Globally Analytic p-adic Representations of the Pro–p Iwahori Subgroup of GL(2) and Base Change, II: A Steinberg Tensor Product Theorem

Abstract

In this paper, which is a sequel to Clozel (Globally analytic p-adic representations of the pro-p Iwahori subgroup of GL(2) and base change, I: Iwasawa algebras and a base change map, to appear in Bull. Iran Math Soc, [4]), we exploit the base change map for globally analytic distributions constructed there, relating distributions on the pro-p Iwahori subgroup of GL(2) over $\mathbb {Q}_p$ and those on the pro-p Iwahori subgroup of GL(2, L) where L is an unramified extension of $\mathbb {Q}_p$. This is used to obtain a functor, the ‘Steinberg tensor product’, relating globally analytic p-adic representations of these two groups. We are led to extend the theory, sketched by Emerton (Locally analytic vectors in representations of locally p-adic analytic groups, [6]), of these globally analytic representations. In the last section we show that this functor exhibits, for principal series, Langlands’ base change (at least for the restrictions of these representations to the pro-p Iwahori subgroups.)

Dedicated to Joachim schwermer on his 66th birthday

A Appendix: “Pathologies”

We review some properties of the distribution algebras on our groups, relative to tensor products or the Nœtherian property. They introduce some difficulties explained in the main text.

1.1 A.1 

Here X, Y are rigid–analytic spaces isomorphic to products of unit balls, over a p–adic field L. Then $\mathcal {A}(X\times Y)$ is naturally isomorphic to $\mathcal {A}(X)\widehat{\otimes }\mathcal {A}(Y)$. Moreover, each space of analytic functions is, as a Banach space, isomorphic to $C_0(M)$ where M is the set of exponents: $M=\mathbb {N}^d$, $d=\dim (X)$. Here $C_0(M)=C_0(M,L)$. We have, for two countable sets M, N.

$$ C_0(M\times N) \cong C_0(M) \widehat{\otimes } C_0(N) $$

(Schneider [9, p. 112]) which yields the requisite (well–known) isomorphism for the Tate algebras. We can assume that our sets M, N are equal to $\mathbb {N}$. We now have the following result, certainly well–known:

Proposition A.1

The natural map

$$ \ell ^\infty (\mathbb {N}) \widehat{\otimes } \ell ^\infty (\mathbb {N}) \longrightarrow \ell ^\infty (\mathbb {N}\times \mathbb {N}) $$

is injective, but is not an isomorphism.

Here $\ell ^\infty (\mathbb {N})=\ell ^\infty (\mathbb {N},L)$ is the Banach space of bounded sequences, the dual of $C_0(\mathbb {N})$. We will denote by $V'$ the dual of a Banach space V, with its strong topology (the topology as a Banach space.) We denote by $\mathcal {L}(V,W)$ the space of continuous linear maps $V\rightarrow W$, again with the topology of the norm. Now we have [2, 2.1.7.2]

$$ \mathcal {L}(V\widehat{\otimes }W,X) {\mathop {\longrightarrow }\limits ^{\approx }} \mathcal {L}(V,\mathcal {L}(W,X)) $$

(isometric isomorphism) for three Banach spaces, the map being the natural one, so

$$ (C_0\widehat{\otimes }C_0)' = \mathcal {L}(C_0 \widehat{\otimes } C_0,L) \cong \mathcal {L}(C_0,\ell ^\infty )\,. $$

Since $(C_0\widehat{\otimes } C_0)' = \ell ^\infty (\mathbb {N}\times \mathbb {N})$, it suffices to check:

Proposition A.2

The natural map

$$ \ell ^\infty \widehat{\otimes } \ell ^\infty \longrightarrow \mathcal {L}(C_0, \ell ^\infty ) $$

obtained by completion from

$$ \ell ^\infty {\otimes } \ell ^\infty \longrightarrow \mathcal {L}(C_0, \ell ^\infty ) $$

is injective, but is not an isomorphism.

Here $\mathcal {L}(C_0,\ell ^\infty )$ is provided with the strong ($=$ Banach) topology. Schneider [9, Proposition 18.2] shows that this map is an isomorphism onto its image, whence the injectivity in Proposition A.1. He also shows that its image is the space

$$ \mathcal {C}\mathcal {C}(C_0,\ell ^\infty ) $$

of completely continuous operators. Thus it suffices to show

$$\begin{aligned} \mathcal {C}\mathcal {C}(C_0,\ell ^\infty ) \not = \mathcal {L}(C_0,\ell ^\infty )\,. \end{aligned}$$

(A.1)

Now $F:C_0\rightarrow \ell ^\infty$ is in $\mathcal {C}\mathcal {C}$ if, and only if, F(B) is compactoid for any bounded set $B\subset C_0(X)$. We can simply consider the unit ball.

Recall also that $\Omega \subset V$ (a Banach space) is compactoid [9, p. 71] if $\forall r>0$ $\exists (v_i)_{i=1,\ldots N}$, $v_i \in V$, such that

$$ \Omega \subset B_V(r) + \sum _1^N \mathcal {O}_L v_i\,. $$

We consider the identity map $F:C_0 \rightarrow \ell ^\infty$ and show that it is not completely continuous. If it were, we would have for $f\in C_0$:

$$ \Vert f\Vert _\infty \le 1 \Longrightarrow \forall r\quad \Vert f - \sum _1^{N(r)} z_if_i\Vert _\infty \le r $$

for some functions $f_i\in \ell ^\infty$, and integers $z_i$, depending on r (but not on f).

For simplicity assume $L=\mathbb {Q}_p$. Fix $r=p^{-n}$, $n\ge 0$. The $f_i$ take values in $p^{-M}\mathbb {Z}_p$, $M\ge 0$. The function f defines $\bar{f}:\mathbb {N}\rightarrow \mathbb {Z}_p/p^n\mathbb {Z}_p$, and

$$\begin{aligned} \bar{f}(x) = \sum _1^{N(r)} \bar{z_i} \bar{f_i}(x)\,, \end{aligned}$$

(A.2)

a linear combination of functions $\mathbb {N}\rightarrow p^{-M}\mathbb {Z}_p/p^n\mathbb {Z}_p$. Since f is arbitrary in $B(1)\subset C_0$, $\bar{f}$ can be an arbitrary function with finite support $\mathbb {N}\rightarrow \mathbb {Z}_p/p^n\mathbb {Z}_p$. However, when $(z_i)$ varies in $\mathbb {Z}_p^{N(r)}$, the set of functions on the right–hand side of (A2) with values in $p^{-M}\mathbb {Z}_p/p^n\mathbb {Z}_p$ is finite, and this is a contradiction.

1.2 A.2 $\mathcal {D}(G)$ (As a Convolution Algebra) is not Nœtherian for Left- or Right- Ideals

Here the rigid–analytic group is assumed to verify the assumptions in Sect. 1. We start with B(1) (rigid–analytic ball) seen as an additive group. Then

$$ \begin{array}{rl} \mathcal {A}(G) &{} =\Big \{\displaystyle \sum _0^\infty a_n x^n,\ a_n \rightarrow 0\Big \}\,, \\ &{} a_n = \dfrac{1}{n!}\Big (\dfrac{d^n}{dx^n}f\Big )(0)\,. \end{array} $$

The algebra $\mathcal {D}(G)$ is isomorphic to $\ell ^\infty$ by taking the basis dual to the $x^n$; we can write a distribution $T\in \mathcal {D}(G)$ as

$$ T = \sum _0^\infty c_n \frac{1}{n!}\Big (\frac{d^n}{dx^n}\Big )_0\,,\quad (c_n)\in \ell ^\infty $$

and then $\mathcal {D}$ is naturally isomorphic to an algebra of divided power series:

$$ T = \sum _0^\infty \frac{c_n}{n!}t^n\,, \quad t=\Big (\frac{d}{dx}\Big )_0\,,\quad (c_n)\in \ell ^\infty \,. $$

As pointed out by Berthelot, this algebra is not Nœtherian.² Since it is a Banach algebra for convolution (which becomes here the product of the series), it suffices to check that there is an ideal which is not closed [2, 3.7.2.2]. In fact:

Lemma A.3

(Berthelot) The ideal $(t)\in \mathcal {D}$ is not closed.

Indeed if $T\in (t)$,

$$ \begin{array}{rl} T=tS &{} =t\Big (\displaystyle \sum _0^\infty \dfrac{c_n}{n!}t^n\Big ) \\ &{} =\displaystyle \sum _1^\infty t^n\dfrac{d_n}{n!} \end{array} $$

with $d_n=nc_{n-1}$, so

$$ T\in (t) \Leftrightarrow d_0=0 \ \mathrm {and}\ \Big |\frac{d_n}{n}\Big | \le C\,. $$

But this subspace of $\mathcal {D}$ is clearly not closed. For instance, if $T=(d_n)$ has support on $\{n=p^{r}\}$ $(r\ge 0$ even) with

$$ \begin{array}{rl} d_{p^r} &{} = p^{r/2}\ (\mathrm {so}\ T\in \ell ^\infty ) \\ \Big |\dfrac{d_{p^r}}{p^r} \Big |&{} =|p^{-r/2}| \longrightarrow \infty \ (\mathrm {so}\ T\notin (t))\,, \end{array} $$

T is the limit in $\mathcal {D}$ of the truncated series $T^\alpha$ with $d_{p^r}^\alpha =p^{r/2}$ $(r\le \alpha )$, $d_{p^r}^\alpha =0$ $(r>\alpha )$ which obviously belong to the ideal.

Consider now a rigid–analytic group G, isomorphic to a product $B(1)^d$ over $\mathbb {Q}_p$ as a rigid–analytic space, the coproduct being then given by Tate series with integral coefficients. We further assume (as is the case in this paper) that the factors are (additive) analytic subgroups, and their distribution algebras are therefore as in the previous proof.

Proposition A.4

Under these assumptions $\mathcal {D}(G)$ is not Nœtherian (for left- or right- ideals).

Indeed we have, as in the commutative case:

Lemma A.5

(Schneider–Teitelbaum) If $\mathcal {D}(G)$ is (left) Nœtherian, any (left) ideal is closed.

See Proposition 2.1 in [12]. For completeness we provide a proof. In the commutative case this is [2, 3.7.2.2]. A glance at their proof shows that it suffices to prove Nakayama’s lemma for the ideal $A^\vee = \{a\in A:\Vert a\Vert <1\}$ in $A^0$, where we have written $A=\mathcal {D}(G)$. We may assume that the (Banach) norm on A is submultiplicative [2, 1.2.1.2]. Since the argument in [2, 1.2.3.6] for Nakayama’s lemma uses determinants, we rephrase it (using moreover $A^\vee$ rather than the set $\check{A}$ of topologically nilpotent elements):

Lemma A.6

Let M be an A–module, and N a submodule of M such that there exist $x_1,\ldots x_n\in M$ with $M\subset N+ \sum \limits _1^n A^\vee x_i$. Then $N=M$.

As in [2], loc. cit., we can write

$$ \underline{x} = \underline{y} + C \underline{x} $$

where $\underline{x}$ is a column vector in $M^n$ with coordinates $x_i$, and $\underline{y}$ has coordinates in N, and the matrix $C\in M_n(A)$ has entries in $A^\vee$. Thus $\underline{y} = (1-C)\underline{x}$. The matrix $1-C$ is invertible: if $M_n(A)$ is endowed with the operator norm, this norm is submultiplicative, and $\Vert C\Vert <1$. This implies that $\underline{x} = (1-C)^{-1}\underline{y} \in N^n$. (It is not clear to us that the argument applies for $\check{A}$.)

We now return to the proof of Proposition A.4. Assume that (as a rigid analytic space)

$$ G=G_1\times \cdots \times G_d $$

where each $G_i$ is a rigid–analytic group isomorphic to the additive unit ball over $\mathbb {Q}_p$.

In particular we have a bijection

$$ \begin{array}{rcl} \mathbb {Z}_p^d = G_1(\mathbb {Q}_p)\times \cdots \times G_d(\mathbb {Q}_p) &{} \longrightarrow &{} G(\mathbb {Q}_p) \\ (g_1,g_2,\ldots , g_d) &{} \longmapsto &{} g_1\cdots g_d\,. \end{array} $$

The Tate algebra $\mathcal {A}_G$ is isomorphic with $\widehat{\bigotimes \limits _{i=1,\ldots d}} \mathcal {A}_{G_i}$ where each $\mathcal {A}_{G_i}$ is the Tate algebra in one variable. Evaluated on the points of $G(\mathbb {Q}_p)$, this yields the map $f\mapsto f(g_1,\ldots , g_d)$ $(f\in \mathcal {A}_G)$.

Each injection $j_i:G_i\longrightarrow G$ is an homomorphism, and the restriction $\mathcal {A}_G\longrightarrow \mathcal {A}_{G_i}$ is therefore compatible with the coproduct. Dually, we get

$$ (j_i)_* : \mathcal {D}_{G_i} \longrightarrow \mathcal {D}_G\,, $$

compatible with convolution. If we denote by $x_i$ the local variable on $G_i$, a function $f\in \mathcal {A}_G$ being then in the Tate algebra in the $x_i$, an element of $\mathcal {D}_G$ can be written

$$\begin{aligned} T=\sum _n c_n \frac{1}{n!}\partial _1^{n_1}\cdots \partial _d^{n_d} := \sum _n c_n \delta _n \end{aligned}$$

(A.3)

with $n=(n_1,\ldots n_d)\in \mathbb {N}^d$, $n!=\Pi (n_i)!$,

$$ \partial _if(x_1,\ldots , x_d) =\frac{d}{dx_i}f(x_1,\ldots , x_d)(0),\ \mathrm {and}\ (c_n)\in \ell ^\infty (\mathbb {N}^d).$$

Let us write $\mathcal {D}_1$ for the subalgebra $(j_1)_*(\mathcal {D}_{G_1})$, given by $c_n=0$ if $n_i>0$ for some $i\ge 2$. This is clearly a closed subalgebra. The element $\partial _1=\delta _1$ - abuse of notation for $\delta _{(1,0, \ldots , 0)}$ - is equal to $(j_1)_*((\frac{d}{dx})_0)$. Moreover, for the convolution product in G, we have

$$ \delta _1*\delta _n =\frac{1}{n!}\frac{d}{dx_1}\Big (\frac{d}{dx_1}\Big )^{n_1}\cdots \Big (\frac{d}{dx_d}\Big )^{n_d} $$

(evaluated at $(0,\ldots , 0))=(n_1+1)\frac{1}{(n_1+1)!n_2!\cdots n_d}(\frac{d}{dx_1})^{n_1+1}\cdots (\frac{d}{dx_d})^{n_d}=(n_1+1)\delta _{n'}$ where $n'=(n_1+1,\ldots , n_d)$.

We will show that the left ideal $\delta _1\mathcal {D}(G)$ is not closed in $\mathcal {D}(G)$. Indeed, if it were, its intersection with the closed subalgebra $\mathcal {D}_1$ would be so. But the subalgebra is isomorphic (as an algebra) with the algebra

$$ \mathcal {D}_{G_1} = \Big \{\sum _{m\ge 0}c_m \delta _m\Big \} = \Big \{\sum _{m\ge 0} c_m \frac{1}{m!}\Big (\frac{d}{dx}\Big )^m\Big \} $$

in one variable, and the previous computation shows that the intersection is the ideal $(\frac{d}{dx})\subset \mathcal {D}_{G_1}$ considered in the first part of the proof. Therefore $\delta _1\mathcal {D}(G)$ is not closed; this completes the proof of Lemma A.6.

1.3 A.3 .

Finally, we note that there is a possible substitute for the consideration of $\mathcal {D}(G)$, which could obviate the problems we encountered. (This was pointed out by Schneider.)³

This algebra was already introduced by Lazard [7, III.3.3.3] who calls in Ala(G). To use Lazard’s results, we keep the assumptions of Sect. A.2 and assume moreover that the factors $G_i(\mathbb {Q}_p)\cong \mathbb {Z}_p$ form a Lazard basis (a “base ordonnée” in the sense of [7, III.2.2.4]. The Iwasawa algebra of G (with integral coefficients) is then given by the series

$$ \sum _n a_n z_1^{n_1} \cdots z_d^{n_d} = \sum a_n z^n $$

with $a_n \in \mathbb {Z}_p$ and $z_i = \delta _{1}-\delta _0$ in the Iwasawa algebra of $G_i(\mathbb {Q}_p) = \mathbb {Z}_p$. Lazard defines the algebra Sat Al(G), where $Al(G)= \Lambda _G$ is the Iwasawa algebra, and $Sat\ Al(G)$ is given by $val(a_n)\ge -|n|$. (Thus $Sat$ $Al(G)$ is contained in the completion of $\Lambda _G \otimes \mathbb {Q}_p$).

Recall that $\Lambda _G \subset \mathcal {D}_G$.

Lazard defines $Ala(G) \subset Sat\ Al(G)$ as the compact $\mathbb {Z}_p$–module generated by the $\dfrac{z^n}{n!}$, with the usual notation. If we recall that the basis elements $z^n$ are dual to the Mahler basis $\begin{pmatrix} x \\ n \end{pmatrix}$ of $\mathcal {C}(G)$, we see that this is the unit ball in the Banach dual of the functions $\sum \limits _n c_nn!\begin{pmatrix} x \\ n \end{pmatrix}$ $(c_n\rightarrow 0)$ in $\mathcal {C}(G)$, with the weak topology. By Amice’s theorem [5, I.4] this space of functions is $\mathcal {A}(G)$. Thus $Ala(G)\subset \mathcal {D}_G$, and Ala(G) is the unit ball in $\mathcal {D}_G$, with the weak topology. Note that both the $n!\begin{pmatrix} x \\ n \end{pmatrix}$ and the $x^n$ $(x=(x_1,\ldots x_d))$ are orthonormal bases of $\mathcal {A}_G$ : see Colmez [5, I.4.3]. Thus this coincides with the description of $\mathcal {D}_G$ as an $\ell ^\infty$ space on the set of powers n.

Now if we consider these integral spaces (isomorphic to $\mathbb {Z}_p^N$, N the set of indices, with the product topology), we obtain indeed, for a product $G\times H$, an isomorphism $Ala(G\times H)\cong Ala(G) \widehat{\bigotimes \limits _{\mathbb {Z}_p}}Ala(H)$. This is stated (but not proven) by Lazard [7, III.3.]. The completed tensor product (over $\mathbb {Z}_p$) is defined in [7, I.3.2.6, I.3.2.9].

Recall that we have assumed that the coproduct for G (and H, a group of the same type) was given by integral Tate expansions. Then $\mathcal {D}_G(\mathbb {Z}_p)$, the distributions with integral coefficients, is the unit ball in $\mathcal {D}_G$, and its weak topology is as we saw, the product topology. The tensor product $\mathcal {D}_G(\mathbb {Z}_p)\times \mathcal {D}_H(\mathbb {Z}_p) \rightarrow \mathcal {D}_{G\times H}(\mathbb {Z}_p)$ is simply given, M being the set of exponents for H, by

$$ (z_n,w_m) \longmapsto (z_n w_m)_{n,m} $$

where $N\times M$ is the set of exponents for $G\times H$. It is easy to see that it yields, as asserted by Lazard, an isomorphism $\mathbb {Z}_p^N\widehat{\otimes }\mathbb {Z}_p^M \rightarrow \mathbb {Z}_p^{N\times M}$. In particular, under our standing assumptions on the groups (in particular, the integrality conditions), we have:

Proposition A.7

Let $\mathcal {D}_G^0$, $\mathcal {D}_H^0$ denote the unit balls of $\mathcal {D}_G$, $\mathcal {D}_H$ with their weak topology. Then

$$ \mathcal {D}_{G\times H}^0 \cong \mathcal {D}_G^0 \widehat{\bigotimes _{\mathbb {Z}_p}} \mathcal {D}_H^0\,. $$

Now assume V, W are globally analytic Banach representations of G, H. The algebra $\mathcal {D}_G$ acts on $V'$, by

$$<Tv',v> = \int <v',gv> dT(g) $$

where we have used the integral notation for T.

Now write $V_0'$ for the unit ball in $V'$, and $V_0$ for the unit ball in V. We may assume, as in Sect. 2.1, that the action of G on V preserves the norm. Then if $v\in V_0$ and $v'\in V_0'$ $<v',gv>$ is a function f on G, given by a Tate series, and such that $\overset{}{\underset{g\in G}{\mathrm {Sup}}} |f(g)|\le 1$.

If moreover V is admissible, the Tate norm $\Vert f\Vert$ is $\le 1$ by Corollary 2.8. Thus $Tv'$ belongs to $V_0'$ for $T \in \mathcal {D}_G^0$. We obtain an action of $\mathcal {D}_G^0$ on the unit ball of $V'$, compatible with the weak topologies.

If now we consider another group H, an admissible globally analytic representation W of H, and the product $G\times H$, we see that $\mathcal {D}_G^0 \widehat{\otimes } \mathcal {D}_H^0$ acts on $V_0'\widehat{\otimes }W_0'$. Replacing the distribution algebras by their unit balls, we are very close to the original construction of Schneider–Teitelbaum [10] for Banach representations.

Since $\mathcal {D}_G$ is not Nœtherian, however, it easily follows that $\mathcal {D}_G^0$ is not Nœtherian.