvalues on the left (where $\mathscr {H}$ is obtained with respect to $\mathscr {U}^p\Gamma _0(d)$) go to identical systems of $\mathscr {H}$-eigenvalues on the right (where $\mathscr {H}$ is obtained with respect to $\mathscr {U}^p\Gamma _0(e)$).

Proof. For the purposes of this proposition, let $X=G(\mathbb {Q})\backslash G(\mathbb {A}_f)$. The left-hand side is the subset of

(1)\begin{equation} (\text{Hom}_{set}(X,\mathbb{C}_p)\otimes_{\mathbb{C}_p}\operatorname{Ind}_B^{\operatorname{Iw}_p,c\text{-loc.alg.}}(t\chi))^{\Gamma_0(e)} \end{equation}

that remains invariant under a set of coset representatives $A$ for $\Gamma _0(e)\backslash \Gamma _0(d)$. This subset has a map by restriction of the second factor to

\[ (\text{Hom}(X,\mathbb{C}_p)\otimes \mathscr{O})^{\Gamma_0(e)}, \]

where $\mathscr {O}$ is the space of functions on $B((p^{e-d}\mathbb {Z}_p)^{n(n-1)/2},c)$ that are algebraic on each ball $B(\underline {a},c)$. The map is an isomorphism: if $\varphi \in (\text {Hom}(X,\mathbb {C}_p)\otimes \mathscr {O})^{\Gamma _0(e)}$, its inverse $\psi$ may be defined by

\[ \psi(x)(z)=\varphi(xa^{-1})(\overline{N}^{-1}(\overline{N}(z)a^{-1}))\text{ for }a\in A\text{ such that }za^{-1}\in B\bigl((p^{e-d}\mathbb{Z}_p)^{n(n-1)/2},c\bigr). \]

In $\overline {N}(z)a^{-1}$, $a$ should be interpreted as a coset representative for $\Gamma _0(e-d+1)\backslash \operatorname {Iw}_p$. Note that this inverse depends on the choice of coset representatives $A$. Now $B((p^{e-d}\mathbb {Z}_p)^{n(n-1)/2},c)$ is isomorphic to $B(\mathbb {Z}_p^{n(n-1)/2},c+d-e)$ via multiplication by $p^{d-e}$, so $(\text {Hom}(X,\mathbb {C}_p)\otimes \mathscr {O})^{\Gamma _0(e)}$ is the desired right-hand side.

To check that the Hecke operator action is preserved, it suffices to note that the Hecke operator action on the left-hand side can be calculated on its image in (1).

Corollary 3.1.3 For all positive integers $c$ and group-like $\underline {d}\in \mathbb {Z}_{\ge 0}^{n(n-1)/2}$, we have a vector space embedding

\[ S_{t\chi,c}(G,\mathscr{U}^p\Gamma_0(\underline{d}))\hookrightarrow \mathscr{S}_{t\chi}(G,U_0(p)) \]

preserving systems of $\mathscr {H}$-eigenvalues.

Proof. Let $d=\max d_{ij}$. Then we have an embedding

\[ S_{t\chi,c}(G,\mathscr{U}^p\Gamma_0(\underline{d}))\hookrightarrow S_{t\chi,c}(G,\mathscr{U}^p\Gamma_0(d)). \]

By Proposition 3.1.2, we have an isomorphism

\[ S_{t\chi,c}(G,\mathscr{U}^p\Gamma_0(d))\cong S_{t\chi,c+d-1}(G,\mathscr{U}^p\Gamma_0(1))=S_{t\chi,c+d-1}(G,\mathscr{U}^p\operatorname{Iw}_p). \]

The space on the right certainly embeds into $\mathscr {S}_{t\chi }(G,\mathscr {U}^p\operatorname {Iw}_p)=\mathscr {S}_{t\chi }(G,U_0(p))$ as discussed.

For future reference, it will be important to note the following distinction between the space $S_{t,1}(G,U_0(p))$ defined here and the space $S_t(k)(G,U_0(p))$ of classical algebraic automorphic forms defined in § 2.1, which is that they are identical except for the normalization of the action of the $U_p$-operator. This is because, as in the beginning of § 2.3, the action of $u^a$ on $S_t=\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,{\rm alg}}t=\operatorname {Ind}_{B(\mathbb {Q}_p)}^{B(\mathbb {Q}_p)\operatorname {Iw}_p,{\rm alg}}t^0$ implicitly arises from the extension of $t$ to $t^0:(\mathbb {Q}_p^\times )^n\to \mathbb {C}$ where $t^0(u^a)=1$, whereas the action of $u^a$ on $S_t(k)$ arises from the algebraic character $t:(\mathbb {Q}_p^\times )^n\to \mathbb {C}$, for which we can compute $t(u^a)=p^{\sum _i a_it_i}$. Thus, we have

\[ U_p^a|S_{t,1}(G,U_0(p))=p^{\sum_i a_it_i}U_p^a|S_t(k)(G,U_0(p)). \]

3.2 A classicality theorem following Bellaïche–Chenevier

This is essentially Proposition 7.3.5 of [Reference Bellaïche and ChenevierBC09]. We just summarize the proof with modifications so that it also works for locally algebraic weights.

Theorem 3.2.1 Let $f\in \mathscr {S}_{t\chi }(G,\mathscr {U})$ where $t\chi =(t_1\chi _1,\dotsc, t_n\chi _n)$, in which the $t_i$ are integers such that $t_1\ge \dotsb \ge t_n$ and the $\chi _i$ are finite, such that $f$ is an eigenform for all operators $U_p^{(a_1,\dotsc,a_n)}$. Let $\lambda _1,\dotsc,\lambda _{n-1}$ be the $\lambda$-values associated to $f$ as defined at the end of § 2.6. If

\[ v(\lambda_1\lambda_2\dotsb\lambda_i)< t_i-t_{i+1}+1 \]

for all $i=1,\dotsc,n-1$, then $f$ is classical (i.e. lies in the image of $S_{t\chi,c}(G,\mathscr {U})$ for any $c$ such that this is well-defined).

Proof. Let $V=\mathbb {Q}_pv\oplus \mathbb {Q}_pR$ be a finite-dimensional vector space generated by a nonzero vector $v$ and a lattice $R$. In some basis whose first vector is $v$, we define the following matrix groups, where a lowercase letter refers to a single matrix entry, an uppercase letter refers to a larger submatrix of the appropriate size, a number refers to a submatrix consisting of copies of that number of the appropriate size, and the subscript after the matrix refers to its coefficient ring:

\begin{gather*} H={GL}_{\mathbb{Q}_p}(V),\hspace{0.1in} P={\begin{pmatrix} {a} & {B} \\ {0} & {D}\end{pmatrix}}_{\mathbb{Q}_p}\subset H,\\ \overline{N}={\begin{pmatrix} {1} & {0} \\ {C} & {I_R}\end{pmatrix}}_{\mathbb{Z}_p},\hspace{0.1in} J={\begin{pmatrix} {a} & {B} \\ {pC} & {D}\end{pmatrix}}_{\mathbb{Z}_p}\subset {GL}_{\mathbb{Z}_p}(V). \end{gather*}

In words, $P$ is the parabolic subgroup of ${GL}_{\mathbb {Q}_p}(V)$ stabilizing the line $\mathbb {Q}_pv\subset V$, $\overline {N}$ is the unipotent radical of the opposite parabolic to $P$, and $J$ is the parahoric subgroup of ${GL}_{\mathbb {Z}_p}(V)$ associated to $P$. We have an Iwasawa decomposition

\[ J=(\overline{N}\cap J)\times(P\cap J)={\begin{pmatrix} {1} & {0} \\ {pC} & {I_R}\end{pmatrix}}_{\mathbb{Z}_p}\times{\begin{pmatrix} {a} & {B} \\ {0} & {D}\end{pmatrix}}_{\mathbb{Z}_p} \]

and an isomorphism $\alpha :\overline {N}\cap J\to R$ given by

\[ \alpha{\begin{pmatrix} {1} & {0} \\ {pC} & {I_R}\end{pmatrix}}=C. \]

We also write

\begin{align*} \mathfrak{U}^-&=\left\{{\begin{pmatrix} {p^k} & {0} \\ {0} & {p^{k'}D}\end{pmatrix}}\mid k\ge k'\in\mathbb{Z}, D\in {GL}_{\mathbb{Z}_p}(R)\right\},\\ \mathfrak{U}^{--}&=\left\{{\begin{pmatrix} {p^k} & {0} \\ {0} & {p^{k'}D}\end{pmatrix}}\mid k> k'\in\mathbb{Z}, D\in {GL}_{\mathbb{Z}_p}(R)\right\}; \end{align*}

these are submonoids of $H={GL}_{\mathbb {Q}_p}(V)$. Finally, we write $\mathfrak {M}$ for the submonoid of $H$ generated by $\mathfrak {U}^-$ and $J$. Let $\chi :P\to \mathbb {Q}_p^\times$ be the character of $P$ acting on $\mathbb {Q}_p v$. We have a $\mathbb {C}_p[H]$-equivariant isomorphism

\begin{gather*} \operatorname{Sym}^m(V\otimes_{\mathbb{Q}_p}\mathbb{C}_p)^\vee\to \operatorname{Ind}_P^{H,{\rm alg}}(\chi^m)\\ \varphi\mapsto(h\mapsto\varphi(h(e))). \end{gather*}

We get a natural $\mathfrak {M}$-equivariant map

\[ \operatorname{Ind}_P^{H,{\rm alg}}(\chi^m)\to\operatorname{Ind}_P^{JP,an}(\chi^m) \]

by restriction. Let $\delta :\mathfrak {M}\to \mathbb {C}_p^\times$ be the character such that $\delta (J)=1$ and $\delta (u)=p^a$ if

\[ u={\begin{pmatrix} {p^a} & {0} \\ {0} & {U}\end{pmatrix}}\in \mathfrak{U}^-. \]

Let $e_1,\dotsc,e_n$ be the standard basis of $\mathbb {Q}_p^n$. Let $V_i=\wedge ^i(\mathbb {Q}_p^n)$, $v_i=e_1\wedge e_1\wedge \dotsb \wedge e_i$, $m_i=t_i-t_{i+1}$ if $i< n$ and $m_n=t_n$, and $R_i$ be the $\mathbb {Z}_p$-span of the elements $e_{j_1}\wedge \dotsb e_{j_i}$ with $j_1<\dotsb < j_i$ and $(j_1,\dotsc,j_i)\neq (1,\dotsc,i)$. Then for $i=1,\dotsc,n$, we get

\[ H_i,P_i,\chi_i,\overline{N}_i,J_i,\alpha_i,\mathfrak{U}_i^-,\mathfrak{U}_i^{--},\mathfrak{M}_i,\delta_i \]

as defined above.

Write $S_i(\mathbb {C}_p)^\vee$ for the space $\operatorname {Ind}_{P_i}^{H_i,{\rm alg}}(\chi _i^{m_i})$ viewed as a representation of $G(\mathbb {Q}_p)$ via $\wedge ^i:G(\mathbb {Q}_p)\to H_i$. Write $\mathscr {S}_i(m_i)$ for the space $\operatorname {Ind}_{P_i}^{H_i,an}(\chi _i^{m_i})\otimes \delta _i^{m_i}$ viewed as a representation of $\mathfrak {M}$ via $\wedge ^i$. We have surjections

\begin{gather*} \bigotimes_{i=1}^mS_i(\mathbb{C}_p)^\vee\to S_t(\mathbb{C}_p)^\vee\\ \widehat{\bigotimes}_{i=1}^m \mathscr{S}_i(m_i)\to \mathscr{S}_t \end{gather*}

which are equivariant with respect to $G(\mathbb {Q}_p)$ and $\mathfrak {M}$ respectively, both given by the formula

\[ (f_1,\dotsc,f_m)\mapsto\biggl(g\mapsto\prod_{i=1}^m f_i(\wedge^i(g))\biggr). \]

Let

\begin{align*} Q &=\mathscr{S}_t/(S_t(\mathbb{C}_p)^\vee\otimes\delta_t)\otimes\biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\operatorname{Iw}_p\cap \Gamma(c)}\chi\biggr),\\ Q' &=\left. \widehat{\bigotimes}_{i=1}^m \mathscr{S}_i(m_i) \right/ \biggl(\bigotimes_{i=1}^mS_i(\mathbb{C}_p)^\vee\otimes\delta_i^{m_i}\biggr)\otimes\biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\operatorname{Iw}_p\cap \Gamma(c)}\chi\biggr),\\ Q_i' &=\biggl(\widehat{\bigotimes}_{j\neq i}\mathscr{S}_j(m_j)\biggr)\otimes\bigl(\mathscr{S}_i(m_i)/S_i(\mathbb{C}_p)^\vee\otimes\delta_i^{m_i}\bigr)\otimes\biggl(\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\operatorname{Iw}_p\cap \Gamma(c)}\chi\biggr). \end{align*}

Then we have a surjection

\[ Q'(G,\mathscr{U})\twoheadrightarrow Q(G,\mathscr{U}) \]

and an injection

\[ Q'(G,\mathscr{U})\hookrightarrow\prod_{i=1}^n Q_i'(G,\mathscr{U}). \]

We wish to show that if $w\in Q(G,\mathscr {U})$ satisfies the hypotheses of the theorem, that is, $u_i(w)=\lambda _i w$ with the $\lambda _i$ satisfying the given inequalities, then $w=0$. We can instead check this claim for $w'\in Q'(G,\mathscr {U})$ satisfying the same condition, and for this it suffices to check that the image $w_i'$ of $w'$ vanishes in $Q_i'(G,\mathscr {U})$ for each $i$. Let $U_i=({\prod _{j=1}^iu_i})/{p^{m_i+1}}$, so that $w_i'$ has $U_i$-eigenvalue $(({\lambda _1\lambda _2\dotsb \lambda _i})/{p^{m_i+1}})w$, which has norm $>1$. Thus it suffices to check that $U_i$ has norm $\le 1$ on $Q_i'(G,\mathscr {U})$, which follows from the claim that any element of the form

\[ \frac{g\bigl(\prod_{j=1}^iu_i\bigr)g'}{p^{m_i+1}} \]

for $g,g'\in \operatorname {Iw}_p$ has norm $\le 1$ on $Q_i'$. This follows from Lemma 7.3.6 of [Reference Bellaïche and ChenevierBC09].

3.3 Automorphic representations associated to automorphic forms of locally algebraic weights

Fix an isomorphism $\iota _p:\overline {\mathbb {Q}}_p\xrightarrow {\sim }\mathbb {C}$. Using $\iota _p$, we may view algebraic representations of ${GL}_n$ over $\overline {\mathbb {Q}}_p$ as representations over $\mathbb {C}$, or vice versa. Let $f\in \mathscr {S}_{t\chi }(G,U_0(p))$ be a $p$-adic automorphic form coming from some classical subspace $S_{t\chi,c}(G,U_0(p))$. Let $W=\operatorname {Ind}_{B(\mathbb {Z}_p)}^{\operatorname {Iw}_p,c-\text {loc.alg.}}(t\chi )$, so that $f$ is a function $G(\mathbb {Q})\backslash G(\mathbb {A}_f)\to W$. Following the proof of Proposition 3.8.1 of [Reference LoefflerLoe10], let $W=W^{{\rm sm},c}(\chi )\otimes S_t(\mathbb {C})$, where, as in § 3.1,

\begin{align*} W^{{\rm sm},c}(\chi) &=\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\operatorname{Iw}_p\cap \Gamma(c)}\chi, \\ S_t(\mathbb{C}) &=\operatorname{Ind}_{B(\mathbb{Z}_p)}^{\operatorname{Iw}_p,{\rm alg}}t, \end{align*}

and let $\rho _{{\rm sm}},\rho _{{\rm alg}}$ denote the actions of $\operatorname {Iw}_p$ on $W^{{\rm sm},c}(\chi )\otimes S_t(\mathbb {C})$ given by acting on only the first factor and only the second factor, respectively. Then we can define a function $f_\infty :G(\mathbb {A})\to W$ by $f_\infty (g)=\rho _{{\rm alg}}(g_\infty ^{-1}\iota _p(g_p))f(g_f)$ which satisfies the relation

\[ f_\infty(gu)=\rho_{{\rm sm}}(u_p^{-1})\rho_{{\rm alg}}(u_\infty^{-1})f_\infty(g) \]

for all $u\in G(\mathbb {R})U_0(p)$. Equivalently, $f_\infty$ can be viewed as the function

\begin{align*} f_\infty^\vee:(W^{{\rm sm},c}(\chi)^\vee\otimes S_t(\mathbb{C})^\vee)\times G(\mathbb{Q})\backslash G(\mathbb{A}) &\to \mathbb{C} \\ (\varphi,x) &\mapsto \varphi(f_\infty(x)), \end{align*}

which satisfies

\[ f_\infty^\vee(\varphi,xu)=\varphi(f_\infty(xu))=\varphi(\rho_{{\rm sm}}(u_p^{-1})f_\infty(x))=f_\infty^\vee(u_p\varphi,x) \]

for all $u\in U_0(p)$. Thus, for each $\varphi \in W^{{\rm sm},c}(\chi )^\vee \otimes S_t(\mathbb {C})^\vee$, the function $f_\infty ^\vee (\varphi,\cdot )$ is an element of $C(G(\mathbb {Q})\backslash G(\mathbb {A}),\mathbb {C})$ which generates under right translation by $\operatorname {Iw}_p$ a representation containing an irreducible component of $W^{{\rm sm},c}(\chi )^\vee$. The right translates of $f_\infty ^\vee (\varphi,\cdot )$ under $G(\mathbb {A})$ generate an automorphic representation $\pi _f$ of $G(\mathbb {A})$ which decomposes as a tensor product $\bigotimes _p'\pi _{f,p}$. We are interested in describing the structure of $\pi _{f,p}$.

Note that this process is reversible: suppose given $\psi \in C(G(\mathbb {Q})\backslash G(\mathbb {A}),\mathbb {C})$ which generates a representation containing an irreducible component of $W^{{\rm sm},c}(\chi )^\vee$ under right translation by $\operatorname {Iw}_p$. Fixing $\varphi \in W^{{\rm sm},c}(\chi )^\vee \otimes S_t(\mathbb {C})^\vee$, we may recover the corresponding $f_\infty ^\vee$ such that $f_\infty ^\vee (\varphi,\cdot )=\psi$ by setting $f_\infty ^\vee (u_p\varphi,x)=\psi (xu_p)$ for all $u_p\in \operatorname {Iw}_p$. Then from $f_\infty ^\vee$ we get $f_\infty :G(\mathbb {A})\to W$, and finally we get $f\in S_{t\chi,c}(G,U_0(p))$ by setting $f(g_f)=\rho _{{\rm alg}}(\iota _p(g_p^{-1}))f_\infty (g_f)$ (where on the right-hand side $g_f$ is the element of $G(\mathbb {A})$ which equals $g_f$ at the places in $\mathbb {A}_f$ and $1$ at $\infty$). Write $f_\psi$ for the element of $S_{t\chi,c}(G,U_0(p))$ associated to $\psi$ in this way.

3.4 Structure of $W^{{\rm sm},c}(\chi )$

We are interested in the representation

\[ W^{{\rm sm},c}(\chi)=\operatorname{Ind}_{B(\mathbb{Z}_p)/B(\mathbb{Z}_p)\cap \Gamma(c)}^{\operatorname{Iw}_p/\Gamma(c)}\chi \]

of $\operatorname {Iw}_p$. Note that there is an obvious embedding $W^{{\rm sm},c}(\chi )\hookrightarrow W^{{\rm sm},c+1}(\chi )$ which takes $f\in W^{{\rm sm},c}(\chi )$ to the composition of $f$ with the reduction map $\operatorname {Iw}_p/\Gamma (c+1)\to \operatorname {Iw}_p/\Gamma (c)$.

Let $J$ be the compact open subgroup of ${GL}_n(\mathbb {Q}_p)$ corresponding to $\chi$ defined in § 3 of [Reference RocheRoc98]; we have $J=\Gamma (\underline {c})$ where

\[ c_{ij}=\begin{cases} 0 & \text{ if }i=j,\\ \left\lfloor\dfrac{\operatorname{cond}(\chi_i\chi_j^{-1})}2\right\rfloor & \text{ if } i< j,\\ \left\lfloor\dfrac{\operatorname{cond}(\chi_i\chi_j^{-1})+1}2\right\rfloor & \text{ if } i>j. \end{cases} \]

Then $\chi$ extends to a character of $J$ which we will also call $\chi$; it is defined by the equation $\chi (j^-jj^+)=\chi (j)$ when $j^-\in J\cap \overline {N}(\mathbb {Z}_p)$, $j\in T(\mathbb {Z}_p)$, and $j^+\in J\cap N(\mathbb {Z}_p)$. Let $U^{{\rm sm}}(\chi ):=\operatorname {Ind}_J^{\operatorname {Iw}_p}\chi$.

Now note that $W^{{\rm sm},c}(\chi )$ contains the vector

\[ f(\overline{x})=\begin{cases} \chi(j)\chi(b) & \text{ if }\overline{x}=\overline{j}\overline{b}\in\operatorname{Iw}_p/\Gamma(c)\text{ with }j\in J\text{ and }b\in B(\mathbb{Z}_p),\\ 0 & \text{otherwise.} \end{cases} \]

Note, furthermore, that for any $j\in J$ and $\overline {x}\in \operatorname {Iw}_p/\Gamma (c)$, we have

\[ (jf)(\overline{x})=f(j^{-1}\overline{x})=\chi(j^{-1})f(\overline{x})=\chi^{-1}(j)f(\overline{x}), \]

so that $f$ is $(J,\chi ^{-1})$-isotypic.

Proof. By Mackey's criterion, it is necessary and sufficient to show that for any $s\in \operatorname {Iw}_p\setminus J$, the characters $\chi$ and $\chi ^s:j\mapsto \chi (sjs^{-1})$ are not identically equal on $J\cap s^{-1}Js$. If $s\in \operatorname {Iw}_p\setminus J$, let $t=s^{-1}$. Then there is a pair $i\neq j$ such that $t_{ji}$ is not divisible by $p^{c_{ji}}$. Among all such $i\neq j$, choose a pair such that either:

1. (a) among the integers $c_{kj}+c_{jk}-v(t_{jk})$, $1\le k\le n$, $k\neq j$, $c_{ij}+c_{ji}-v(t_{ji})$ is the unique maximal one; or

2. (b) if this is not possible, among the integers $c_{kj}+c_{jk}-v(t_{jk})$, $1\le k\le n$, $k\neq j$, $c_{ij}+c_{ji}-v(t_{ji})$ is maximal and $i$ is minimal such that this is the case.

Let $x\in J$ be the identity except for the $ij$th entry; let $x_{ij}=b$. Note that we must have $p^{c_{ij}}|b$. We show that we can choose $b$ such that $sxt\in J$ and $1=\chi (x)\neq \chi (sxt)$ and, hence, $\chi (x)\neq \chi ^s(x)$, as desired.

The matrix $xt$ is the same as $t$ except for the $i$th row, which is

\[ (t_{i1}+bt_{j1},\dotsc,t_{in}+bt_{jn}). \]

The $kk$th entry of $sxt$ is

\[ s_{k1}t_{1k}+\dotsb+s_{ki}(t_{ik}+bt_{jk})+\dotsb+s_{kn}t_{nk}=s_{k1}t_{1k}+\dotsb+s_{kn}t_{nk}+bs_{ki}t_{jk}=1+bs_{ki}t_{jk}. \]

Because of condition (i), one can check that for all $j\in J$, we have $\chi (j)=\chi _1(j_{11})\cdot \dotsb \cdot \chi _n(j_{nn})$. Thus, we wish to choose $b$ such that

\[ \chi_1(1+bs_{1i}t_{j1})\dotsb\chi_i(1+bs_{ii}t_{ji})\dotsb\chi_j(1+bs_{ji}t_{jj})\dotsb\chi_n(1+bs_{ni}t_{jn})\neq1. \]

Note that for all $k\neq i,j$, we have

\[ v(s_{ki})+c_{ij}+c_{ji}-v(t_{ji})>c_{jk}+c_{kj}-v(t_{jk}). \]

This is just because we chose $i,j$ such that $c_{ij}+c_{ji}-v(t_{ji})\ge c_{jk}+c_{kj}-v(t_{jk})$, and such that if equality holds then $k>i$, in which case $v(s_{ki})\ge 1$ since $s\in \operatorname {Iw}_p$. Thus, if we choose $b$ such that $v(b)=c_{ij}+c_{ji}-v(t_{ji})-1\ge c_{ij}$, then we have

\[ v(bs_{ki}t_{jk})\ge c_{jk}+c_{kj}=\operatorname{cond}(\chi_k\chi_j^{-1}) \]

for all $k\neq i,j$, hence $\chi _k(1+bs_{ki}t_{jk})=1$. Then we have

\begin{align*} &\chi_1(1+bs_{1i}t_{j1})\dotsb\chi_i(1+bs_{ii}t_{ji})\dotsb\chi_j(1+bs_{ji}t_{jj})\dotsb\chi_n(1+bs_{ni}t_{jn})\\ &\qquad =\chi_i(1+bs_{ii}t_{ji})\chi_j(1+bs_{ji}t_{jj})=\chi_i(1+bs_{ii}t_{ji})\chi_j\biggl(1+b\biggl(\sum_{k\neq i}s_{ki}t_{jk}\biggr)\biggr) \end{align*}

since $v(bs_{ki}t_{jk})\ge \operatorname {cond}(\chi _j)$, but this is

\[ \chi_i(1+bs_{ii}t_{ji})\chi_j(1-bs_{ii}t_{ji}) \]

since $\sum _{k}s_{ki}t_{jk}=\sum _k t_{jk}s_{ki}=(ts)_{ji}=0$, and this can be rewritten as

\[ \frac{\chi_i}{\chi_j}(1+bs_{ii}t_{ji})\chi_j(1-b^2s_{ii}^2t_{ji}^2)=\frac{\chi_i}{\chi_j}(1+bs_{ii}t_{ji}) \]

because if $i>j$, then $v(b^2)\ge 2c_{ij}\ge c_{ij}+c_{ji}$, and if $i< j$, then $v(b^2)\ge 2c_{ij}\ge c_{ij}+c_{ji}-1$ and $v(t_{ji})\ge 1$. But since $v(b)< c_{ij}+c_{ji}-v(t_{ji})$, we have $v(bs_{ii}t_{ji})<\operatorname {cond}(\chi _i\chi _j^{-1})$, so we can choose $b$ to make $({\chi _i}/{\chi _j})(1+bs_{ii}t_{ji})\neq 1$.

Finally, we verify that for this choice of $b$, we actually have $sxt\in J$. The $kl$th entry of $sbt$ is

\[ s_{k1}t_{1l}+\dotsb+s_{ki}(t_{il}+bt_{jl})+\dotsb+s_{kn}t_{nk}=\delta_{kl}+bs_{ki}t_{jl}. \]

We have

\begin{align*} v(bs_{ki}t_{jl}) &=c_{ij}+c_{ji}-v(t_{ji})-1+v(s_{ki})+v(t_{jl}) \\ &=c_{ij}+c_{ji}-v(t_{ji})-(c_{lj}+c_{jl}-v(t_{jl}))+c_{lj}+c_{jl}-1+v(s_{ki}) \\ &\ge c_{lj}+c_{jl}-1+v(s_{ki})\ge c_{kl} \end{align*}

by condition (ii).