Banach-Hecke Algebras

and p-Adic Galois Representations

P. Schneider, J. Teitelbaum

Received: October 31, 2005 Revised: December 28, 2005

Wir lassen vom Geheimnis uns erheben Der magischen Formelschrift, in deren Bann Das Uferlose, Stürmende, das Leben Zu klaren Gleichnissen gerann.

Hermann Hesse

Dedicated to John Coates

Abstract. In this paper, we take some initial steps towards illuminating the (hypothetical) p-adic local Langlands functoriality principle relating Galois representations of a p-adic field L and admissible unitary Banach space representations of G(L) when G is a split reductive group over L.

2000 Mathematics Subject Classification: 11F80, 11S37, 22E50 Keywords and Phrases: Satake isomorphism, Iwahori-Hecke algebra, Banach-Hecke algebra, filtered isocrystal, crystalline Galois representation, p-adic local Langlands correspondence

In this paper, we take some initial steps towards illuminating the (hypothetical) p-adic local Langlands functoriality principle relating Galois representations of a p-adic field L and admissible unitary Banach space representations of G(L) when G is a split reductive group over L. The outline of our work is derived from Breuil’s remarkable insights into the nature of the correspondence between 2-dimensional crystalline Galois representations of the Galois group of Q_{p} and Banach space representations of GL_{2}(Q_{p}).

In the first part of the paper, we study the p-adic completion B(G, ρ) of the Hecke algebra H(G, ρ) of bi-equivariant compactly supported End(ρ)-valued functions on a totally disconnected, locally compact group G derived from a finite dimensional continuous representation ρ of a compact open subgroup U of G. (These are the “Banach-Hecke algebras” of the title). After describing some general features of such algebras we study in particular the case where G is split reductive and U = U_{0} is a special maximal compact or U = U_{1} is an Iwahori subgroup of G and ρ is the restriction of a finite dimensional algebraic representation of G to U_{0} or U_{1}.

In the smooth theory for trivial ρ = 1_{U}, by work of Bernstein, the maximal commutative subalgebra of the Iwahori-Hecke algebra is isomorphic to the group ring K[Λ] where Λ is the cocharacter group of a maximal split torus T of G, and the spherical Hecke algebra is isomorphic by the Satake isomorphism to the ring K[Λ]^{W} of Weyl group invariants. At the same time the algebra K[Λ] may be viewed as the ring of algebraic functions on the dual maximal torus T′ in the dual group G′. Together, these isomorphisms allow the identification of characters of the spherical Hecke algebra with semisimple conjugacy classes in G′. On the one hand, the Hecke character corresponds to a certain parabolically induced smooth representation; on the other, the conjugacy class in G′ determines the Frobenius in an unramified Weil group representation of the field L. This is the unramified local Langlands correspondence (the Satake parametrization) in the classical case.

With these principles in mind, we show that the completed maximal commutative subalgebra of the Iwahori-Hecke algebra for ρ is isomorphic to the affinoid algebra of a certain explicitly given rational subdomain T′_{ρ} in the dual torus T′. The spectrum of this algebra therefore corresponds to certain points of T′. We also show that the quotient of this subdomain by the Weyl group action is isomorphic to the corresponding completion of the spherical Hecke algebra; this algebra, for most groups G, turns out to be a Tate algebra. These results may be viewed as giving a p-adic completion of the Satake isomorphism, though our situation is somewhat complicated by our reluctance to introduce a square root of q as is done routinely in the classical case. These computations take up the first four sections of the paper.

In the second part of the paper, we let G = GL_{d+1}(L). We relate the subdomain of T′ determined by the completion B(G, ρ) to isomorphism classes of a certain kind of crystalline Dieudonne module. This relationship follows Breuil’s theory, which puts a 2-dimensional irreducible crystalline representation V of Gal(\overline{Q_{p}}/Q_{p}) with coefficients in a field K into correspondence with a topologically irreducible admissible unitary representation of GL_{2}(Q_{p}) in a K-Banach space. Furthermore, this Banach space representation is a completion of a locally algebraic representation whose smooth factor comes from D_{cris}(V) viewed as a Weil group representation and whose algebraic part is determined by the Hodge-Tate weights of V.

To state our relationship, let V be a d+1-dimensional crystalline representation of Gal(\overline{L}/L) in a K-vector space, where L and K are finite extensions of Q_{p}. In this situation, D_{cris}(V) has a K-vector space structure. Suppose further that: i. L is embeddable into K, and fix once and for all such an embedding L ⊆ K;

ii. the eigenvalues of the Frobenius on D_{cris}(V) lie in K;

iii. the (negatives of) the Hodge-Tate weights of D_{cris}(V) are multiplicity free and are separated from one another by at least [L : Q_{p}];

iv. V is special, meaning that the kernel of the natural map

C_{p} ⊗_{Q_{p}} V → C_{p} ⊗_{L} V

is generated by its Gal(\overline{L}/L) invariants.

It follows from the Colmez-Fontaine theory that the category of such special representations is equivalent to a category of “K-isocrystals”, which are Kvector spaces with a K-linear Frobenius and a filtration that is admissible in a sense very close to the usual meaning.

Given such a representation, we extract from the associated K-isocrystal its Frobenius, which we view as an element of the dual group G′(K) determined up to conjugacy. The semi-simple part ζ of this element determines a point of T′(K) up to the Weyl group action. From the Hodge-Tate weights, we extract a dominant cocharacter of G′ and hence a highest weight ξ determining an algebraic representation ρ = ρ_{ξ} for G. (In fact, the highest weight is a modification of the Hodge-Tate weights, but we avoid this complication in this introduction). Put together, this data yields a completion of the Iwahori-Hecke algebra, determined by the highest weight, and a character of its maximal commutative subalgebra, determined up to the Weyl group action. In other words, we obtain a simple module K_{ζ} for the completed spherical Hecke algebra B(G, ρ_{ξ}|U_{0}).

Our main result is that the existence of an admissible filtration on D_{cris}(V) translates into the condition that the point of T′ determined by the Frobenius lives in the subdomain T′_{ρ}. Conversely, we show how to reverse this procedure and, from a point of T′_{ρ}(K) (up to Weyl action), make an isocrystal that admits an admissible filtration of Hodge-Tate type determined by ρ. See Section 5 (esp. Proposition 5.2) for the details.

It is crucial to realize that the correspondence between points of T′_{ρ} and isocrystals outlined above does not determine a specific filtration on the isocrystal. Except when d = 1 there are infinitely many choices of filtration compatible with the given data. Consequently the “correspondence” we describe is a very coarse version of a p-adic local Langlands correspondence.

To better understand this coarseness on the “representation-theoretic” side, recall that to a Galois representation V of the type described above we associate a simple module K_{ζ} for the completion B(G, ρ|U_{0}) of the spherical Hecke algebra. There is an easily described sup-norm on the smooth compactly induced representation ind_{U_{0}}^{G}(ρ|U_{0}); let B_{U_{0}}^{G}(ρ|U_{0}) be the completion of this representation. We show that the completed Hecke algebra acts continuously on this space. By analogy with the Borel-Matsumoto theory constructing parabolically induced representations from compactly induced ones, and following also Breuil’s approach for GL_{2}(Q_{p}), it is natural to consider the completed tensor product

B_{ξ,ζ} := K_{ζ} ˆ⊗_{B(G,ρ_{ξ}|U_{0})} B_{U_{0}}^{G}(ρ_{ξ}|U_{0}) .

A deep theorem of Breuil-Berger ([BB]) says that, in the GL_{2}(Q_{p})-case, this representation in most cases is nonzero, admissible, and irreducible, and under Breuil’s correspondence it is the Banach representation associated to V. In our more general situation, we do not know even that B_{ξ,ζ} is nonzero. Accepting, for the moment, that it is nonzero, we do not expect it to be admissible or irreducible, because it is associated to the entire infinite family of representations having the same Frobenius and Hodge-Tate weights as V but different admissible filtrations. We propose that B_{ξ,ζ} maps, with dense image, to each of the Banach spaces coming from this family of Galois representations. We discuss this further in Section 5.

In the last section of this paper (Section 6) we consider the shape of a p-adic local Langlands functoriality for a general L-split reductive group G over L, with Langlands dual group G′ also defined over L. Here we rely on ideas from the work of Kottwitz, Rapoport-Zink, and Fontaine-Rapoport. Recall that a cocharacter ν of the dual group G′ defined over K allows one to put a filtration F il_{ρ′∘ν}^{·} E on every K-rational representation space (ρ′, E) of G′. Using (a modified version of) a notion of Rapoport-Zink, we say that a pair (ν, b) consisting of an element b of G′(K) and a K-rational cocharacter ν of G′ is an “admissible pair” if, for any K-rational representation (ρ′, E) of G′, the K-isocrystal (E, ρ′(b), F il_{ρ′∘ν}^{·} E) is admissible. Such an admissible pair defines a faithful tensor functor from the neutral Tannakian category of K-rational representations of G′ to that of admissible filtered K-isocrystals. Composing this with the Fontaine functor one obtains a tensor functor to the category of “special” Gal(\overline{L}/L) representations of the type described earlier. The Tannakian formalism therefore constructs from an admissible pair an isomorphism class of representations of the Galois group of L in G′(\overline{K}).

Now suppose given an irreducible algebraic representation ρ of G. Its highest weight may be viewed as a cocharacter of G′. Under a certain technical condition, we prove in this section that there is an admissible pair (ν, b) where ν is conjugate by G′(K) to a (modification of) the highest weight, and b is an element of G′(K), if and only if the semisimple part of b is conjugate to an element of the affinoid domain T′_{ρ}(K) (See Proposition 6.1). Thus in some sense this domain is functorial in the group G′.

Our work in this section relies on a technical hypothesis on G. Suppose that η is half the sum of the positive roots of G. We need [L : Q_{p}]η to be an integral weight of G. This happens, for example, if L has even degree over Q_{p}, and in general for many groups, but not, for example, when G = P GL_{2}(Q_{p}). This complication has its origin in the normalization of the Langlands correspondence. Because of the square root of q issue the p-adic case seems to force the use of the “Hecke” or the “Tate” correspondence rather than the traditional unitary correspondence; but even for smooth representations this is not functorial (cf. [Del] (3.2.4-6)). It turns out that without the above integrality hypothesis one even has to introduce a square root of a specific continuous Galois character (for L = Q_{p} it is the cyclotomic character). This leads to isocrystals with a filtration indexed by half-integers. Although it seems possible to relate these to Galois representations this has not been done yet in the literature. We hope to come back to this in the future.

The authors thank Matthew Emerton for pointing out that the conditions which define our affinoid domains T′_{ρ} are compatible with the structure of his Jacquet functor on locally algebraic representations ([Em1] Prop. 3.4.9 and Lemma 4.4.2, [Em2] Lemma 1.6). We thank Laurent Berger, Christophe Breuil, and especially Jean-Marc Fontaine for their very helpful conversations about these results. We also want to stress that our computations in Section 4 rely in an essential way on the results of Marie-France Vigneras in [Vig]. The first author gratefully acknowledges support from UIC and CMI. During the final stages of this paper he was employed by the Clay Mathematics Institute as a Research Scholar. The second author was supported by National Science Foundation Grant DMS-0245410.

We dedicate this paper to John Coates on the occasion of his sixtieth birthday. His constant support and unrelenting enthusiasm was and is an essential source of energy and inspiration for us over all these years.

Throughout this paper K is a fixed complete extension field of Q_{p} with absolute value | |.

Added in proof: In a forthcoming joint paper by C. Breuil and P. Schneider the technical restrictions of the present paper – that L ⊆ K, that the crystalline Galois representations V have to be special, and that [L : Q_{p}]η has to be an integral weight for the split group G – will be removed. In fact, this forces a renormalization of the picture in the present paper.

1. Banach-Hecke algebras

In this section G denotes a totally disconnected and locally compact group, and U ⊆ G is a fixed compact open subgroup. We let (ρ, E) be a continuous representation of U on a finite dimensional K-vector space E, and we fix a U-invariant norm ∥ ∥ on E.

The Hecke algebra H(G, ρ) is the K-vector space of all compactly supported functions ψ : G ⟶ End_{K}(E) satisfying

ψ(u_{1}gu_{2}) = ρ(u_{1}) ∘ ψ(g) ∘ ρ(u_{2}) for any u_{1}, u_{2} ∈ U and g ∈ G .

It is a unital associative K-algebra via the convolution

ψ_{1} ∗ ψ_{2}(h) := ∑_{g∈G/U} ψ_{1}(g) ∘ ψ_{2}(g^{−1}h) .

Its unit element is the function

ψ_{e}(h) := \{ \begin{matrix}ρ(h) & if h ∈ U, \\ 0 & otherwise. \end{matrix}

We note that any function ψ in H(G, ρ) necessarily is continuous. We now introduce the norm

∥ψ∥ := sup_{g∈G} ∥ψ(g)∥

on H(G, ρ) where on the right hand side ∥ ∥ refers to the operator norm on End_{K}(E) with respect to the original norm ∥ ∥ on E. This norm on H(G, ρ) evidently is submultiplicative. By completion we therefore obtain a unital K-Banach algebra B(G, ρ), called in the following the Banach-Hecke algebra, with submultiplicative norm. As a Banach space B(G, ρ) is the space of all continuous functions ψ : G ⟶ End_{K}(E) vanishing at infinity and satisfying

ψ(u_{1}gu_{2}) = ρ(u_{1}) ∘ ψ(g) ∘ ρ(u_{2}) for any u_{1}, u_{2} ∈ U and g ∈ G .

In the special case where ρ = 1_{U} is the trivial representation H(G, 1_{U}), resp. B(G, 1_{U}), is the vector space of all K-valued finitely supported functions, resp. functions vanishing at infinity, on the double coset space U\G/U.

A more intrinsic interpretation of these algebras can be given by introducing the compactly induced G-representation ind_{U}^{G}(ρ). This is the K-vector space of all compactly supported functions f : G ⟶ E satisfying

f(gu) = ρ(u^{−1})(f(g)) for any u ∈ U and g ∈ G

with G acting by left translations. Again we note that any function f in ind_{U}^{G}(ρ) is continuous. We equip ind_{U}^{G}(ρ) with the G-invariant norm

∥f∥ := sup_{g∈G} ∥f(g)∥

and let B_{U}^{G}(ρ) denote the corresponding completion. The G-action extends isometrically to the K-Banach space B_{U}^{G}(ρ), which consists of all continuous functions f : G ⟶ E vanishing at infinity and satisfying

f(gu) = ρ(u^{−1})(f(g)) for any u ∈ U and g ∈ G .

Lemma 1.1: The G-action on B_{U}^{G}(ρ) is continuous.

Proof: Since G acts isometrically it remains to show that the orbit maps

\begin{matrix}c_{f} : G & ⟶ & B_{U}^{G}(ρ) \\ g & ⟼ & gf , \end{matrix}

for any f ∈ B_{U}^{G}(ρ), are continuous. In case f ∈ ind_{U}^{G}(ρ) the map c_{f} even is locally constant. In general we write f = lim_{i→∞} as the limit of a sequence (f_{i})_{i∈IN}

in ind_{U}^{G}(ρ). Because of

∥(c_{f} − c_{f_{i}})(g)∥ = ∥g(f − f_{i})∥ = ∥f − f_{i}∥

the map c_{f} is the uniform limit of the locally constant maps c_{f_{i}} and hence is continuous.

One easily checks that the pairing

(1) \begin{matrix}H(G, ρ) × ind_{U}^{G}(ρ) & ⟶ & ind_{U}^{G}(ρ) \\ (ψ, f) & ⟼ & (ψ ∗ f)(g) := ∑_{h∈G/U} ψ(g^{−1}h)(f(h)) \end{matrix}

makes ind_{U}^{G}(ρ) into a unital left H(G, ρ)-module and that this module structure commutes with the G-action.

Lemma 1.2: The map

\begin{matrix}H(G, ρ) & ⟶^{≅} & End_{G}(ind_{U}^{G}(ρ)) \\ ψ & ⟼ & A_{ψ}(f) := ψ ∗ f \end{matrix}

is an isomorphism of K-algebras.

Proof: For a smooth representation ρ this can be found in [Kut]. Our more general case follows by the same argument. But since we will need the notations anyway we recall the proof. The map in question certainly is a homomorphism of K-algebras. We now introduce, for any w ∈ E, the function

f_{w}(g) := \{ \begin{matrix}ρ(g^{−1})(w) & if g ∈ U, \\ 0 & otherwise \end{matrix}

in ind_{U}^{G}(ρ). We have

(2)  A_{ψ}(f_{w})(g) = (ψ ∗ f_{w})(g) = ψ(g^{−1})(w) for any ψ ∈ H(G, ρ) .

This shows that the map in question is injective. To see its surjectivity we fix an operator A_{0} ∈ End_{G}(ind_{U}^{G}(ρ)) and consider the function

\begin{matrix}ψ_{0} : G & ⟶ & End_{K}(E) \\ g & ⟼ & [w ↦ A_{0}(f_{w})(g^{−1})] . \end{matrix}

It clearly has compact support. Furthermore, for u_{1}, u_{2} ∈ U, we compute

ψ_{0}(u_{1}gu_{2})(w) = A_{0}(f_{w})(u_{2}^{−1}g^{−1}u_{1}^{−1}) = ρ(u_{1})[A_{0}(f_{w})(u_{2}^{−1}g^{−1})]
= ρ(u_{1})[(u_{2}(A_{0}(f_{w})))(g^{−1})] = ρ(u_{1})[A_{0}(u_{2}(f_{w}))(g^{−1})]
= ρ(u_{1})[A_{0}(f_{ρ(u_{2})(w)})(g^{−1})] = ρ(u_{1})[ψ_{0}(ρ(u_{2})(w))]
= [ρ(u_{1}) ∘ ψ_{0} ∘ ρ(u_{2})](w) .

Hence ψ_{0} ∈ H(G, ρ). Moreover, for any f ∈ ind_{U}^{G}(ρ) we have

f = ∑_{h∈G/U} h(f_{f(h)})

and therefore

A_{ψ_{0}}(f)(g) = (ψ_{0} ∗ f)(g) = ∑_{h∈G/U} ψ_{0}(g^{−1}h)(f(h))

= ∑_{h∈G/U} A_{0}(f_{f(h)})(h^{−1}g) = A_{0}( ∑_{h∈G/U} h(f_{f(h)}))(g)

= A_{0}(f)(g) .

Hence A_{ψ_{0}} = A_{0}.

We evidently have ∥ψ ∗ f∥ ≤ ∥ψ∥ · ∥f∥. By continuity we therefore obtain a continuous left action of the Banach algebra B(G, ρ) on the Banach space B_{U}^{G}(ρ) which is submultiplicative in the corresponding norms and which commutes with the G-action. This action is described by the same formula (1), and we therefore continue to denote it by ∗.

Lemma 1.3: The map

\begin{matrix}B(G, ρ) & ⟶^{≅} & End_{G}^{cont}(B_{U}^{G}(ρ)) \\ ψ & ⟼ & A_{ψ}(f) := ψ ∗ f \end{matrix}

is an isomorphism of K-algebras and is an isometry with respect to the operator norm on the right hand side.

Proof: (The superscript “cont” refers to the continuous endomorphisms.) By the previous discussion the map ψ ⟼ A_{ψ} is well defined, is a homomorphism of K-algebras, and is norm decreasing. Using the notations from the proof of Lemma 1.2 the formula (2), by continuity, holds for any ψ ∈ B(G, ρ). Using that ∥f_{w}∥ = ∥w∥ we now compute

∥A_{ψ}∥ ≥ sup_{w≠0} \frac{∥ψ ∗ f_{w}∥}{∥f_{w}∥} = sup_{w≠0} sup_{g} \frac{∥ψ(g^{−1})(w)∥}{∥w∥} = sup_{g} ∥ψ(g^{−1})∥
= ∥ψ∥ ≥ ∥A_{ψ}∥ .

It follows that the map in the assertion is an isometry and in particular is injective. To see its surjectivity we fix an A_{0} ∈ End_{G}^{cont}(B_{U}^{G}(ρ)) and define

\begin{matrix}ψ_{0} : G & ⟶ & End_{K}(E) \\ g & ⟼ & [w ↦ A_{0}(f_{w})(g^{−1})] . \end{matrix}

Since each A_{0}(f_{w}) is continuous and vanishing at infinity on G it follows that ψ_{0} is continuous and vanishing at infinity. By exactly the same computations as in the proof of Lemma 1.2 one then shows that in fact ψ_{0} ∈ B(G, ρ) and that A_{ψ_{0}} = A_{0}.

We end this section by considering the special case where (ρ, E) is the restriction to U of a continuous representation ρ of G on a finite dimensional K-vector space E. It is easy to check that then the map

\begin{matrix}ι_{ρ} : H(G, 1_{U}) & ⟶ & H(G, ρ) \\ ψ & ⟼ & (ψ · ρ)(g) := ψ(g)ρ(g) \end{matrix}

is an injective homomorphism of K-algebras. There are interesting situations where this map in fact is an isomorphism. We let L be a finite extension of Q_{p} contained in K, and we assume that G as well as (ρ, E) are locally L-analytic.

Lemma 1.4: Suppose that, for the derived action of the Lie algebra 𝔤 of G, the K ⊗_{L} 𝔤-module E is absolutely irreducible; then the homomorphism ι_{ρ} is bijective.

Proof: Using Lemma 1.2 and Frobenius reciprocity we have

H(G, ρ) = End_{G}(ind_{U}^{G}(ρ)) = Hom_{U}(E, ind_{U}^{G}(ρ))
= Hom_{U}(E, ind_{U}^{G}(1) ⊗_{K} E)
= [ind_{U}^{G}(1) ⊗_{K} E^{∗} ⊗_{K} E]^{U}

where the last term denotes the U-fixed vectors in the tensor product with respect to the diagonal action. This diagonal action makes the tensor product equipped with the finest locally convex topology into a locally analytic G-representation. Its U-fixed vectors certainly are contained in the vectors annihilated by the derived action of 𝔤. Since G acts smoothly on ind_{U}^{G}(1) we have

(ind_{U}^{G}(1) ⊗_{K} E^{∗} ⊗_{K} E)^{𝔤=0} = ind_{U}^{G}(1) ⊗_{K} (E^{∗} ⊗_{K} E)^{𝔤=0}
= ind_{U}^{G}(1) ⊗_{K} End_{K⊗_{L}𝔤}(E) .

Our assumption on absolute irreducibility implies that End_{K⊗_{L}𝔤}(E) = K. We therefore see that

H(G, ρ) = [ind_{U}^{G}(1) ⊗_{K} E^{∗} ⊗_{K} E]^{U} = ind_{U}^{G}(1)^{U} = H(G, 1_{U}) .

2. Weights and affinoid algebras

For the rest of this paper L/Q_{p} is a finite extension contained in K, and G denotes the group of L-valued points of an L-split connected reductive group over L. Let | |_{L} be the normalized absolute value of L, val_{L} : K^{×} ⟶ IR the unique additive valuation such that val_{L}(L^{×}) = ZZ, and q the number of elements in the residue class field of L. We fix a maximal L-split torus T in G and a Borel subgroup P = T N of G with Levi component T and unipotent radical N. The Weyl group of G is the quotient W = N(T)/T of the normalizer N(T) of T in G by T. We also fix a maximal compact subgroup U_{0} ⊆ G which is special with respect to T (i.e., is the stabilizer of a special vertex x_{0} in the apartment corresponding to T, cf. [Car]§3.5). We put T_{0} := U_{0} ∩ T and N_{0} := U_{0} ∩ N. The quotient Λ := T/T_{0} is a free abelian group of rank equal to the dimension of T and can naturally be identified with the cocharacter group of T. Let λ : T ⟶ Λ denote the projection map. The conjugation action of N(T) on T induces W -actions on T and Λ which we denote by t ⟼ ^{w}t and λ ⟼ ^{w}λ, respectively. We also need the L-torus T′ dual to T. Its K-valued points are given by

T′(K) := Hom(Λ, K^{×}) .

The group ring K[Λ] of Λ over K naturally identifies with the ring of algebraic functions on the torus T′. We introduce the “valuation map”

\xymatrix{val : T′(K) = Hom(Λ, K^{×}) \ar[r]^{val_{L} ∘} & Hom(Λ, IR) =: V_{IR} .}

If X^{∗}(T) denotes the algebraic character group of the torus T then, as |χ(T_{0})| = {1}, we have the embedding

X^{∗}(T) ⟶ Hom(Λ, IR)
χ ⟼ val_{L} ∘χ

which induces an isomorphism

X^{∗}(T) ⊗ IR →^{≅} V_{IR} .

We therefore may view V_{IR} as the real vector space underlying the root datum of G with respect to T. Evidently any λ ∈ Λ defines a linear form in the dual vector space V_{IR}^{∗} also denoted by λ. Let Φ denote the set of roots of T in G and let Φ^{+} ⊆ Φ be the subset of those roots which are positive with respect to P. As usual, α̌ ∈ Λ denotes the coroot corresponding to the root α ∈ Φ. The subset Λ^{−−} ⊆ Λ of antidominant cocharacters is defined to be the image Λ^{−−} := λ(T^{−−}) of

T^{−−} := {t ∈ T : |α(t)|_{L} ≥ 1 for any α ∈ Φ^{+}} .

Hence

Λ^{−−} = {λ ∈ Λ : val_{L} ∘α(λ) ≤ 0 for any α ∈ Φ^{+}} .

We finally recall that Λ^{−−} carries the partial order ≤ given by

μ ≤ λ if λ − μ ∈ ∑_{α∈Φ^{+}} IR_{≥0} · (−α̌) ⊆ Λ ⊗ IR .

In this section we will investigate certain Banach algebra completions of the group ring K[Λ] together with certain twisted W -actions on them. We will proceed in an axiomatic way and will give ourselves a cocycle on W with values in T′(K), i.e., a map

γ : W × Λ ⟶ K^{×}

such that

(a)  γ(w, λμ) = γ(w, λ)γ(w, μ)  for any w ∈ W and λ, μ ∈ Λ

and

(b)  γ(vw, λ) = γ(v,^{w}λ)γ(w, λ)  for any v, w ∈ W and λ ∈ Λ .

Moreover we impose the positivity condition

(c)  |γ(w, λ)| ≤ 1  for any w ∈ W and λ ∈ Λ^{−−}

as well as the partial triviality condition

(d)  γ(w, λ) = 1  for any w ∈ W and λ ∈ Λ such that ^{w}λ = λ .

The twisted action of W on K[Λ] then is defined by

\begin{matrix}W × K[Λ] & ⟶ & K[Λ] \\ (w, ∑_{λ} c_{λ}λ) & ⟼ & w · (∑_{λ} c_{λ}λ) := ∑_{λ} γ(w, λ)c_{λ}^{w}λ . \end{matrix}

By (a), each w ∈ W acts as an algebra automorphism, and the cocycle condition (b) guarantees the associativity of this W -action. The invariants with respect to this action will be denoted by K[Λ]^{W,γ}. Since Λ^{−−} is a fundamental domain for the W -action on Λ it follows that K[Λ]^{W,γ} has the K-basis {σ_{λ}}_{λ∈Λ^{−−}} defined by

σ_{λ} := ∑_{w∈W/W (λ)} w · λ = ∑_{w∈W/W (λ)} γ(w, λ)^{w}λ

where W (λ) ⊆ W denotes the stabilizer of λ and where the sums are well defined because of (d). Next, again using (d), we define the map

\begin{matrix}γ^{dom} : Λ & ⟶ & K^{×} \\ λ & ⟼ & γ(w, λ) if ^{w}λ ∈ Λ^{−−} , \end{matrix}

and we equip K[Λ] with the norm

∥ ∑_{λ} c_{λ}λ∥_{γ} := sup_{λ∈Λ} |γ^{dom}(λ)c_{λ}| .

The cocycle condition (b) implies the identity

(1)  γ^{dom}(^{w}λ)γ(w, λ) = γ^{dom}(λ)

from which one deduces that the twisted W -action on K[Λ] is isometric in the norm ∥ ∥_{γ} and hence extends by continuity to a W -action on the completion K⟨Λ; γ⟩ of K[Λ] with respect to ∥ ∥_{γ}. Again we denote the corresponding W - invariants by K⟨Λ; γ⟩^{W,γ}. One easily checks that {σ_{λ}}_{λ∈Λ^{−−}} is an orthonormal basis of the Banach space (K⟨Λ; γ⟩^{W,γ}, ∥ ∥_{γ}).

Lemma 2.1: i. |γ^{dom}(λ)| ≥ 1 for any λ ∈ Λ;

ii. |γ^{dom}(λμ)| ≤ |γ^{dom}(λ)||γ^{dom}(μ)| for any λ, μ ∈ Λ.

Proof: i. If ^{w}λ ∈ Λ^{−−} then γ^{dom}(λ) = γ(w, λ) = γ(w^{−1},^{w}λ)^{−1}. The claim therefore is a consequence of the positivity condition (c). ii. If ^{w}(λμ) ∈ Λ^{−−} then, using (1), we have

γ^{dom}(λμ) = γ^{dom}(^{w}λ)^{−1}γ^{dom}(^{w}μ)^{−1}γ^{dom}(λ)γ^{dom}(μ) .

Hence the claim follows from the first assertion.

It is immediate from Lemma 2.1.i that the norm ∥ ∥_{γ} is submultiplicative. Hence K⟨Λ; γ⟩ is a K-Banach algebra containing K[Λ] as a dense subalgebra. Moreover, since the twisted W -action on K⟨Λ; γ⟩ is by algebra automorphisms, K⟨Λ; γ⟩^{W,γ} is a Banach subalgebra of K⟨Λ; γ⟩.

In order to compute the Banach algebra K⟨Λ; γ⟩ we introduce the subset

T′_{γ}(K) := {ζ ∈ T′(K) : |ζ(λ)| ≤ |γ^{dom}(λ)| for any λ ∈ Λ}

of T′(K). We obviously have

T′_{γ}(K) = val^{−1}(V_{IR}^{γ})

with

V_{IR}^{γ} := {z ∈ V_{IR} : λ(z) ≥ val_{L}(γ^{dom}(λ)) for any λ ∈ Λ} .

By (a), our cocycle γ defines the finitely many points

z_{w} := −val(γ(w^{−1}, .)) for w ∈ W

in V_{IR}. The cocycle condition (b) implies that

(2)  z_{vw} = ^{v}z_{w} + z_{v}  for any v, w ∈ W

and the positivity condition (c) that

(3)  λ(z_{w}) ≤ 0  for any w ∈ W and λ ∈ Λ^{−−} .

Remark 2.2: {z ∈ V_{IR} : λ(z) ≤ 0 for any λ ∈ Λ^{−−}} = ∑_{α∈Φ^{+}} IR_{≥0} · val_{L} ∘α.

Proof: This reduces to the claim that the (closed) convex hull of Λ^{−−} in V_{IR}^{∗} is equal to the antidominant cone

(V_{IR}^{∗})^{−−} = {z^{∗} ∈ V_{IR}^{∗} : z^{∗}(z) ≤ 0 for any z ∈ ∑_{α∈Φ^{+}} IR_{≥0} · val_{L} ∘α} .

Let Z ⊆ G denote the connected component of the center of G. Then G/Z is semisimple and the sequence

0 ⟶ Z/Z_{0} ⟶ T/T_{0} ⟶ (T/Z)/(T/Z)_{0} ⟶ 0

is exact. Hence the fundamental antidominant coweights for the semisimple group G/Z can be lifted to elements ω_{1}, …, ω_{d} ∈ V_{IR}^{∗} in such a way that, for some m ∈ IN, we have mω_{1}, …, mω_{d} ∈ Λ^{−−}. It follows that

(V_{IR}^{∗})^{−−} = (Z/Z_{0}) ⊗ IR + ∑_{i=1}^{d} IR_{≥0} · ω_{i}

and

Λ^{−−} ⊇ Z/Z_{0} + m · ∑_{i=1}^{d} ZZ_{≥0} · ω_{i} .

We therefore obtain from (3) that

(4)  z_{w} ∈ ∑_{α∈Φ^{+}} IR_{≥0} · val_{L} ∘α  for any w ∈ W .

In terms of these points z_{w} the set V_{IR}^{γ} is given as

{z ∈ V_{IR} : λ(z) ≥ λ(−z_{w^{−1}}) for any λ ∈ Λ, w ∈ W such that ^{w}λ ∈ Λ^{−−}} = {z ∈ V_{IR} : ^{w^{−1}}λ(z) ≥ ^{w^{−1}}λ(−z_{w^{−1}}) for any w ∈ W and λ ∈ Λ^{−−}} = {z ∈ V_{IR} : λ(^{w}z) ≥ λ(z_{w}) for any w ∈ W and λ ∈ Λ^{−−}}

where the last identity uses (2). Obviously V_{IR}^{γ} is a convex subset of V_{IR}. Using the partial order ≤ on V_{IR} defined by Φ^{+} (cf. [B-GAL] Chap. VI §1.6) we obtain from Remark 2.2 that

V_{IR}^{γ} = {z ∈ V_{IR} : ^{w}z ≤ z_{w} for any w ∈ W } .

Lemma 2.3: V_{IR}^{γ} is the convex hull in V_{IR} of the finitely many points −z_{w} for w ∈ W.

Proof: From (2) and (4) we deduce that

wz_{v} + z_{w} = z_{wv} ≥ 0 and hence ^{w}(−z_{v}) ≤ z_{w}

for any v, w ∈ W. It follows that all −z_{v} and therefore their convex hull is contained in V_{IR}^{γ}. For the reverse inclusion suppose that there is a point z ∈ V_{IR}^{γ} which does not lie in the convex hull of the −z_{w}. We then find a linear form ℓ ∈ V_{IR}^{∗} such that ℓ(z) < ℓ(−z_{w}) for any w ∈ W. Choose v ∈ W such that ℓ_{0} := ^{v}ℓ is antidominant. It follows that ^{v^{−1}}ℓ_{0}(z) < ^{v^{−1}}ℓ_{0}(−z_{w}) and hence, using (2), that

ℓ_{0}(^{v}z) < ℓ_{0}(−^{v}z_{w}) = ℓ_{0}(z_{v}) − ℓ_{0}(z_{vw})

for any w ∈ W. For w := v^{−1} we in particular obtain

ℓ_{0}(^{v}z) < ℓ_{0}(z_{v}) .

On the other hand, since z ∈ V_{IR}^{γ}, we have

λ(^{v}z) ≥ λ(z_{v})

for any λ ∈ Λ^{−−} and hence for any λ in the convex hull of Λ^{−−}. But as we have seen in the proof of Remark 2.2 the antidominant ℓ_{0} belongs to this convex hull which leads to a contradiction.

Proposition 2.4: i. T′_{γ}(K) is the set of K-valued points of an open K-affinoid subdomain T′_{γ} in the torus T′;

ii. the Banach algebra K⟨Λ; γ⟩ is naturally isomorphic to the ring of analytic functions on the affinoid domain T′_{γ};

iii. the affinoid domain T′_{γ} is the preimage, under the map “val”, of the convex hull of the finitely many points −z_{w} ∈ V_{IR} for w ∈ W;

iv. K⟨Λ; γ⟩^{W,γ} is an affinoid K-algebra.

Proof: It follows from Gordan’s lemma ([KKMS] p. 7) that the monoid Λ^{−−} is finitely generated. Choose a finite set of generators F^{−−}, and let

F := {^{w}λ : λ ∈ F^{−−}} .

Using the fact that, by construction, the function γ^{dom} is multiplicative within Weyl chambers we see that the infinitely many inequalities implicit in the definition of T′_{γ}(K) can in fact be replaced by finitely many:

T′_{γ}(K) = {ζ ∈ T′(K) : |ζ(λ)| ≤ |γ^{dom}(λ)| for any λ ∈ F }.

We therefore define T′_{γ} to be the rational subset in T′ given by the finitely many inequalities |γ^{dom}(λ)^{−1}λ(ζ)| ≤ 1 for λ ∈ F and obtain point i. of our assertion.

Now choose indeterminates T_{λ} for λ ∈ F and consider the commutative diagram of algebra homomorphisms

\xymatrix{o_{K}[T_{λ} : λ ∈ F] \ar[d]_{⊆} \ar[r] & K[Λ]^{0} \ar[d]^{⊆} \\ K[T_{λ} : λ ∈ F] \ar[d]_{⊆} \ar[r] & K[Λ] \ar[d]^{⊆} \\ K⟨T_{λ} : λ ∈ F ⟩ \ar[r] & K⟨Λ; γ⟩}

where the horizontal arrows send T_{λ} to γ^{dom}(λ)^{−1}λ, where o_{K} is the ring of integers of K, and where K[Λ]^{0} denotes the unit ball with respect to ∥ ∥_{γ} in K[Λ]. Again, the multiplicativity of γ^{dom} within Weyl chambers shows that all three horizontal maps are surjective. The lower arrow gives a presentation of K⟨Λ; γ⟩ as an affinoid algebra. The middle arrow realizes the dual torus T′ as a closed algebraic subvariety

\begin{matrix}ι : T′ & ⟶ & A A^{f} \\ ζ & ⟼ & (ζ(λ))_{λ∈F} \end{matrix}

in the affine space A A^{f} where f denotes the cardinality of the set F. The surjectivity of the upper arrow shows that the norm ∥ ∥_{γ} on K[Λ] is the quotient norm of the usual Gauss norm on the polynomial ring K[T_{λ} : λ ∈ F]. Hence the kernel of the lower arrow is the norm completion of the kernel I of the middle arrow. Since any ideal in the Tate algebra K⟨T_{λ} : λ ∈ F ⟩ is closed we obtain

K⟨Λ; γ⟩ = K⟨T_{λ} : λ ∈ F ⟩/IK⟨T_{λ} : λ ∈ F ⟩ .

This means that the affinoid variety Sp(K⟨Λ; γ⟩) is the preimage under ι of the affinoid unit polydisk in A A^{f}. In particular, Sp(K⟨Λ; γ⟩) is an open subdomain in T′ which is reduced and coincides with the rational subset T′_{γ} (cf. [FvP] Prop. 4.6.1(4)). This establishes point ii. of the assertion. The point iii. is Lemma 2.3. For point iv., as the invariants in an affinoid algebra with respect to a finite group action, K⟨Λ; γ⟩^{W,γ} is again affinoid (cf. [BGR] 6.3.3 Prop. 3).

Suppose that the group G is semisimple and adjoint. Then the structure of the affinoid algebra K⟨Λ; γ⟩^{W,γ} is rather simple. The reason is that for such a group the set Λ^{−−} is the free commutative monoid over the fundamental antidominant cocharacters λ_{1}, …, λ_{d}. As usual we let K⟨X_{1}, …, X_{d}⟩ denote the Tate algebra in d variables over K. Obviously we have a unique continuous algebra homomorphism

K⟨X_{1}, …, X_{d}⟩ ⟶ K⟨Λ; γ⟩^{W,γ}

sending the variable X_{i} to σ_{λ_{i}}.

We also need a general lemma about orthogonal bases in normed vector spaces. Let (Y, ∥ ∥) be a normed K-vector space and suppose that Y has an orthogonal basis of the form {x_{ℓ}}_{ℓ∈I}. Recall that the latter means that

∥ ∑_{ℓ} c_{ℓ}x_{ℓ}∥ = sup_{ℓ} |c_{ℓ}| · ∥x_{ℓ}∥

for any vector ∑_{ℓ} c_{ℓ}x_{ℓ} ∈ Y. We suppose moreover that there is given a partial order ≤ on the index set I such that:

– Any nonempty subset of I has a minimal element;

– for any k ∈ I the set {ℓ ∈ I : ℓ ≤ k} is finite.

(Note that the partial order ≤ on Λ^{−−} has these properties.)

Lemma 2.5: Suppose that ∥x_{ℓ}∥ ≤ ∥x_{k}∥ whenever ℓ ≤ k; furthermore, let elements c_{ℓk} ∈ K be given, for any ℓ ≤ k in I, such that |c_{ℓk}| ≤ 1; then the vectors

y_{k} := x_{k} + ∑_{ℓ<k} c_{ℓk}x_{ℓ}

form another orthogonal basis of Y, and ∥y_{k}∥ = ∥x_{k}∥.

Proof: We have

∥y_{k}∥ = max(∥x_{k}∥, max_{ℓ<k} |c_{ℓk}| · ∥x_{ℓ}∥) = ∥x_{k}∥

as an immediate consequence of our assumptions. We also have

x_{k} = y_{k} + ∑_{ℓ<k} b_{ℓk}y_{ℓ}

where (b_{ℓk}) is the matrix inverse to (c_{ℓk}) (over the ring of integers in K; cf. [B-GAL] Chap. VI §3.4 Lemma 4). Let now x = ∑_{k} c_{k}x_{k} be an arbitrary vector in Y. We obtain

x = ∑_{k} c_{k}x_{k} = ∑_{k} c_{k}(∑_{ℓ≤k} b_{ℓk}y_{ℓ}) = ∑_{ℓ}(∑_{ℓ≤k} c_{k}b_{ℓk})y_{ℓ} .

Clearly ∥x∥ ≤ sup_{ℓ} | ∑_{ℓ≤k} c_{k}b_{ℓk}| · ∥y_{ℓ}∥. On the other hand we compute

sup_{ℓ} | ∑_{ℓ≤k} c_{k}b_{ℓk}| · ∥y_{ℓ}∥ ≤ sup_{ℓ} sup_{ℓ≤k} |c_{k}| · ∥y_{ℓ}∥ = sup_{ℓ} sup_{ℓ≤k} |c_{k}| · ∥x_{ℓ}∥
≤ sup_{k} |c_{k}| · ∥x_{k}∥ = ∥x∥ .

Proposition 2.6: If the group G is semisimple and adjoint then the above map is an isometric isomorphism K⟨X_{1}, …, X_{d}⟩ ⟶^{≅} K⟨Λ; γ⟩^{W,γ}.

Proof: We write a given λ ∈ Λ^{−−} as λ = λ_{1}^{m_{1}} … λ_{d}^{m_{d}} and put

σ̃_{λ} := σ_{λ_{1}}^{m_{1}} · . . . · σ_{λ_{d}}^{m_{d}} .

It suffices to show that these {σ̃_{λ}}_{λ∈Λ^{−−}} form another orthonormal basis of K⟨Λ; γ⟩^{W,γ}. One checks that the arguments in [B-GAL] Chap. VI §§3.2 and 3.4 work, over the ring of integers in K, equally well for our twisted W -action and show that we have

σ̃_{λ} = σ_{λ} + ∑_{μ<λ} c_{μλ}σ_{μ}

with |c_{μλ}| ≤ 1. So we may apply Lemma 2.5.

We finish this section with a discussion of those examples of a cocycle γ which will be relevant later on.

Example 1: We fix a prime element π_{L} of L. Let ξ ∈ X^{∗}(T) be a dominant integral weight and put

γ(w, λ(t)) := π_{L}^{val_{L}(ξ(^{w}t))−val_{L}(ξ(t))} for t ∈ T .

This map γ obviously has the properties (a),(b), and (d). For t ∈ T^{−−} we have λ(^{w}t) ≤ λ(t) by [B-GAL] Chap. VI §1.6 Prop. 18; since ξ is dominant we obtain

val_{L} ∘ξ(\frac{t}{^{w}t}) ≤ 0 .

This means that |γ(w, λ)| ≤ 1 for λ ∈ Λ^{−−} which is condition (c). We leave it as an exercise to the reader to check that the resulting Banach algebra K⟨Λ; γ⟩ together with the twisted W -action, up to isomorphism, is independent of the choice of the prime element π_{L}.

Example 2: A particular case of a dominant integral weight is the determinant of the adjoint action of T on the Lie algebra Lie(N) of the unipotent radical N

Δ(t) := det(ad(t); Lie(N)) .

Its absolute value satisfies

δ(t) = |Δ(t)|_{L}^{−1}

where δ : P ⟶ Q^{×} ⊆ K^{×} is the modulus character of the Borel subgroup P. We let K_{q}/K denote the splitting field of the polynomial X^{2} − q and we fix a root q^{1/2} ∈ K_{q}^{×}. Then the square root δ^{1/2} : Λ ⟶ K_{q}^{×} of the character δ is well defined. For a completely analogous reason as in the first example the cocycle

γ(w, λ) := \frac{δ^{1/2}(^{w}λ)}{δ^{1/2}(λ)}

has the properties (a) − (d). Moreover, using the root space decomposition of Lie(N) one easily shows that

γ(w, λ(t)) = ∏_{α∈Φ^{+}\^{w^{−1}} Φ^{+}} |α(t)|_{L} .

Hence the values of this cocycle γ are integral powers of q and therefore lie in K.

Example 3: Obviously the properties (a) − (d) are preserved by the product of two cocycles. For any dominant integral weight ξ ∈ X^{∗}(T) therefore the cocycle

γ_{ξ}(w, λ(t)) := \frac{δ^{1/2}(^{w}λ)}{δ^{1/2}(λ)} · π_{L}^{val_{L}(ξ(^{w}t))−val_{L}(ξ(t))}

is K-valued and satisfies (a) − (d). We write

V_{IR}^{ξ} := V_{IR}^{γ_{ξ}}  and  T′_{ξ} := T′_{γ_{ξ}} .

Let η ∈ V_{IR} denote half the sum of the positive roots in Φ^{+} and put

η_{L} := [L : Q_{p}] · η .

Let

ξ_{L} := val_{L} ∘ξ ∈ V_{IR} .

For the points z_{w} ∈ V_{IR} corresponding to the cocycle γ_{ξ} we then have

z_{w} = (η_{L} + ξ_{L}) −^{w}(η_{L} + ξ_{L}) .

In particular, V_{IR}^{ξ} is the convex hull of the points ^{w}(η_{L} + ξ_{L}) − (η_{L} + ξ_{L}) for w ∈ W. Note that, since γ_{ξ} has values in L^{×}, the affinoid variety T′_{ξ} is naturally defined over L. Given any point z ∈ V_{IR}, we will write z^{dom} for the unique dominant point in the W -orbit of z.

Lemma 2.7: V_{IR}^{ξ} = {z ∈ V_{IR} : (z + η_{L} + ξ_{L})^{dom} ≤ η_{L} + ξ_{L}}.

Proof: Using the formula before Lemma 2.3 we have

V_{IR}^{ξ} = {z ∈ V_{IR} : ^{w}z ≤ (η_{L} + ξ_{L}) −^{w}(η_{L} + ξ_{L}) for any w ∈ W }
= {z ∈ V_{IR} : ^{w}(z + η_{L} + ξ_{L}) ≤ η_{L} + ξ_{L} for any w ∈ W } .

It remains to recall ([B-GAL] Chap. VI §1.6 Prop. 18) that for any z ∈ V_{IR} and any w ∈ W one has ^{w}z ≤ z^{dom}.

The γ_{ξ} in Example 3 are the cocycles which will appear in our further investigation of specific Banach-Hecke algebras. In the following we explicitly compute the affinoid domain T′_{ξ} in case of the group G := GL_{d+1}(L). (In case ξ = 1 compare also [Vig] Chap. 3.) We let P ⊆ G be the lower triangular Borel subgroup and T ⊆ P be the torus of diagonal matrices. We take U_{0} := GL_{d+1}(o_{L}) where o_{L} is the ring of integers of L. If π_{L} ∈ o_{L} denotes a prime element then

Λ^{−−} = \{( \begin{matrix}π_{L}^{m_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & π_{L}^{m_{d+1}} \end{matrix} ) T_{0} : m_{1} ≥ … ≥ m_{d+1}\} .

For 1 ≤ i ≤ d + 1 define the diagonal matrix

t_{i} := ( \begin{matrix}π_{L} & & & & & & & & & 0 \\ & . & . & . \\ & & & & π_{L} \\ & & & & & 1 \\ & & & & & & . & . & . \\ 0 & & & & & & & & & 1 \end{matrix} ) with i diagonal entries equal to π_{L} .

As a monoid Λ^{−−} is generated by the elements λ_{1}, …, λ_{d+1}, λ_{d+1}^{−1} where λ_{i} := λ(t_{i}). For any nonempty subset I = {i_{1}, …, i_{s}} ⊆ {1, …, d + 1} let λ_{I} ∈ Λ be the cocharacter corresponding to the diagonal matrix having π_{L} at the places i_{1}, …, i_{s} and 1 elsewhere. Moreover let, as usual, |I| := s be the cardinality of I and put ht(I) := i_{1} +…+i_{s}. These λ_{I} together with λ_{{1,...,d+1}}^{−1} form the W - orbit of the above monoid generators. From the proof of Prop. 2.4 we therefore know that T′_{ξ} as a rational subdomain of T′ is described by the conditions

|ζ(λ_{I})| ≤ |γ_{ξ}^{dom}(λ_{I})|

for any I and

|ζ(λ_{{1,...,d+1}})| = |γ_{ξ}^{dom}(λ_{{1,...,d+1}})| .

One checks that

|γ_{1}^{dom}(λ_{I})| = |q|^{|I|(|I|+1)/2−ht(I)} .

If the dominant integral weight ξ ∈ X^{∗}(T) is given by

( \begin{matrix}g_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & g_{d+1} \end{matrix} ) ⟼ ∏_{i=1}^{d+1} g_{i}^{a_{i}}

with (a_{1}, …, a_{d+1}) ∈ ZZ^{d+1} then

|γ_{ξ}^{dom}(λ_{I})| = |q|^{|I|(|I|+1)/2−ht(I)}|π_{L}|^{∑_{j=1}^{|I|} a_{j}−∑_{i∈I} a_{i}} .

We now use the coordinates

\begin{matrix}T′(K) & ⟶ & (K^{×})^{d+1} \\ ζ & ⟼ & (ζ_{1}, …, ζ_{d+1}) with ζ_{i} := q^{i−1}π_{L}^{a_{i}}ζ(λ_{\{i\}}) \end{matrix}

on the dual torus. In these coordinates T′_{ξ} is the rational subdomain of all (ζ_{1}, …, ζ_{d+1}) ∈ (K^{×})^{d+1} such that

∏_{i∈I} |ζ_{i}| ≤ |q|^{|I|(|I|−1)/2}|π_{L}|^{∑_{i=1}^{|I|} a_{i}}

for any proper nonempty subset I ⊆ {1, …, d + 1} and

∏_{i=1}^{d+1} |ζ_{i}| = |q|^{d(d+1)/2}|π_{L}|^{∑_{i=1}^{d+1} a_{i}} .

The advantage of these variables is the following. As usual we identify the Weyl group W with the symmetric group on the set {1, …, d + 1}. One checks that

γ_{ξ}(w, λ_{{i}}) = q^{w(i)−i}π_{L}^{a_{w(i)}−a_{i}}

for any w ∈ W and 1 ≤ i ≤ d + 1. This implies that the twisted W -action on the affinoid algebra K⟨Λ; γ_{ξ}⟩ is induced by the permutation action on the coordinates ζ_{1}, …, ζ_{d+1} of the affinoid domain T′_{ξ}. In fact, the above identity means that the cocycle γ_{ξ} can be written as the coboundary of an element in T′(K). This is more generally possible for any group G whose derived group is simply connected (cf. [Gro] §8). We do not pursue this point of view systematically, though, since it is not compatible with general Langlands functoriality. But the problem of “splitting” the cocycle and the difficulty of reconciling the normalization of the Satake isomorphism will reappear as a technical complication in our attempt, in section 6, to treat Langlands functoriality.

3. The p-adic Satake isomorphism

Keeping the notations and assumptions introduced in the previous section we now consider a locally L-analytic representation (ρ, E) of G of the form

E = K_{χ} ⊗_{L} E_{L}

where

– K_{χ} is a one dimensional representation of G given by a locally L-analytic character χ : G ⟶ K^{×}, and

– E_{L} is an L-rational irreducible representation ρ_{L} of G of highest weight ξ.

Let

E_{L} = ⊕_{β∈X^{∗}(T)}E_{L,β}

be the decomposition into weight spaces for T. According to [BT] II.4.6.22 and Prop. II.4.6.28(ii) the reductive group G has a smooth connected affine model G over the ring of integers o_{L} in L such that G(o_{L}) = U_{0}. We fix once and for all a U_{0}-invariant o_{L}-lattice M in E_{L} ([Jan] I.10.4) and let ∥ ∥ be the corresponding U_{0}-invariant norm on E. The following fact is well-known.

Lemma 3.1: We have M = ⊕_{β∈X^{∗}(T)} M_{β} with M_{β} := M ∩ E_{L,β}.

Proof: For the convenience of the reader we sketch the argument. Fix a weight β ∈ X^{∗}(T). It suffices to construct an element Π_{β} in the algebra of distributions Dist(G) which acts as a projector

Π_{β} : E_{L} ⟶ E_{L,β} .

Let B be the finite set of weights ≠ β which occur in E_{L}. Also we need the Lie algebra elements

H_{i} := (dμ_{i})(1) ∈ Lie(G)

where μ_{1}, …, μ_{r} is a basis of the cocharacter group of T. We have

\underline{γ} := (dγ(H_{1}), …, dγ(H_{r})) ∈ ZZ^{r} for any γ ∈ X^{∗}(T) .

According to [Hum] Lemma 27.1 we therefore find a polynomial Π ∈ Q[y_{1},…,y_{r}] such that Π(ZZ^{r}) ⊆ ZZ, Π(\underline{β}) = 1, and Π(\underline{γ}) = 0 for any γ ∈ B. Moreover [Hum] Lemma 26.1 says that the polynomial Π is a ZZ-linear combination of polynomials of the form

( \begin{matrix}y_{1} \\ b_{1} \end{matrix} ) · . . . · ( \begin{matrix}y_{r} \\ b_{r} \end{matrix} ) with integers b_{1}, …, b_{r} ≥ 0 .

Then [Jan] II.1.12 implies that

Π_{β} := Π(H_{1}, …, H_{r})

lies in Dist(G). By construction Π_{β} induces a projector from E_{L} onto E_{L,β}.

It follows that, for any t ∈ T, the operator norm of ρ_{L}(t) on E_{L} is equal to

∥ρ_{L}(t)∥ = max{|β(t)| : β ∈ X^{∗}(T) such that E_{L,β} ≠ 0}.

Lemma 3.2: For any t ∈ T we have ∥ρ(t)∥ = |χ(t)| · |ξ(^{w}t)| with w ∈ W such that ^{w}t ∈ T^{−−}.

Proof: Consider first the case t ∈ T^{−−} with w = 1. For any weight β occurring in E_{L} one has ξ = αβ where α is an appropriate product of simple roots. But by definition of T^{−−} we have |α(t)|_{L} ≥ 1 for any simple root α. For general t ∈ T and w ∈ W as in the assertion we then obtain

|ξ(^{w}t)| = max{|β(^{w}t)| : E_{L,β} ≠ 0}
= max{|β(t)| : E_{L,β} ≠ 0}
= ∥ρ_{L}(t)∥ .

Here the second identity is a consequence of the fact that the set of weights of E_{L} is W -invariant.

Collecting this information we first of all see that Lemma 1.4 applies and gives, for any open subgroup U ⊆ U_{0}, the isomorphism

H(G, 1_{U}) ≅ H(G, ρ|U) .

But the norm ∥ ∥ on H(G, ρ|U) corresponds under this isomorphism to the norm ∥ ∥_{χ,ξ} on H(G, 1_{U}) defined by

∥ψ∥_{χ,ξ} := sup_{g∈G} |ψ(g)χ(g)| · ∥ρ_{L}(g)∥ .

If |χ|= 1 (e.g., if the group G is semisimple) then the character χ does not affect the norm ∥ ∥_{ξ} := ∥ ∥_{χ,ξ}. In general χ can be written as a product χ = χ_{1}χ_{un} of two characters where |χ_{1}| = 1 and χ_{un}|U_{0} = 1. Then

\begin{matrix}(H(G, 1_{U}), ∥ ∥_{ξ}) & ⟶^{≅} & (H(G, 1_{U}), ∥ ∥_{χ,ξ}) \\ ψ & ⟼ & ψ · χ_{un}^{−1} \end{matrix}

is an isometric isomorphism. We therefore have the following fact.

Lemma 3.3: The map

∥ ∥_{ξ}-completion of H(G, 1_{U}) ⟶^{≅} B(G, ρ|U)
ψ ⟼ ψ · χ_{un}^{−1}ρ

is an isometric isomorphism of Banach algebras.

In this section we want to compute these Banach-Hecke algebras in the case U = U_{0}. By the Cartan decomposition G is the disjoint union of the double cosets U_{0}tU_{0} with t running over T^{−−}/T_{0}. Let therefore ψ_{λ(t)} ∈ H(G, 1_{U_{0}}) denote the characteristic function of the double coset U_{0}tU_{0}. Then {ψ_{λ}}_{λ∈Λ^{−−}} is a K-basis of H(G, 1_{U_{0}}). According to Lemma 3.2 the norm ∥ ∥_{ξ} on H(G, 1_{U_{0}}) is given by

∥ψ∥_{ξ} := sup_{t∈T^{−−}} |ψ(t)ξ(t)| .

The {ψ_{λ}}_{λ∈Λ^{−−}} form a ∥ ∥_{ξ}-orthogonal basis of H(G, 1_{U_{0}}) and hence of its ∥ ∥_{ξ}-completion.

The Satake isomorphism computes the Hecke algebra H(G, 1_{U_{0}}). For our purposes it is important to consider the renormalized version of the Satake map given by

\begin{matrix}S_{ξ} : H(G, 1_{U_{0}}) & ⟶ & K[Λ] \\ ψ & ⟼ & ∑_{t∈T/T_{0}} π_{L}^{val_{L}(ξ(t))}( ∑_{n∈N/N_{0}} ψ(tn))λ(t) . \end{matrix}

On the other hand we again let K_{q}/K be the splitting field of the polynomial X^{2} − q and we temporarily fix a root q^{1/2} ∈ K_{q}. Satake’s theorem says (cf. [Car]§4.2) that the map

\begin{matrix}S^{norm} : H(G, 1_{U_{0}}) ⊗_{K} K_{q} & ⟶ & K_{q}[Λ] \\ ψ & ⟼ & ∑_{t∈T/T_{0}} δ^{−1/2}(t)( ∑_{n∈N/N_{0}} ψ(tn))λ(t) \end{matrix}

induces an isomorphism of K_{q}-algebras

H(G, 1_{U_{0}}) ⊗_{K} K_{q} ⟶^{≅} K_{q}[Λ]^{W} .

Here the W -invariants on the group ring K_{q}[Λ] are formed with respect to the W -action induced by the conjugation action of N(T) on T. Since π_{L}^{val_{L} ∘ξ}δ^{1/2} defines a character of Λ it is clear that S_{ξ} is a homomorphism of algebras as well and a simple Galois descent argument shows that S_{ξ} induces an isomorphism of K-algebras

H(G, 1_{U_{0}}) ⟶^{≅} K[Λ]^{W,γ_{ξ}}

where γ_{ξ} is the cocycle from Example 3 in section 2. The left hand side has the ∥ ∥_{ξ}-orthogonal basis {ψ_{λ}}_{λ∈Λ^{−−}} with

∥ψ_{λ(t)}∥_{ξ} = |ξ(t)| .

The right hand side has the ∥ ∥_{γ_{ξ}}-orthonormal basis {σ_{λ}}_{λ∈Λ^{−−}} where

σ_{λ} = ∑_{w∈W/W (λ)} γ_{ξ}(w, λ)^{w}λ

(cf. section 2). Since the maps

\begin{matrix}N/N_{0} & ⟶^{≃} & NtU_{0}/U_{0} \\ nN_{0} & ⟼ & tnU_{0} \end{matrix}

are bijections we have

∑_{n∈N/N_{0}} ψ_{λ(s)}(tn) = |(NtU_{0} ∩ U_{0}sU_{0})/U_{0}| =: c(λ(t), λ(s)) for any s, t ∈ T .

It follows that

S_{ξ}(ψ_{μ}) = ∑_{t∈T/T_{0}} π_{L}^{val_{L}(ξ(t))}c(λ(t), μ)λ(t)

= ∑_{λ∈Λ^{−−}} π_{L}^{val_{L} ∘ξ(λ)}c(λ, μ)σ_{λ} for any μ ∈ Λ^{−−} .

and

π_{L}^{val_{L} ∘ξ(^{w}λ)}c(^{w}λ, μ) = γ_{ξ}(w, λ)π_{L}^{val_{L} ∘ξ(λ)}c(λ, μ)

for any λ ∈ Λ^{−−}, μ ∈ Λ, and w ∈ W.

The reason for the validity of Satake’s theorem lies in the following properties of the coefficients c(λ, μ).

Lemma 3.4: For λ, μ ∈ Λ^{−−} we have:

i. c(μ, μ) = 1;

ii. c(λ, μ) = 0 unless λ ≤ μ.

Proof: [BT] Prop. I.4.4.4.

Proposition 3.5: The map S_{ξ} extends by continuity to an isometric isomorphism of K-Banach algebras

∥ ∥_{ξ}-completion of H(G, 1_{U_{0}}) ⟶^{≅} K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} .

Proof: Define

ψ̃_{λ} := π_{L}^{− val_{L} ∘ξ(λ)}ψ_{λ}

for λ ∈ Λ^{−−}. The left, resp. right, hand side has the ∥ ∥_{ξ}-orthonormal, resp. ∥ ∥_{γ_{ξ}}-orthonormal, basis {ψ̃_{λ}}_{λ∈Λ^{−−}}, resp. {σ_{λ}}_{λ∈Λ^{−−}}. We want to apply Lemma 2.5 to the normed vector space (K[Λ]^{W,γ_{ξ}}, ∥ ∥_{γ_{ξ}}), its orthonormal basis {σ_{λ}}, and the elements

S_{ξ}(ψ̃_{μ}) = σ_{μ} + ∑_{λ<μ} π_{L}^{val_{L} ∘ξ(λ)−val_{L} ∘ξ(μ)}c(λ, μ)σ_{λ}

(cf. Lemma 3.4). The coefficients c(λ, μ) are integers and therefore satisfy |c(λ, μ)| ≤ 1. Moreover, λ < μ implies, since ξ is dominant, that val_{L} ∘ξ(μ) ≤ val_{L} ∘ξ(λ). Hence the assumptions of Lemma 2.5 indeed are satisfied and we obtain that {S_{ξ}(ψ̃_{λ})} is another orthonormal basis for (K[Λ]^{W,γ_{ξ}}, ∥ ∥_{γ_{ξ}}).

Corollary 3.6: The Banach algebras B(G, ρ|U_{0}) and K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} are isometrically isomorphic.

If ξ = 1 then, in view of Lemma 2.7, the reader should note the striking analogy between the above result and the computation in [Mac] Thm. (4.7.1) of the spectrum of the algebra of integrable complex valued functions on U_{0}\G/U_{0}. The methods of proof are totally different, though. In fact, in our case the spherical function on U_{0}\G/U_{0} corresponding to a point in T′_{1} in general is not bounded.

Suppose that the group G is semisimple and adjoint. We fix elements t_{1}, …, t_{d} ∈ T^{−−} such that λ_{i} := λ(t_{i}) are the fundamental antidominant cocharacters. In Prop. 2.6 we have seen that then K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} is a Tate algebra in the variables σ_{λ_{1}} …, σ_{λ_{d}}. Hence B(G, ρ|U_{0}) is a Tate algebra as well. But it seems complicated to compute explicitly the variables corresponding to the σ_{λ_{i}}. Instead we may repeat our previous reasoning in a modified way.

Proposition 3.7: Suppose that G is semisimple and adjoint; then B(G, ρ|U_{0}) is a Tate algebra over K in the variables \frac{ψ_{λ_{1}} ·ρ}{ξ(t_{1})}, …, \frac{ψ_{λ_{d}} ·ρ}{ξ(t_{d})}.

Proof: By Lemma 3.3 and Prop. 3.5 it suffices to show that K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} is a Tate algebra in the variables ξ(t_{i})^{−1}S_{ξ}(ψ_{λ_{i}}). We write a given λ ∈ Λ^{−−} as λ = λ_{1}^{m_{1}} … λ_{d}^{m_{d}} and put

σ̃_{λ} := S_{ξ}(ψ̃_{λ_{1}})^{m_{1}} · . . . · S_{ξ}(ψ̃_{λ_{d}})^{m_{d}} = S_{ξ}(ψ̃_{λ_{1}}^{m_{1}} ∗ … ∗ ψ̃_{λ_{d}}^{m_{d}})

using notation from the proof of Prop. 3.5. Similarly as in the proof of Prop. 2.7 one checks that the arguments in [B-GAL] Chap. VI §§3.2 and 3.4 work, over the ring of integers in K, equally well for our twisted W -action (note that, in the language of loc. cit. and due to Lemma 3.4, the unique maximal term in S_{ξ}(ψ̃_{λ_{i}}) is λ_{i}) and show that we have

σ̃_{λ} = σ_{λ} + ∑_{μ<λ} c_{μλ}σ_{μ}

with |c_{μλ}| ≤ 1. So we may apply again Lemma 2.5 and obtain that {σ̃_{λ}} is another orthonormal basis for K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}}. It remains to note that ξ(t_{i}) and π_{L}^{val_{L}(ξ(t))} only differ by a unit.

Example: Consider the group G := GL_{d+1}(L). Cor. 3.6 applies to G but Prop. 3.7 does not. Nevertheless, with the same notations as at the end of section 2 a simple modification of the argument gives

B(G, ρ|U_{0}) = K⟨\frac{ψ_{λ_{1}} · χ_{un}^{−1}ρ}{ξ(t_{1})}, …, \frac{ψ_{λ_{d}} · χ_{un}^{−1}ρ}{ξ(t_{d})}, (\frac{ψ_{λ_{d+1}} · χ_{un}^{−1}ρ}{ξ(t_{d+1})})^{±1}⟩ .

Moreover in this case the λ_{i} are minimal with respect to the partial order ≤ so that we do have

ξ(t_{i})^{−1}S_{ξ}(ψ_{λ_{i}}) = σ_{λ_{i}} .

Hence the above representation of B(G, ρ|U_{0}) as an affinoid algebra corresponds to the representation

K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} = K⟨σ_{λ_{1}} …, σ_{λ_{d}}, σ_{λ_{d+1}}^{±1}⟩ .

On affinoid domains this corresponds to a map

T′_{ξ} ⟶ {(ω_{1}, …, ω_{d+1}) ∈ K^{d+1} : |ω_{1}|, …, |ω_{d}| ≤ 1, |ω_{d+1}| = 1}

which, using our choice of coordinates on T′ from section 2, is given by

(ζ_{1}, …, ζ_{d+1}) ⟼ (…, q^{− \frac{(i−1)i}{2}} ξ(t_{i})^{−1}Σ_{i}(ζ_{1}, …, ζ_{d+1}), …)

where

Σ_{1}(ζ_{1}, …, ζ_{d+1}) = ζ_{1} + … + ζ_{d+1}, …, Σ_{d+1}(ζ_{1}, …, ζ_{d+1}) = ζ_{1} · . . . · ζ_{d+1}

denote the elementary symmetric polynomials.

Let us further specialize to the case G = GL_{2}(L). Then E_{L} is the k-th symmetric power, for some k ≥ 0, of the standard representation of GL_{2}. The

highest weight of E_{L} is ξ(( \begin{matrix}t_{1} & 0 \\ 0 & t_{2} \end{matrix} )) = t_{2}^{k}. We obtain

B(G, ρ|U_{0}) = K⟨X_{1}, (π_{L}^{−k}X_{2})^{±1}⟩ .

with the variables X_{i} := ψ_{λ_{i}} · χ_{un}^{−1}ρ. The above map between affinoid domains becomes

(ζ_{1}, ζ_{2}) ⟶ (ζ_{1} + ζ_{2}, q^{−1}π_{L}^{−k}ζ_{1}ζ_{2}) .

4. p-adic Iwahori-Hecke algebras

With the same assumptions and notations as in the previous section we now let U_{1} ⊆ U_{0} be the Iwahori subgroup such that U_{1} ∩P = U_{0} ∩P. In this section we will compute the Banach-Hecke algebras B(G, ρ|U_{1}). By Lemma 3.3 this means, similarly as before, computing the ∥ ∥_{ξ}-completion of H(G, 1_{U_{1}}).

The extended affine Weyl group W̃ of G is given by

W̃ := N(T)/T_{0} .

Since the Weyl group W lifts to U_{0} ∩ N(T)/T_{0} ⊆ W̃ we see that W̃ is the semidirect product of W and Λ. The Bruhat-Tits decomposition says that G is the disjoint union of the double cosets U_{1}xU_{1} with x running over W̃. Therefore, if we let τ_{x} ∈ H(G, 1_{U_{1}}) denote the characteristic function of the double coset U_{1}xU_{1}, then {τ_{x}}_{x∈_{W}} is a K-basis of H(G, 1_{U_{1}}). The τ_{x} are known to be invertible in the algebra H(G, 1_{U_{1}}). As a consequence of Lemma 3.2 the ∥ ∥_{ξ}-norm is given by

∥ψ∥_{ξ} = sup_{v,w∈W} sup_{t∈T^{−−}} |ψ(vλ(^{w}t))ξ(t)| .

In particular, {τ_{x}}_{x∈_{W}} is an ∥ ∥_{ξ}-orthogonal basis of H(G, 1_{U_{1}}) such that

∥τ_{x}∥_{ξ} = |ξ(^{w}t)| if v, w ∈ W and t ∈ T such that x = vλ(t) and ^{w}t ∈ T^{−−} .

We let C be the unique Weyl chamber corresponding to P in the apartment corresponding to T with vertex x_{0} (cf. [Car]§3.5). The Iwahori subgroup U_{1} fixes pointwise the unique chamber C ⊆ C with vertex x_{0}. The reflections at the walls of C generate the Weyl group W. Let s_{0}, …, s_{e} ∈ W̃ be the reflections at all the walls of C and let W_{aff} denote the subgroup of W̃ generated by s_{0}, …, s_{e}. This affine Weyl group W_{aff} with the generating set {s_{0}, …, s_{e}} is a Coxeter group. In particular we have the corresponding length function ℓ : W_{aff} ⟶ IN∪{0} and the corresponding Bruhat order ≤ on W_{aff}. If Ω ⊆ W̃ is the subgroup which fixes the chamber C then W̃ also is the semidirect product of Ω and W_{aff}. We extend the length function ℓ to W̃ by ℓ(ωw) := ℓ(w) for ω ∈ Ω and w ∈ W_{aff}. The Bruhat order is extended to W̃ by the rule ωw ≤ ω′w′, for w, w′ ∈ W_{aff} and ω, ω′ ∈ Ω, if and only if ω = ω′ and w ≤ w′. One of the basic relations established by Iwahori-Matsumoto is:

(1) For any x, y ∈ W̃ such that ℓ(xy) = ℓ(x) + ℓ(y) we have τ_{xy} = τ_{x} ∗ τ_{y}.

It easily implies that, for any λ ∈ Λ, the element

Θ(λ) := τ_{λ_{1}} ∗ τ_{λ_{2}}^{−1} ∈ H(G, 1_{U_{1}})

where λ = λ_{1}λ_{2}^{−1} with λ_{i} ∈ Λ^{−−} is independent of the choice of λ_{1} and λ_{2}. Moreover Bernstein has shown that the map

\begin{matrix}Θ : K[Λ] & ⟶ & H(G, 1_{U_{1}}) \\ λ & ⟼ & Θ(λ) \end{matrix}

is an embedding of K-algebras.

Comment: It is more traditional (cf. [HKP] §1) to consider the embedding of K_{q}-algebras (with K_{q}/K and q^{1/2} ∈ K_{q} be as before)

\begin{matrix}Θ^{norm} : K_{q}[Λ] & ⟶ & H(G, 1_{U_{1}}) ⊗_{K} K_{q} \\ λ & ⟼ & δ^{−1/2}(λ)τ_{λ_{1}} ∗ τ_{λ_{2}}^{−1} \end{matrix}

where λ = λ_{1}λ_{2}^{−1} with dominant λ_{i}. The modified map Θ^{+} := δ^{1/2} · Θ^{norm} already is defined over K. On K[Λ] we have the involution ι_{λ} defined by ι_{Λ}(λ) := λ^{−1}, and on H(G, 1_{U_{1}}) there is the anti-involution ι defined by ι(ψ)(g) := ψ(g^{−1}). We then have

Θ = ι ∘ Θ^{+} ∘ ι_{Λ} .

In the following we consider the renormalized embedding of K-algebras

\begin{matrix}Θ_{ξ} : K[Λ] & ⟶ & H(G, 1_{U_{1}}) \\ λ & ⟼ & π_{L}^{− val_{L} ∘ξ(λ)}Θ(λ) . \end{matrix}

In order to compute the norm induced, via Θ_{ξ}, by ∥ ∥_{ξ} on K[Λ] we introduce the elements

θ_{x} := q^{(ℓ(x)−ℓ(w)−ℓ(λ_{1})+ℓ(λ_{2}))/2}τ_{w} ∗ τ_{λ_{1}} ∗ τ_{λ_{2}}^{−1} .

for any x ∈ W̃ written as x = wλ_{1}λ_{2}^{−1} with w ∈ W and λ_{i} ∈ Λ^{−−}. Since ℓ(w) + ℓ(λ_{1}) = ℓ(wλ_{1}) (cf. [Vig] App.) we obtain from (1) that

θ_{x} = q^{(ℓ(wλ_{1}λ_{2}^{−1})−ℓ(wλ_{1})+ℓ(λ_{2}))/2}τ_{wλ_{1}} ∗ τ_{λ_{2}}^{−1} .

On the other hand [Vig] Lemma 1.2 (compare also [Hai] Prop. 5.4) says that, for any x, y ∈ W̃, the number

(ℓ(xy^{−1}) − ℓ(x) + ℓ(y))/2

is an integer between 0 and ℓ(y) and that

τ_{x} ∗ τ_{y}^{−1} = q^{−(ℓ(xy^{−1})−ℓ(x)+ℓ(y))/2}(τ_{xy^{−1}} + Q_{x,y})

where Q_{x,y} is a linear combination with integer coefficients of τ_{z} with z < xy^{−1}. It follows that for any x ∈ W̃ we have

(2)  θ_{x} = τ_{x} + Q_{x}

where Q_{x} is a linear combination with integer coefficients of τ_{z} with z < x.

Lemma 4.1: Consider two elements x = w′λ and y = v′μ in W̃ where w′, v′ ∈ W and λ, μ ∈ Λ; let w, v ∈ W such that ^{w}λ,^{v}μ ∈ Λ^{−−}; if x ≤ y then we have:

i. ^{v}μ −^{w}λ ∈ ∑_{α∈Φ^{+}} IN_{0} · (−α̌);

ii. ∥τ_{x}∥_{ξ} ≤ ∥τ_{y}∥_{ξ}.

Proof: i. Let w_{0} ∈ W denote the longest element. We will make use of the identity

{x′ ∈ W̃ : x′ ≤ w_{0}(^{w_{0}v}μ)} = ⋃_{λ′} W λ′W

where λ′ ranges over all elements in Λ^{−−} such that ^{v}μ − λ′ ∈ ∑_{α∈Φ^{+}} IN_{0} · (−α̌) (see [Ka2] (4.6) or [HKP] 7.8). Since y ∈ W (^{v}μ)W this identity implies first that x ≤ y ≤ (^{w_{0}v}μ)w_{0} and then that x ∈ W λ′W for some λ′ ∈ Λ^{−−} such that vμ−λ′ ∈ ∑_{α∈Φ^{+}} IN_{0} ·(−α̌). Obviously we must have λ′ = ^{v}λ. ii. Let λ = λ(t_{1}) and μ = λ(t_{2}). We have ∥τ_{x}∥_{ξ} = |ξ(^{w}t_{1})| and ∥τ_{y}∥_{ξ} = |ξ(^{v}t_{2})|. Since highest weights are dominant we obtain from i. that |ξ(^{v}t_{2}(^{w}t_{1})^{−1})| ≥ 1.

It follows from Lemma 4.1.ii and formula (2) that Lemma 2.5 is applicable showing that {θ_{x}}_{x∈_{W}} is another ∥ ∥_{ξ}-orthogonal basis of H(G, 1_{U_{1}}) with

∥θ_{x}∥_{ξ} = ∥τ_{x}∥_{ξ} .

For any λ(t) = λ = λ_{1}λ_{2}^{−1} ∈ Λ with λ_{i} ∈ Λ^{−−} we have

θ_{λ} = q^{(ℓ(λ)−ℓ(λ_{1})+ℓ(λ_{2}))/2}π_{L}^{val_{L}(ξ(t))}Θ_{ξ}(λ)

and

∥θ_{λ}∥_{ξ} = ∥τ_{λ}∥_{ξ} = |ξ(^{w}t)|

where w ∈ W such that ^{w}t ∈ T^{−−}. In particular {θ_{λ}}_{λ∈Λ} is a ∥ ∥_{ξ}-orthogonal basis of im(Θ_{ξ}).

Lemma 4.2: With the above notations we have

q^{−(ℓ(λ)−ℓ(λ_{1})+ℓ(λ_{2}))/2} = \frac{δ^{1/2}(^{w}λ)}{δ^{1/2}(λ)} .

Proof: Write t = t_{1}t_{2}^{−1} with λ(t_{i}) = λ_{i}. According to the explicit formula for the length ℓ in [Vig] App. we have

q^{ℓ(λ)} = ∏_{α∈Φ^{+},|α(t)|_{L}≥1} |α(t)|_{L} · ∏_{α∈Φ^{+},|α(t)|_{L}≤1} |α(t)|_{L}^{−1}

and

q^{ℓ(λ_{i})} = ∏_{α∈Φ^{+}} |α(t_{i})|_{L} .

It follows that

q^{−(ℓ(λ)−ℓ(λ_{1})+ℓ(λ_{2}))/2} = ∏_{α∈Φ^{+},|α(t)|_{L}≤1} |α(t)|_{L}

Since ^{w}t ∈ T^{−−} we have |^{w^{−1}}α(t)|_{L} ≥ 1 for any α ∈ Φ^{+}. Hence {α ∈ Φ^{+} : |α(t)|_{L} < 1} ⊆ Φ^{+} \^{w^{−1}}Φ^{+}. By the last formula in Example 2 of section 2 the above right hand side therefore is equal to \frac{δ^{1/2}(^{w}λ)}{δ^{1/2}(λ)}.

It readily follows that

∥Θ_{ξ}(λ)∥_{ξ} = |γ_{ξ}^{dom}(λ)| for any λ ∈ Λ .

In other words

Θ_{ξ} : (K[Λ], ∥ ∥_{γ_{ξ}}) ⟶ (H(G, 1_{U_{1}}), ∥ ∥_{ξ})

is an isometric embedding. Combining all this with Lemma 3.3 we obtain the following result.

Proposition 4.3: i. The map

\begin{matrix}K⟨Λ; γ_{ξ}⟩ & ⟶ & B(G, ρ|U_{1}) \\ λ & ⟼ & Θ_{ξ}(λ) · χ_{un}^{−1}ρ \end{matrix}

is an isometric embedding of Banach algebras;

ii. the map

\begin{matrix}H(U_{0}, 1_{U_{1}}) ⊗_{K} K⟨Λ; γ_{ξ}⟩ & ⟶^{≅} & B(G, ρ|U_{1}) \\ τ_{w} ⊗ λ & ⟼ & (τ_{w} ∗ Θ_{ξ}(λ)) · χ_{un}^{−1}ρ \end{matrix}

is a K-linear isomorphism.

Remarks: 1) A related computation in the case ξ = 1 is contained in [Vig] Thm. 4(suite).

2) It is worth observing that the “twisted” W -action on K⟨Λ; γ_{ξ}⟩ corresponds under the isomorphism Θ_{ξ} to the W -action on im(Θ_{ξ}) given by

(w, θ_{λ}) ⟼ θ_{^{w}λ} .

The results of this section and of the previous section are compatible in the following sense.

Proposition 4.4: The diagram

\xymatrix{K⟨Λ; γ_{ξ}⟩ \ar[r]^{Θ_{ξ}(.)·χ_{un}^{−1}ρ} & B(G, ρ|U_{1}) \ar[d]^{(ψ_{λ(1)}·χ_{un}^{−1}ρ)∗.} \\ K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} \ar[u]^{⊆} \ar[r]^{S_{ξ}^{−1}(.)·χ_{un}^{−1}ρ} & B(G, ρ|U_{0})}

is commutative. Moreover, the image of K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}} under the map Θ_{ξ}(.) · χ_{un}^{−1}ρ lies in the center of B(G, ρ|U_{1}).

Proof: We recall that the upper, resp. lower, horizontal arrow is an isometric unital monomorphism by Prop. 4.3.i, resp. by Lemma 3.3 and Prop. 3.5. The right perpendicular arrow is a continuous linear map respecting the unit elements. It suffices to treat the case of the trivial representation ρ = 1. By continuity we therefore are reduced to establishing the commutativity of the diagram

\xymatrix{K[Λ] \ar[r]^{Θ} & H(G, 1_{U_{1}}) \ar[d]^{ψ_{λ(1)}∗.} \\ K[Λ]^{W,γ_{ξ}} \ar[u]^{⊆} \ar[r]^{S_{1}^{−1}} & H(G, 1_{U_{0}})}

as well as the inclusion

Θ(K[Λ]^{W,γ_{ξ}}) ⊆ center of H(G, 1_{U_{1}}) .

It is known (cf. [HKP] Lemma 2.3.1, section 4.6, and Lemma 3.1.1) that:

– Θ^{norm}(K_{q}[Λ]^{W}) = center of H(G, 1_{U_{1}}) ⊗_{K} K_{q};

– ψ_{λ(1)} ∗ Θ^{norm} ∘ S^{norm} = id on H(G, 1_{U_{0}}) ⊗_{K} K_{q};

– Θ^{norm} = ι ∘ Θ^{norm} ∘ ι_{Λ} on K_{q}[Λ]^{W}. The first identity implies the asserted inclusion. We further deduce that

Θ ∘ S_{1} = ι ∘ (δ^{1/2} · Θ^{norm}) ∘ ι_{Λ} ∘ (δ^{1/2} · S^{norm})
= ι ∘ Θ^{norm} ∘ ι_{Λ} ∘ S^{norm}
= Θ^{norm} ∘ S^{norm} on H(G, 1_{U_{0}}) ⊗_{K} K_{q}

and hence that

ψ_{λ(1)} ∗ (Θ ∘ S_{1}) = id on H(G, 1_{U_{0}}) .

5. Crystalline Galois representations

We go back to the example of the group G := GL_{d+1}(L) which we have discussed already at the end of section 3. But we now want to exploit Lemma 2.7. As before we fix a dominant integral weight ξ ∈ X^{∗}(T) that is given by

( \begin{matrix}g_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & g_{d+1} \end{matrix} ) ⟼ ∏_{i=1}^{d+1} g_{i}^{a_{i}}

with (a_{1}, …, a_{d+1}) ∈ ZZ^{d+1}. Note that the dominance means that

a_{1} ≤ … ≤ a_{d+1} .

Equally as before we use the coordinates

\begin{matrix}T′(K) & ⟶ & (K^{×})^{d+1} \\ ζ & ⟼ & (ζ_{1}, …, ζ_{d+1}) with ζ_{i} := q^{i−1}π_{L}^{a_{i}}ζ(λ_{\{i\}}) \end{matrix}

on the dual torus. Some times we view ζ as the diagonal matrix in GL_{d+1}(K) with diagonal entries (ζ_{1}, …, ζ_{d+1}). On the other hand, on the root space we use the coordinates

V_{IR} = Hom(Λ, IR) ⟶ IR^{d+1}
z ⟼ (z_{1}, …, z_{d+1}) with z_{i} := z(λ_{{i}}) .

In these coordinates we have:

1) The points η_{L} and ξ_{L} from Example 3 in section 2 correspond to

\frac{[L : Q_{p}]}{2}(−d, −(d − 2), …, d − 2, d) and (a_{1}, …, a_{d+1}) ,

respectively.

2) The map val : T′(K) ⟶ V_{IR} corresponds to the map

(K^{×})^{d+1} ⟶ IR^{d+1}
(ζ_{1}, …, ζ_{d+1}) ⟼ (val_{L}(ζ_{1}), …, val_{L}(ζ_{d+1})) − ξ_{L} − η̃_{L} .

where

η̃_{L} := [L : Q_{p}](0, 1, …, d) = η_{L} + \frac{[L : Q_{p}]}{2}(d, …, d) .

3) On IR^{d+1} the partial order defined by Φ^{+} is given by

(z_{1}, …, z_{d+1}) ≤ (z′_{1}, …, z′_{d+1})

if and only if

z_{d+1} ≤ z′_{d+1} , z_{d} + z_{d+1} ≤ z′_{d} + z′_{d+1} , …, z_{2} + … + z_{d+1} ≤ z′_{2} + … + z′_{d+1}

and

z_{1} + … + z_{d+1} = z′_{1} + … + z′_{d+1} .

4) The map z ⟼ z^{dom} corresponds in IR^{d+1} to the map which rearranges the coordinates in increasing order and which we will also denote by (.)^{dom}.

It now is a straightforward computation to show that Lemma 2.7 amounts to

T′_{ξ} = {ζ ∈ T′ : (val_{L}(ζ_{1}), …, val_{L}(ζ_{d+1}))^{dom} ≤ ξ_{L} + η̃_{L}}.

Even more explicitly, T′_{ξ} is the domain of all (ζ_{1}, …, ζ_{d+1}) ∈ (K^{×})^{d+1} such that

(val_{L}(ζ_{1}), …, val_{L}(ζ_{d+1}))^{dom} ≤ (a_{1}, a_{2} + [L : Q_{p}], …, a_{d+1} + d[L : Q_{p}]) .

For any increasing sequence \underline{r} = (r_{1} ≤ … ≤ r_{d+1}) of real numbers we denote by P(\underline{r}) the convex polygon in the plane through the points

(0, 0), (1, r_{1}), (2, r_{1} + r_{2}), …, (d + 1, r_{1} + … + r_{d+1}) .

We then may reformulate the above description of T′_{ξ} as follows.

Lemma 5.1: T′_{ξ} is the subdomain of all ζ ∈ T′ such that P(val(ζ)^{dom}) lies above P(ξ_{L} + η̃_{L}) and both polygons have the same endpoint.

We recall that a filtered K-isocrystal is a triple \underline{D} = (D, φ, F il^{·}D) consisting of a finite dimensional K-vector space D, a K-linear automorphism φ of D – the “Frobenius” – , and an exhaustive and separated decreasing filtration F il^{·}D on D by K-subspaces. In the following we fix the dimension of D to be equal to d + 1 and, in fact, the vector space D to be the d + 1-dimensional standard vector space D = K^{d+1}. We then may think of φ as being an element in the group G′(K) := GL_{d+1}(K). The (filtration) type type(\underline{D}) ∈ ZZ^{d+1} is the sequence (b_{1}, …, b_{d+1}), written in increasing order, of the break points b of the filtration F il^{·}D each repeated dim_{K} gr^{b}D many times. We put

t_{H}(\underline{D}) := ∑_{b∈Z Z} b · dim_{K} gr^{b} D .

Then (d + 1, t_{H}(\underline{D})) is the endpoint of the polygon P(type(\underline{D})). On the other hand we define the Frobenius type s(\underline{D}) of \underline{D} to be the conjugacy class of the semisimple part of φ in G′(K). We put

t_{N}^{L}(\underline{D}) := val_{L}(det_{K}(φ)) .

The filtered K-isocrystal \underline{D} is called weakly L-admissible if t_{H}(\underline{D}) = t_{N}^{L}(\underline{D}) and t_{H}(\underline{D}′) ≤ t_{N}^{L}(\underline{D}′) for any filtered K-isocrystal \underline{D}′ corresponding to a φinvariant K-subspace D′ ⊆ D with the induced filtration.

Proposition 5.2: Let ζ ∈ T′(K) and let ξ be a dominant integral weight of G; then ζ ∈ T′_{ξ}(K) if and only if there is a weakly L-admissible filtered K-isocrystal \underline{D} such that type(\underline{D}) = ξ_{L} + η̃_{L} and ζ ∈ s(\underline{D}).

Proof: Let us first suppose that there exists a filtered K-isocrystal \underline{D} with the asserted properties. Then P(type(\underline{D})) = P(ξ_{L} +η̃_{L}) is the Hodge polygon of \underline{D} and P(val(ζ)^{dom}) is its Newton polygon (relative to val_{L}). By [Fon] Prop. 4.3.3 (the additional assumptions imposed there on the field K are irrelevant at this point) the weak admissibility of \underline{D} implies that its Newton polygon lies above its Hodge polygon with both having the same endpoint. Lemma 5.1 therefore implies that ζ ∈ T′_{ξ}(K).

We now assume vice versa that ζ ∈ T′_{ξ}(K). We let φ_{ss} be the semisimple automorphism of the standard vector space D given by the diagonal matrix with diagonal entries (ζ_{1}, …, ζ_{d+1}). Let

D = D_{1} + … + D_{m}

be the decomposition of D into the eigenspaces of φ_{ss}. We now choose the Frobenius φ on D in such a way that φ_{ss} is the semisimple part of φ and that any D_{j} is φ-indecomposable. In this situation D has only finitely many φ-invariant subspaces D′ and each of them is of the form

D′ = D′_{1} + … + D′_{m}

with D′_{j} one of the finitely many φ-invariant subspaces of D_{j}. By construction the Newton polygon of (D, φ) is equal to P(val(ζ)^{dom}). To begin with consider any filtration F il^{·}D of type ξ_{L} + η̃_{L} on D and put \underline{D} := (D, φ, F il^{·}D). The corresponding Hodge polygon then is P(ξ_{L} + η̃_{L}). By Lemma 5.1 the first polygon lies above the second and both have the same endpoint. The latter already says that

t_{H}(\underline{D}) = t_{N}^{L}(\underline{D}) .

It remains to be seen that we can choose the filtration F il^{·}D in such a way that t_{H}(\underline{D}′) ≤ t_{N}^{L}(\underline{D}′) holds true for any of the above finitely many φ-invariant subspaces D′ ⊆ D. The inequality between the two polygons which we have does imply that

a_{1} + (a_{2} + [L : Q_{p}]) + … + (a_{dimD′} + (dim D′ − 1)[L : Q_{p}]) ≤ t_{N}^{L}(\underline{D}′) .

Hence it suffices to find the filtration in such a way that we have

t_{H}(\underline{D}′) ≤ a_{1} + (a_{2} + [L : Q_{p}]) + … + (a_{dimD′} + (dim D′ − 1)[L : Q_{p}])

for any D′. But it is clear that for any filtration (of type ξ_{L} + η̃_{L}) in general position we actually have

t_{H}(\underline{D}′) = a_{1} + (a_{2} + [L : Q_{p}]) + … + (a_{dimD′} + (dim D′ − 1)[L : Q_{p}])

for the finitely many D′.

In order to connect this to Galois representations we have to begin with a different kind of filtered isocrystal (cf. [BM] §3.1). First of all we now suppose that K is a finite extension of Q_{p} (as always containing L). Then a filtered isocrystal over L with coefficients in K is a triple \underline{M} = (M, ϕ, F il^{·}M_{L}) consisting of a free L_{0} ⊗_{Q_{p}} K-module M of finite rank, a σ-linear automorphism ϕ of M – the “Frobenius” – , and an exhaustive and separated decreasing filtration F il^{·}M_{L} on M_{L} := L ⊗_{L_{0}} M by L ⊗_{Q_{p}} K-submodules. Here L_{0} denotes the maximal unramified subextension of L and σ its Frobenius automorphism. By abuse of notation we also write σ for the automorphism σ ⊗ id of L_{0} ⊗_{Q_{p}} K. We put

t_{H}(\underline{M}) := ∑_{b∈Z Z} b · dim_{L} gr^{b} M_{L} = [K : L] · ∑_{b∈Z Z} b · dim_{K} gr^{b} M_{L} .

The equality is a consequence of the fact that for any finitely generated L⊗_{Q_{p}}Kmodule M′ the identity

dim_{L}M′ = [K : L] · dim_{K}M′

holds true. By semisimplicity this needs to be verified only for a simple module which must be isomorphic to a field into which L and K both can be embedded and in which case this identity is obvious.

The number t_{N}(\underline{M}) is defined as val_{Q_{p}}(ϕ(x)/x) where x is an arbitrary nonzero element in the maximal exterior power of M as an L_{0}-vector space. But we have

t_{N}(\underline{M}) = val_{Q_{p}}(ϕ(x)/x)

= \frac{1}{[L_{0} : Q_{p}]} · val_{Q_{p}}(det_{L_{0}}(ϕ^{[L_{0}:Q_{p}]}))

= \frac{1}{[L_{0} : Q_{p}]} · val_{Q_{p}}(Norm_{K/L_{0}}(det_{K}(ϕ^{[L_{0}:Q_{p}]})))

= val_{Q_{p}}(Norm_{K/L_{0}}(det_{K}(ϕ)))

= [K : L_{0}] · val_{Q_{p}}(det_{K}(ϕ))

= [K : L] · val_{L}(det_{K}(ϕ)) .

The filtered isocrystal \underline{M} over L with coefficients in K is called weakly admissible (cf. [BM] Prop. 3.1.1.5) if t_{H}(\underline{M}) = t_{N}(\underline{M}) and t_{H}(\underline{M}′) ≤ t_{N}(\underline{M}′) for any subobject \underline{M}′ of \underline{M} corresponding to a ϕ-invariant L_{0} ⊗_{Q_{p}} K-submodule M′ ⊆ M with the induced filtration on L ⊗_{L_{0}} M′.

By the main result of [CF] there is a natural equivalence of categories V ⟼ D_{cris}(V) between the category of K-linear crystalline representations of the absolute Galois group Gal(\overline{L}/L) of the field L and the category of weakly admissible filtered isocrystals over L with coefficients in K. It has the property that

dim_{K} V = rank_{L_{0}⊗_{Q_{p}} K} D_{cris}(V) .

To avoid confusion we recall that a K-linear Galois representation is called crystalline if it is crystalline as a Q_{p}-linear representation. We also recall that the jump indices of the filtration on D_{cris}(V)_{L} are called the Hodge-Tate coweights of the crystalline Galois representation V (they are the negatives of the Hodge-Tate weights). Moreover, we will say that V is K-split if all eigenvalues of the Frobenius on D_{cris}(V) are contained in K. This is a small technical condition which always can be achieved by extending the coefficient field K. More important is the following additional condition. We let C_{p} denote the completion of the algebraic closure \overline{L}. We may view V as an L-vector space through the inclusion L ⊆ K.

Definition: A K-linear crystalline representation V of Gal(\overline{L}/L) is called special if the kernel of the natural map C_{p} ⊗_{Q_{p}} V ⟶ C_{p} ⊗_{L} V is generated, as a C_{p}-vector space, by its Gal(\overline{L}/L)-invariants (for the diagonal action).

On the full subcategory of special crystalline Galois representations we have a simplified form of the above equivalence of categories. This is well known (see [FR] Remark 0.3). But since we have not found any details in the literature we include them here for the convenience of the reader. We will speak of a K-isocrystal and an isocrystal over L with coefficients in K, respectively, if no filtration is prescribed. Suppose that (M, ϕ) is an isocrystal over L with coefficients in K. We then have the L_{0}-isotypic decomposition

M = ⊕_{τ∈Δ} M_{τ}

where Δ := Gal(L_{0}/Q_{p}) and where M_{τ} is the K-subspace of M on which L_{0} acts via the embedding τ : L_{0} ↪ K. One has

ϕ(M_{τ}) = M_{τσ^{−1}}

so that ϕ^{f} with f := |Δ| is an L_{0} ⊗_{Q_{p}} K-linear automorphism of M which respects the above decomposition. We see that (M_{1}, ϕ^{f}|M_{1}) is a K-isocrystal with dim_{K} M_{1} = rank_{L_{0}⊗_{Q_{p}} K}M.

Lemma 5.3: The functor

category of isocrystals over L ⟶^{∼} category of K-isocrystals

with coefficients in K

(M, ϕ) ⟼ (M_{1}, ϕ^{f}|M_{1})

is an equivalence of categories.

Proof: Let I denote the functor in question. To define a functor J in the opposite direction let (D, φ) be a K-isocrystal. We put M := L_{0} ⊗_{Q_{p}} D and ϕ := (σ ⊗ 1) ∘ ϕ′ with

ϕ′|M_{τ} := \{ \begin{matrix}φ & if τ = 1, \\ id & otherwise. \end{matrix}

Here we have used the K-linear composed isomorphism

D ⟶ L_{0} ⊗_{Q_{p}} D = M ⟶^{pr} M_{1}

to transport φ from D to M_{1}. At the same time it provides a natural isomorphism id ≃ I ∘ J. The opposite natural isomorphism id ≃ J ∘ I is given by the composed maps

M_{σ^{i}} ⟶^{ϕ^{i}} M_{1} ⟶^{≅}(L_{0} ⊗_{Q_{p}} M_{1})_{1} ⟶^{σ^{−i}⊗ϕ^{−f}}(L_{0} ⊗_{Q_{p}} M_{1})_{σ^{i}}

for 0 ≤ i ≤ f − 1.

Suppose now that M_{L} carries a filtration F il^{·}M_{L} making \underline{M} := (M, ϕ, F il^{·}M_{L}) into a filtered isocrystal over L with coefficients in K. Let

M_{L} = ⊕_{β} M_{L,β}

where β runs over the Gal(\overline{K}/K)-orbits in Hom_{Q_{p}}(L, \overline{K}) be the L-isotypic decomposition of the L ⊗_{Q_{p}} K-module M_{L}. The filtration on M_{L} induces a filtration F il^{·}M_{L,β} on each M_{L,β} and by the naturality of the decomposition we have

F il^{·}M_{L} = ⊕_{β} F il^{·}M_{L,β} .

Moreover, let β_{0} denote the orbit of the inclusion map L ⊆ K. Then M_{L,β_{0}} is the K-subspace of M_{L} on which L acts through the inclusion L ⊆ K. The composite map

M_{1} ⟶^{⊆} M ⟶ L ⊗_{L_{0}} M = M_{L} ⟶^{pr} M_{L,β_{0}}

is a K-linear isomorphism which we may use to transport the filtration F il^{·}M_{L,β_{0}} to a filtration F il^{·}M_{1} on M_{1}. In this way we obtain the filtered K-isocrystal \underline{D} := (M_{1}, ϕ^{f}|M_{1}, F il^{·}M_{1}). Obviously the full original filtration F il^{·}M_{L} can be recovered from F il^{·}M_{1} if and only if it satisfies

(∗)  gr^{0}M_{L,β} = M_{L,β}  for any β ≠ β_{0} .

Let us suppose that the condition (∗) is satisfied. Since gr^{0}, by definition, does not contribute to the number t_{H}(.) we obviously have

t_{H}(\underline{M}) = [K : L] · t_{H}(\underline{D}) .

On the other hand, using a normal basis of L_{0} over Q_{p} as well as the inverse functor in the proof of Lemma 5.3, we compute

t_{N}(\underline{M}) = [K : L] · val_{L}(det_{K}(ϕ))

= [K : L] · val_{L}(det_{K}((σ ⊗ 1) ∘ (ϕ^{f}|M_{1} ⊕ id_{M_{σ}} ⊕ … ⊕ id_{M_{σ^{f−1}}} )))

= [K : L] · val_{L}(det_{K}(ϕ^{f}|M_{1}))

= [K : L] · t_{N}^{L}(\underline{D}) .

With \underline{M} any of its subobjects also satisfies the condition (∗). Moreover, by Lemma 5.3, the subobjects of \underline{M} are in one to one correspondence with the subobjects of \underline{D}). It follows that \underline{M} is weakly admissible if and only if \underline{D} is weakly L-admissible. Hence we have the induced equivalence of categories

category of weakly admissible  category of weakly
filtered isocrystals over L with ⟶^{∼} L-admissible filtered
coefficients in K satisfying (∗)  K-isocrystals.

Suppose now that \underline{M} = D_{cris}(V) of some K-linear crystalline representation of Gal(\overline{L}/L). By the general theory of crystalline Galois representations we have the comparison isomorphism

ker(C_{p} ⊗_{Q_{p}} V ⟶ C_{p} ⊗_{L} V) ≅ ⊕_{i∈Z Z} (C_{p}(−i) ⊗_{L} ( ⊕_{β≠β_{0}} gr^{i}M_{L,β})) .

It is Galois equivariant with Gal(\overline{L}/L) acting diagonally on the left and through the first factors on the right. For the Galois invariants we therefore obtain the formula

ker(C_{p} ⊗_{Q_{p}} V ⟶ C_{p} ⊗_{L} V)^{Gal(\overline{L}/L)} ≅ ⊕_{β≠β_{0}} gr^{0}M_{L,β} .

It follows that the isocrystal D_{cris}(V) satisfies the condition (∗) if and only if the crystalline Galois representation V is special. Altogether we obtain that the functor V ⟼ D_{cris}(V)_{1} induces an equivalence of categories

category of K-linear special  category of weakly
crystalline representations ⟶^{∼} L-admissible filtered
of Gal(\overline{L}/L)  K-isocrystals.

It satisfies

dim_{K} V = dim_{K} D_{cris}(V)_{1} .

Finally suppose that V is a

(+)

(d + 1)-dimensional K-linear K-split special crystalline representation of Gal(\overline{L}/L) all of whose Hodge-Tate coweights have multiplicity one and increase at least by [L : Q_{p}] in each step.

Precisely in this situation there is a dominant integral ξ = (a_{1}, …, a_{d+1}) such that the Hodge-Tate coweights of V are ξ_{L} + η̃_{L}. By Prop. 5.2 we find an up to permutation unique point ζ ∈ T′_{ξ}(K) such that ζ ∈ s(D_{cris}(V)). This means we have constructed a surjection

set of isomorphism classes of V ’s with (+) ⟶ ⋃_{ξ dominant}^{·} W \T′_{ξ}(K) .

Let us again fix a dominant ξ = (a_{1}, …, a_{d+1}) and let ρ_{ξ} denote the irreducible rational representation of G of highest weight ξ. By Prop. 2.4 and Cor. 3.6 we have an identification

W \T′_{ξ}(K) ⊆ (W \T′_{ξ})(K) ≃ Sp(B(G, ρ_{ξ}|U_{0}))(K)

where Sp(B(G, ρ|U_{0}))(K) the space of K-rational points of the affinoid variety B(G, ρ_{ξ}|U_{0}), i.e., the space of K-valued characters of the Banach-Hecke algebra B(G, ρ_{ξ}|U_{0}). Our map therefore becomes a map

set of isomorphism classes of (d + 1)-dimensional K-linear K-split

special crystalline representations of ⟶ Sp(B(G, ρ_{ξ}|U_{0}))(K)

Gal(\overline{L}/L) with Hodge-Tate coweights (a_{1}, a_{2} + [L : Q_{p}], …, a_{d+1} + d[L : Q_{p}])

which we write as V ⟼ ζ(V). We point out that in this form our map is canonical in the sense that it does not depend on the choice of the prime element π_{L}: This choice entered into our normalization of the Satake map S_{ξ} and into the coordinates on T′ which we used; it is easy to check that the two cancel each other out. We also note that in the limit with respect to K this map is surjective.

We finish this section with a speculation in which way the map which we have constructed above might be an approximation of a true p-adic local Langlands correspondence. We view a point ζ ∈ Sp(B(G, ρ_{ξ}|U_{0}))(K) as a character ζ : B(G, ρ_{ξ}|U_{0}) ⟶ K. Correspondingly we let K_{ζ} denote the one dimensional K-vector space on which B(G, ρ_{ξ}|U_{0}) acts through the character ζ. We may “specialize” the “universal” Banach B(G, ρ_{ξ}|U_{0})-module B_{U_{0}}^{G}(ρ_{ξ}|U_{0}) from section 1 to ζ by forming the completed tensor product

B_{ξ,ζ} := K_{ζ} ˆ⊗_{B(G,ρ_{ξ}|U_{0})} B_{U_{0}}^{G}(ρ_{ξ}|U_{0}) .

By construction the K-Banach space B_{ξ,ζ} still carries a continuous and isometric (for the quotient norm) action of G. A future p-adic local Langlands correspondence should provide us with a distinguished correspondence (being essentially bijective) between the fiber of our map in ζ (i.e., all V of the kind under consideration such that ζ(V) = ζ) and the isomorphism classes of all topologically irreducible “quotient” representations of B_{ξ,ζ}. Unfortunately it is not even clear that the Banach spaces B_{ξ,ζ} are nonzero.

In order to describe the existing evidence for this picture we first have to recall how the characters of the Hecke algebra H(G, 1_{U_{0}}) can be visualized representation theoretically. Any element ζ ∈ T′(K) can be viewed as a character ζ : T → Λ → K^{×}, and correspondingly we may form the unramified principal series representation

Ind_{P}^{G}(ζ)^{∞} := space of all locally constant functions F : G ⟶ K such that
F (gtn) = ζ(t)^{−1}F (g) for any g ∈ G, t ∈ T, n ∈ N

of G. The latter is a smooth G-representation of finite length. By the Iwasawa decomposition G = U_{0}P the subspace of U_{0}-invariant elements in Ind_{P}^{G}(ζ)^{∞} is one dimensional so that the action of H(G, 1_{U_{0}}) on it is given by a character ω_{ζ}. On the other hand ζ defines in an obvious way a character of the algebra K[Λ] which we also denote by ζ. Using the Satake isomorphism from section 3 one then has (cf. [Ka1] Lemma 2.4(i))

ω_{ζ} = ζ ∘ S_{1} = (ζ · π_{L}^{− val_{L}(ξ(.))}) ∘ S_{ξ} .

By [Ka1] Thm. 2.7 the “specialization” in ω_{ζ}

H_{1,ζ} := K_{ω_{ζ}} ⊗_{H(G,1_{U_{0}} )} ind_{U_{0}}^{G}(1_{U_{0}}) .

of the “universal” H(G, 1_{U_{0}})-module ind_{U_{0}}^{G}(1_{U_{0}}) from section 1 is an admissible smooth G-representation. Since it also is visibly finitely generated it is, in fact, of finite length. Since ind_{U_{0}}^{G}(1_{U_{0}}) as a G-representation is generated by its U_{0}- fixed vectors the same must hold true for any of its quotient representations, in particular for any quotient of H_{1,ζ}. But the subspace of U_{0}-invariant vectors in H_{1,ζ} is one dimensional. It follows that H_{1,ζ} possesses a single irreducible quotient representation V_{1,ζ} – the so called spherical representation for ζ. One has the G-equivariant map

\begin{matrix}H_{1,ζ} & ⟶ & Ind_{P}^{G}(ζ)^{∞} \\ 1 ⊗ f & ⟼ & f ∗ 1_{ζ} := ∑_{g∈G/U_{0}} f(g)g(1_{ζ}) \end{matrix}

where 1_{ζ} ∈ Ind_{P}^{G}(ζ)^{∞} denotes the unique U_{0}-invariant function with value one in 1 ∈ G. Hence V_{ζ} can also be viewed as the, up to isomorphism, unique irreducible constituent of Ind_{P}^{G}(ζ)^{∞} with a nonzero U_{0}-fixed vector.

Bringing in again the dominant integral weight ξ we have the K-linear isomorphism

\begin{matrix}ind_{U_{0}}^{G}(1_{U_{0}}) ⊗_{K} ρ_{ξ} & ⟶^{≅} & ind_{U_{0}}^{G}(ρ_{ξ}|U_{0}) \\ f ⊗ x & ⟼ & f_{x}(g) := f(g)g^{−1}x . \end{matrix}

It is G-equivariant if, on the left hand side, we let G act diagonally. On the left, resp. right, hand side we also have the action of the Hecke algebra H(G, 1_{U_{0}}) through the first factor, resp. the action of the Hecke algebra H(G, ρ_{ξ}|U_{0}). Relative to the isomorphism ι_{ρ_{ξ}} between these two algebras discussed in section 1 the above map is equivariant for these Hecke algebra actions as well. (Warning: But this map does not respect our norms on both sides.) By abuse of notation we will use the same symbol to denote characters of these two Hecke algebras which correspond to each other under the isomorphism ι_{ρ_{ξ}}. We obtain an induced G-equivariant isomorphism

H_{1,ζ} ⊗_{K} ρ_{ξ} ⟶^{≅} H_{ξ,ζ} := K_{ω_{ζ}} ⊗_{H(G,ρ_{ξ}|U_{0})} ind_{U_{0}}^{G}(ρ_{ξ}|U_{0})

between “specializations”. Since with V_{1,ζ} also

V_{ξ,ζ} := V_{1,ζ} ⊗_{K} ρ_{ξ}

is irreducible as a G-representation ([ST1] Prop. 3.4) we see that V_{ξ,ζ} is the unique irreducible quotient of H_{ξ,ζ} and is also the, up to isomorphism, unique irreducible constituent of Ind_{P}^{G}(ζ)^{∞}⊗_{K}ρ_{ξ} which as a U_{0}-representation contains ρ_{ξ}|U_{0}.

Assuming once more that ζ ∈ T′_{ξ}(K) we, of course, have that

B_{ξ,ζ} = Hausdorff completion of H_{ξ,ζ}

with respect to the quotient seminorm from ind_{U_{0}}^{G}(ρ_{ξ}|U_{0}). We remark that the unit ball in ind_{U_{0}}^{G}(ρ_{ξ}|U_{0}) and a fortiori its image in H_{ξ,ζ} are finitely generated over the group ring o_{K}[G]. Hence in order to prove that the quotient topology on H_{ξ,ζ} is Hausdorff, i.e., that the canonical map H_{ξ,ζ} ⟶ B_{ξ,ζ} is injective it suffices to exhibit some bounded open G-invariant o_{K}-submodule in H_{ξ,ζ}.

Example 1: Let G = GL_{2}(Q_{p}), ξ = (a_{1}, a_{2}) a dominant weight, and ζ = (ζ_{1}, ζ_{2}) ∈ (K^{×})^{2}. By the discussion at the end of section 2 the defining conditions for the affinoid domain T′_{ξ} are

|ζ_{i}| ≤ |p|^{a_{1}} for i = 1, 2 and |ζ_{1}ζ_{2}| = |p|^{a_{1}+a_{2}+1} .

The complete list of the weakly Q_{p}-admissible filtered K-isocrystals with a Frobenius φ whose semisimple part is given by ζ is well known (cf. [BB] end of section 3.1): Up to conjugation we may assume that |ζ_{1}| ≥ |ζ_{2}|. Case 1: |ζ_{1}| = |p|^{a_{1}} and |ζ_{2}| = |p|^{a_{2}+1}; then φ is semisimple, and there are (up to isomorphism) exactly two weakly Q_{p}-admissible filtrations; one corresponds to a decomposable and the other to a reducible but indecomposable Galois representation. Case 2: ζ_{1} ≠ ζ_{2} with |ζ_{i}| < |p|^{a_{1}} for i = 1, 2; then φ is semisimple, and there is (up to isomorphism) exactly one weakly Q_{p}-admissible filtration; it corresponds to an irreducible Galois representation. Case 3: ζ_{1} = ζ_{2} with |ζ_{i}| < |p|^{a_{1}}; then φ is not semisimple, and there is (up to isomorphism) exactly one weakly Q_{p}-admissible filtration; it corresponds to an irreducible Galois representation. In particular, the fiber of our above surjection consists of two elements in case 1 and of one element in cases 2 and 3.

On the other hand for |ζ_{1}| ≥ |ζ_{2}| the map H_{ξ,ζ} ⟶^{≅} Ind_{P}^{G}(ζ)^{∞}⊗_{K} ρ_{ξ} always is an isomorphism. It therefore follows from [BB] Thm. 4.3.1 that our B_{ξ,ζ} coincides in Case 2 with the representation denoted by Π(V) in loc. cit. Moreover, still in Case 2, by [BB] Cor.s 5.4.1/2/3 the representation of G in the Banach space B_{ξ,ζ} is topologically irreducible (in particular nonzero) and admissible in the sense of [ST2] §3. In Case 3 the same assertions are shown in [Bre] Thm. 1.3.3 under the restriction that a_{2} − a_{1} < 2p − 1 and a_{1} + a_{2} ≠ −3 if p ≠ 2, resp. a_{2} − a_{1} < 2 and a_{1} + a_{2} ≠ −1 if p = 2.

We mention that in contrast to B_{ξ,ζ} the representation Ind_{P}^{G}(ζ)^{∞} ⊗_{K} ρ_{ξ} (or equivalently Ind_{P}^{G}(ζ)^{∞}) is irreducible if and only if ζ_{2} ≠ pζ_{1}. Hence reducibility can only occur for a_{1} = a_{2} in Case 1 and for a_{1} < a_{2} in Case 2.

It was Breuil’s fundamental idea that the two dimensional crystalline Galois representations of Gal(\overline{Q}_{p}/Q_{p}) with distinct Hodge-Tate weights should correspond to the Banach representations B_{ξ,ζ} of GL_{2}(Q_{p}). Our general speculation therefore should be seen as an attempt to extend his picture. But we warn the reader that the case of GL_{2} is atypical insofar as in general, given a pair (ξ, ζ), there will be infinitely many possibilities for a weakly admissible filtration.

Example 2. The unit ball ind_{U_{0}}^{G}(1_{U_{0}})^{0} in the normed space ind_{U_{0}}^{G}(1_{U_{0}}) is a module for the unit ball H(G, 1_{U_{0}})^{0} in the Hecke algebra H(G, 1_{U_{0}}) (for the sup-norm in both cases). For the two groups G = GL_{2}(L) and G = GL_{3}(L) it is known that ind_{U_{0}}^{G}(1_{U_{0}})^{0} is free as an H(G, 1_{U_{0}})^{0}-module. For G = GL_{2}(L) this is a rather elementary computation on the tree and for G = GL_{3}(L) it is the main result in [BO] Thm. 3.2.4 (see also the paragraph after Thm. 1.5; we point out that the arguments in this paper actually prove freeness and not only flatness). Let {b_{j}}_{j∈IN} be a basis. Then {1 ⊗ b_{j}}_{j} is a basis of H_{1,ζ} as a K-vector space, and ∑_{j} o_{K} · (1 ⊗ b_{j}) is open in H_{1,ζ} for the quotient topology provided ζ ∈ T′_{1}(K). Hence the quotient topology on H_{1,ζ} is Hausdorff which means that the natural map H_{1,ζ} ⟶ B_{1,ζ} is injective. In particular, B_{1,ζ} is nonzero.

Example 3: Let G = GL_{d+1}(L) be general but assume that ζ ∈ Hom(Λ, o_{K}^{×}) ⊆ T′(K). Then, for any element F ∈ Ind_{P}^{G}(ζ)^{∞} the function |F |(g) := |F (g)| is right P -invariant. Since G/P is compact we therefore may equip Ind_{P}^{G}(ζ)^{∞} with the G-invariant norm

∥F ∥ := sup_{g∈G} |F |(g) .

Moreover, our above map

ind_{U_{0}}^{G}(1_{U_{0}}) ⟶ H_{1,ζ} ⟶ Ind_{P}^{G}(ζ)^{∞}

then is continuous. Assuming in addition that ζ ∈ T′_{ξ}(K) we obtain by completion a G-equivariant continuous K-linear map

B_{1,ζ} ⟶ Ind_{P}^{G}(ζ)^{c} .

The completion Ind_{P}^{G}(ζ)^{c} of Ind_{P}^{G}(ζ)^{∞} is explicitly given by

Ind_{P}^{G}(ζ)^{c} := space of all continuous functions F : G ⟶ K such that
F (gtn) = ζ(t)^{−1}F (g) for any g ∈ G, t ∈ T, n ∈ N

It is easy to show that Ind_{P}^{G}(ζ)^{c} as a representation of G in a K-Banach space is admissible.

Conjecture: If ζ is regular then the representation of G in the Banach space Ind_{P}^{G}(ζ)^{c} is topologically irreducible.

Suppose therefore that ζ is regular, i.e., is not fixed by any 1 ≠ w ∈ W for the conjugation action of W on T′). It is then well known that:

– The smooth G-representation Ind_{P}^{G}(ζ)^{∞} is irreducible (for example by the Bernstein-Zelevinsky classification).

– The above map H_{1,ζ} ⟶^{≅} Ind_{P}^{G}(ζ)^{∞} is an isomorphism ([Ka1] Thm. 3.2 and Remark 3.3 or [Dat] Lemma 3.1). The latter in particular implies that the quotient topology on H_{1,ζ} is Hausdorff and that the map B_{1,ζ} ⟶ Ind_{P}^{G}(ζ)^{c} has dense image. In this context we also mention, without proof, the following result.

Proposition 5.4: For any two ζ, ζ′ ∈ Hom(Λ, o_{K}^{×}) the vector space of all Gequivariant continuous linear maps from Ind_{P}^{G}(ζ)^{c} to Ind_{P}^{G}(ζ′)^{c} is zero if ζ ≠ ζ′ and is K · id if ζ = ζ′.

For G = GL_{2}(Q_{p}) the above conjecture follows from a combination of [ST1] §4 and [ST3] Thm. 7.1. If ζ = ζ_{(1)} ⊗ … ⊗ ζ_{(d+1)} with unramified characters ζ_{(i)} : L^{×}/o_{L}^{×} ⟶ o_{K}^{×} such that ζ_{(i)} ≢ ζ_{(i+1)} mod π_{K} for any 1 ≤ i ≤ d then the above conjecture is a consequence of an irreducibility result in characteristic p in the thesis of R. Ollivier.

6. Weakly admissible pairs and functoriality

In the traditional Langlands program the irreducible smooth representations of a general group G over L are put into correspondence with continuous homomorphisms from the Galois group Gal(\overline{L}/L) (or rather the Weil-Deligne group of L) into the Langlands dual group G′ of G. In order to do something in this spirit in our setting it is useful to slightly change our point of view which we motivate by looking once again at the GL_{d+1}-case. We started from a dominant weight ξ ∈ X^{∗}(T) and an element ζ ∈ T′(K) in the dual torus. Viewing ζ, by our particular choice of coordinates, as a diagonal matrix ζ_{c} in G′(K) = GL_{d+1}(K) we considered the K-isocrystals (K^{d+1}, φ) such that ζ_{c} lies in the conjugacy class of the semisimple part of φ. The weight ξ was used to prescribe the type of the filtration which would make these isocrystals into filtered isocrystals. Our basic result then was that among all these filtered K-isocrystals there is at least one weakly L-admissible one if and only if ζ ∈ T′_{ξ}(K). Now we observe that ξ actually can be used to define a model filtration on K^{d+1}. Quite generally, for any K-rational cocharacter ν : G_{m} ⟶ G′ we decompose K^{d+1} into weight spaces

K^{d+1} = ⊕_{i∈Z Z}(K^{d+1})_{i}

with respect to ν and put

F il_{ν}^{i} K^{d+1} := ⊕_{j≥i}(K^{d+1})_{j} .

Because of X^{∗}(T) = X_{∗}(T′) ⊆ X_{∗}(G′) this in particular applies to ξη̃_{L}. Of course, the filtration F il_{ξη̃_{L}}^{·} K^{d+1} has no reason to be weakly L-admissible. But any other filtration of the same type as F il_{ξη̃_{L}}^{·} K^{d+1} is of the form gF il_{ξη̃_{L}}^{·} K^{d+1} = F il_{g(ξη̃_{L})}^{·} K^{d+1} for some g ∈ G′(K). Hence we may express our basic result also by saying that, given the pair (ξ, ζ), there is a pair (ν, φ) ∈ X_{∗}(G′)(K) × G′(K) such that

– ν lies in the G′(K)-orbit of ξη̃_{L},

– the semisimple part of φ is conjugate to ζ_{c} in G′(K), and

– the filtered K-isocrystal (K^{d+1}, φ, F il_{ν}^{·} K^{d+1}) is weakly L-admissible if and only if ζ ∈ T′_{ξ}(K).

Let now G be again a general L-split reductive group. We denote by G′ its Langlands dual group which we consider to be defined over L as well (cf. [Bor]). In particular, T′ is a maximal L-split torus in G′. We view our dominant ξ ∈ X^{∗}(T) = X_{∗}(T′) ⊆ X_{∗}(G′), as above, as a K-rational cocharacter ξ : G_{m} ⟶ G′ and ζ ∈ T′(K) ⊆ G′(K). For a general pair (ν, b) ∈ X_{∗}(G′)(K) × G′(K) we introduce some constructions and terminology which is borrowed from [RZ] Chap. 1. Let REP_{K}(G′) denote the category of K-rational representations of G′ and let F IC_{K} denote the category of filtered K-isocrystals. Both are additive tensor categories. The pair (ν, b) gives rise to the tensor functor

\begin{matrix}I_{(ν,b)} : REP_{K}(G′) & ⟶ & F IC_{K} \\ (ρ, E) & ⟼ & (E, ρ(b), F il_{ρ∘ν}^{·} E) . \end{matrix}

Definition: The pair (ν, b) is called weakly L-admissible if the filtered Kisocrystal I_{(ν,b)}(ρ, E), for any (ρ, E) in REP_{K}(G′), is weakly L-admissible.

Suppose that (ν, b) is weakly L-admissible. Then I_{(ν,b)} can be viewed as a functor

I_{(ν,b)} : REP_{K}(G′) ⟶ F IC_{K}^{L−adm}

into the full subcategory F IC_{K}^{L−adm} of weakly L-admissible filtered K-isocrys-tals which, in fact, is a Tannakian category (the shortest argument for this probably is to observe that for a Galois representation the property of being special crystalline is preserved by tensor products and to use the Colmez-Fontaine equivalence of categories). Moreover, letting Rep_{K}^{con}(Gal(\overline{L}/L)) denote the category of finite dimensional K-linear continuous representations of Gal(\overline{L}/L) we know from the last section that the inverse of the functor D_{cris}(.)_{1} induces a tensor functor between neutral Tannakian categories

F IC_{K}^{L−adm} ⟶ Rep_{K}^{con}(Gal(\overline{L}/L)) .

By composing these two functors we therefore obtain a faithful tensor functor

Γ_{(ν,b)} : REP_{K}(G′) ⟶ Rep_{K}^{con}(Gal(\overline{L}/L))

which possibly is no longer compatible with the obvious fiber functors. This is measured by a G′-torsor over K ([DM] Thm. 3.2). By Steinberg’s theorem ([Ste] Thm. 1.9) that H^{1}(K^{nr}, G′) = 0 over the maximal unramified extension K^{nr} of K this torsor is trivial over K^{nr}. It follows then from the general formalism of neutral Tannakian categories ([DM] Cor. 2.9, Prop. 1.13) that the functor Γ_{(ν,b)} gives rise to a K^{nr}-homomorphism in the opposite direction between the affine group schemes of the two categories which is unique up to conjugation in the target group. For REP_{K}(G′) this affine group scheme of course is G′ ([DM] Prop. 2.8). For Rep_{K}^{con}(Gal(\overline{L}/L)) we at least have that the K-rational points of this affine group scheme naturally contain the Galois group Gal(\overline{L}/L). Hence by restriction we obtain a continuous homomorphism of groups

γ_{ν,b} : Gal(\overline{L}/L) ⟶ G′(K^{nr})

which is determined by the functor Γ_{(ν,b)} up to conjugation in G′(K^{nr}). So we see that any weakly L-admissible pair (ν, b) determines an isomorphism class of “Galois parameters” γ_{ν,b}. We remark that if the derived group of G′ is simply connected Kneser ([Kne]) showed that H^{1}(K, G′) = 0 so that in this case the Galois parameter γ_{ν,b} already has values in G′(K). Following [RZ] p. 14 and [Win] one probably can establish an explicit formula for the cohomology class in H^{1}(K, G′) of the torsor in question.

We indicated already earlier that Langlands functoriality (for smooth representations) requires to work with the normalized Satake isomorphism S^{norm}. This forces us to assume in this section that our coefficient field K contains a square root of q and to pick one once and for all. As a consequence we also have a preferred square root δ^{1/2} ∈ T′(K) of δ ∈ T′(K). Being able to work with the normalized Satake map we do not have to consider the twisted W -action on K[Λ]. But, of course, we still have a norm in the picture which depends on ξ and which is the following. We consider the automorphism of K-algebras

\begin{matrix}a_{ξ} : K[Λ] & ⟶ & K[Λ] \\ λ = λ(t) & ⟼ & δ^{1/2}(λ)π_{L}^{val_{L}(ξ(t))}λ \end{matrix}

which intertwines the conjugation action by W on the source with the twisted action on the target. Pulling back along a_{ξ} the norm ∥ ∥_{γ_{ξ}} gives the norm

∥ ∑_{λ∈Λ} c_{λ}λ∥_{ξ}^{norm} := sup_{λ=λ(t)} |δ^{1/2}(^{w}λ)π_{L}^{val_{L}(ξ(^{w}t))}c_{λ}|

on K[Λ] with w ∈ W for each λ being chosen in such a way that ^{w}λ ∈ Λ^{−−}. Let K⟨Λ; ξ⟩ denote the corresponding Banach algebra completion of K[Λ]. It follows from Prop. 2.4 that K⟨Λ; ξ⟩ is the affinoid algebra of the affinoid subdomain T′_{ξ,norm} obtained by pulling back T′_{ξ} along a_{ξ}. Since a_{ξ} induces on T′ the map ζ ⟼ δ^{1/2}π_{L}^{val_{L} ∘ξ}ζ we deduce from Lemma 2.7 that

T′_{ξ,norm}(K) = val^{−1}(V_{IR}^{ξ,norm})

with

V_{IR}^{ξ,norm} := {z ∈ V_{IR} : z^{dom} ≤ η_{L} + ξ_{L}} .

We have the commutative diagram

\xymatrix{ & B(G, ρ_{ξ}|U_{0}) \ar[d] \\  & ∥ ∥_{ξ}-completion of H(G, 1_{U_{0}}) \ar[dl]^{S^{norm}} \ar[dr]_{S_{ξ}} \\ K⟨Λ; ξ⟩^{W} \ar[rr]^{a_{ξ}} &  & K⟨Λ; γ_{ξ}⟩^{W,γ_{ξ}}}

in which, as a consequence of Lemma 3.3 and Prop. 3.5, all maps are isomorphisms of Banach algebras. In this section we use the left hand sequence of arrows to identify B(G, ρ_{ξ}|U_{0}) with the algebra of analytic functions on the affinoid space W \T′_{ξ,norm}. In particular, this identifies (W \T′_{ξ,norm})(K) with the set of K-valued (continuous) characters of the Banach-Hecke algebra B(G, ρ_{ξ}|U_{0}).

Remark: Using that δ(λ_{{i}}) = q^{−d+2(i−1)} the statement of Prop. 5.2 for the group G = GL_{d+1}(L) becomes: ζ ∈ T′_{ξ,norm}(K) if and only if there is a weakly L-admissible filtered K-isocrystal \underline{D} such that type(\underline{D}) = ξ_{L} + η̃_{L} and the semisimple part of its Frobenius is given by the diagonal matrix with entries q^{d/2}ζ(λ_{{i}}).

We note that in the case where η_{L} happens to be integral (i.e., if d[L : Q_{p}] is even) we can go one step further, can remove completely normalizations accidental to the group GL_{d+1}(L), and can restate the above remark equivalently as follows. We have ζ ∈ T′_{ξ,norm}(K) if and only if there is a weakly L-admissible filtered K-isocrystal \underline{D} such that type(\underline{D}) = ξ_{L} + η_{L} and the semisimple part of its Frobenius is given by the diagonal matrix with entries ζ(λ_{{i}}). Passing now to a general G this unfortunately forces us at present to work under the technical hypothesis that η_{L} ∈ X^{∗}(T) = X_{∗}(T′). This, for example, is the case if [L : Q_{p}] is even or if the group G is semisimple and simply connected. To emphasize that η_{L} then will be considered primarily as a rational cocharacter of T′ we will use multiplicative notation and write ξη_{L} for the product of the rational cocharacters ξ and η_{L}. In this setting and for general G the analog of Prop. 5.2 is the following.

Proposition 6.1: Suppose that η_{L} is integral, let ξ ∈ X^{∗}(T) be dominant, and let ζ ∈ T′(K); then there exists a weakly L-admissible pair (ν, b) (and hence a Galois parameter γ_{ν,b}) such that ν lies in the G′(K)-orbit of ξη_{L} and b has semisimple part ζ if and only if ζ ∈ T′_{ξ,norm}(K).

Proof: First let (ν, b) be a weakly L-admissible pair as in the assertion. Further let ρ : G′ ⟶ GL(E) be any K-rational representation. We then have the weakly L-admissible filtered K-isocrystal (E, ρ(b), F il_{ρ∘ν}^{·} E). Furthermore ρ∘ν is conjugate to ρ ∘ (ξη_{L}) in GL(E)(K) and ρ(ζ) is the semisimple part of ρ(b). We fix a K-rational Borel subgroup P_{E} ⊆ GL(E) and a maximal K-split torus T_{E} ⊆ P_{E} such that ρ(ζ) ∈ T_{E}(K). There is a unique K-rational cocharacter (ρ ∘ ν)^{dom} : G_{m} ⟶ T_{E} which is dominant with respect to P_{E} and which is conjugate to ρ ∘ ν in GL(E)(K). Then (ρ ∘ ν)^{dom} = (ρ ∘ (ξη_{L}))^{dom} corresponds to the type of the filtration F il_{ρ∘ν}^{·} E in the sense of section 5. As in the first part of the proof of Prop. 5.2 we know from [Fon] Prop. 4.3.3 that the weak L-admissibility of our filtered isocrystal implies that the Newton polygon P((ρ(val(ζ)))^{dom}) lies above the Hodge polygon P((ρ ∘ (ξη_{L}))^{dom}) with both having the same endpoint. But, as discussed before Lemma 5.1, this means that

(ρ(val(ζ)))^{dom} ≤ (ρ ∘ (ξη_{L}))^{dom} .

According to [FR] Lemma 2.1 the latter implies that

val(ζ)^{dom} ≤ (ξη_{L})^{dom} = ξη_{L} , i.e., that ζ ∈ T′_{ξ,norm}(K) .

For the reverse implication we first recall that, given any pair (ν, b) and any Krational representation ρ : G′ ⟶ GL(E), the associated filtered K-isocrystal (E, ρ(b), F il_{ρ∘ν}^{·} E) carries the canonical HN-filtration by subobjects (cf. [RZ] Prop. 1.4). The latter is stabilized by a unique parabolic subgroup P_{(ν,b)}^{ρ} ⊆ GL(E). We obviously have

ρ(b) ∈ P_{(ν,b)}^{ρ}(K) .

The HN-filtrations, being functorial, equip our functor I_{(ν,b)} in fact with the structure of an exact ⊗-filtration in the sense of [Saa] IV.2.1.1. The exactness is trivial since the category REP_{K}(G′) is semisimple. The compatibility with the tensor product is a theorem of Faltings and Totaro (independently). It then follows from [Saa] Prop. IV.2.2.5 and Thm. IV.2.4 that

P_{(ν,b)} := ⋂_{ρ} ρ^{−1}(P_{(ν,b)}^{ρ})

is a K-rational parabolic subgroup of G′. Since [Saa] only considers filtrations indexed by integers this requires the following additional observation. The category REP_{K}(G′) has a generator ([Saa] II.4.3.2) and is semisimple. From this one deduces that the jump indices in the HN-filtrations on all the values of our functor can be written with a common denominator. Hence all these HN-filtrations can be reindexed simultaneously in such a way that they become integral, and [Saa] applies. We emphasize that, denoting by ID the protorus with character group Q, one actually has a (not unique) K-rational homomorphism ι_{(ν,b)} : ID ⟶ G′ whose weight spaces define the HN-filtration on the functor I_{(ν,b)}. Its centralizer in G′ is a Levi subgroup of P_{(ν,b)}.

Note that we have

b ∈ P_{(ν,b)}(K) .

After these preliminaries we make our choice of the element b.

Lemma 6.2: There is a regular element b ∈ G′(K) with semisimple part ζ.

Proof: Let M′ ⊆ G′ denote the connected component of the centralizer of ζ in G′. We have:

– M′ is connected reductive ([Ste] 2.7.a);

– M′ is K-split of the same rank as G′ (since T′ ⊆ M′);

– ζ ∈ T′(K) ⊆ M′(K); in fact, ζ lies in the center of M′. The regular unipotent conjugacy class in M′, by its unicity ([Ste] Thm. 3.3), is defined over K. Since M′ is K-split it therefore contains a point u ∈ M′(K) ([Kot] Thm. 4.2). We put b := ζu ∈ G′(K). The centralizer of b in G′ contains with finite index the centralizer of u in M′. Hence b is regular in G′ with semisimple part ζ.

We now fix b ∈ G′(K) to be regular with semisimple part ζ.

Lemma 6.3: There are only finitely many K-rational parabolic subgroups Q ⊆ G′ such that b ∈ Q(K).

Proof: Obviously it suffices to prove the corresponding statement over the algebraic closure \overline{K} of K. By [Ste] Thm. 1.1 there are only finitely many Borel subgroups Q_{0} ⊆ G′ such that b ∈ Q_{0}(\overline{K}). Let Q ⊆ G′ be any parabolic subgroup with b ∈ Q(\overline{K}). It suffices to find a Borel subgroup Q_{0} ⊆ Q such that b ∈ Q_{0}(\overline{K}). Consider the Levi quotient \overline{M} of Q and the image \overline{b} ∈ \overline{M}(\overline{K}) of b. Then \overline{b} is contained in some Borel subgroup \overline{Q}_{0} ⊆ \overline{M} (cf. [Hu1] Thm. 22.2) and we can take for Q_{0} the preimage of \overline{Q}_{0} in Q.

It follows that with ν varying over the G′(K)-orbit Ξ ⊆ X_{∗}(G′) of ξη_{L} the family of parabolic subgroups P_{(ν,b)} actually is finite. Let P_{1}, …, P_{m} denote these finitely many parabolic subgroups and write

Ξ = Ξ_{1} ∪ … ∪ Ξ_{m} with Ξ_{i} := {ν ∈ Ξ : P_{(ν,b)} = P_{i}} .

We want to show that ν ∈ Ξ can be chosen in such a way that P_{(ν,b)} = G′. Because then the homomorphism ι_{(ν,b)} : ID ⟶ G′ factorizes through the center of G′. Since by Schur’s lemma the center of G′ acts through scalars on any irreducible K-rational representation ρ of G′ it follows that the HN-filtration on the filtered isocrystal (E, ρ(b), F il_{ρ∘ν}^{·} E) for irreducible ρ has only one step. On the other hand, our assumption that ζ ∈ T′_{ξ,norm}(K) together with [FR] Lemma 2.1 imply that this filtered isocrystal, for any ρ, has HN-slope zero. Hence it is weakly L-admissible, first for irreducible ρ and then by passing to direct sums also for arbitrary ρ. This proves that the pair (ν, b) is weakly L-admissible.

We argue by contradiction and assume that all P_{1}, …, P_{m} ≠ G′ are proper parabolic subgroups. By [FR] Lemma 2.2.i we then find, for any 1 ≤ i ≤ m, an irreducible K-rational representation ρ_{i} : G′ ⟶ GL(E_{i}) and a K-line ℓ_{i} ⊆ E_{i} such that

P_{i} = stabilizer in G′ of ℓ_{i}

(in particular, ℓ_{i} ≠ E_{i}). We claim that P_{(ν,b)}^{ρ_{i}}, for each ν ∈ Ξ_{i}, stabilizes the line ℓ_{i}. To see this we have to recall the actual construction of ρ_{i} in loc. cit. Fix a maximal K-split torus T_{i} in a Levi subgroup M_{i} of P_{i} and fix a Borel subgroup T_{i} ⊆ B_{i} ⊆ P_{i}. By conjugation we may assume that all the homomorphisms ι_{(ν,b)}, for ν ∈ Ξ_{i}, factorize through the center of M_{i}. Recall that M_{i} then is equal to the centralizer of ι_{(ν,b)} in G′. Hence we may view these ι_{(ν,b)} as elements in X_{∗}(T_{i}) ⊗ Q which lie in the interior of the facet defined by P_{i} (the latter follows from [Saa] Prop. IV.2.2.5.1)). Pick on the other hand a B_{i}-dominant character λ_{i} ∈ X^{∗}(T_{i}) which lies in the interior of the facet corresponding to P_{i} and let ρ_{i} be the rational representation of highest weight λ_{i}. Then, according to [FR], the highest weight space ℓ_{i} ⊆ E_{i} has the required property that P_{i} is its stabilizer in G′. Let λ ∈ X^{∗}(T_{i}) be any weight in E_{i} different from λ_{i}. Then λ_{i} −λ is a nonzero linear combination with nonnegative integral coefficients of B_{i}-simple roots.

Claim: (λ_{i} − λ)(ι_{(ν,b)}) > 0

Proof: Let {α_{j} : j ∈ Δ} ⊆ X^{∗}(T_{i}) be the set of B_{i}-simple roots and let J ⊆ Δ denote the subset corresponding to P_{i}. The highest weight λ_{i} then satisfies

λ_{i}(α̌_{j}) \{ \begin{matrix}= 0 & if j ∈ J, \\ > 0 & if j ∉ J \end{matrix}

where the α̌_{j} ∈ X_{∗}(T_{i}) denote the simple coroots. On the other hand the connected center of M_{i} is equal to (⋂_{j∈J} ker(α_{j}))^{∘}, and we have

α_{j}(ι_{(ν,b)}) \{ \begin{matrix}= 0 & if j ∈ J, \\ > 0 & if j ∉ J. \end{matrix}

We may write

λ_{i} − λ = ∑_{j∈Δ} c_{j}α_{j} with c_{j} ∈ ZZ_{≥0} .

Hence

(+)  (λ_{i} − λ)(ι_{(ν,b)}) = ∑_{j∉J} c_{j}α_{j}(ι_{(ν,b)}) ≥ 0

and we have to show that c_{j}, for at least one j ∉ J, is nonzero. Let λ′ ∈ X^{∗}(T_{i}) denote the unique dominant element in the orbit of λ under the Weyl group of T_{i}. Then λ′ also is a weight occurring in E_{i} and we have

λ_{i} − λ′ = ∑_{j∈Δ} d_{j}α_{j} and λ′ − λ = ∑_{j∈Δ} e_{j}α_{j} with d_{j}, e_{j} ∈ ZZ_{≥0} .

In particular, d_{j} + e_{j} = c_{j}. Suppose first that λ_{i} ≠ λ′. Then it suffices to find a j ∉ J such that d_{j} > 0. By the proof of [Hum] 13.4 Lemma B we obtain λ′ from λ_{i} by successively subtracting simple roots while remaining inside the weights occurring in E_{i} in each step. But because of λ_{i}(α̌_{j}) = 0 if j ∈ J we know ([Hum] 21.3) that λ_{i} − α_{j} cannot be a weight occurring in E_{i} for any j ∈ J. This means of course that we have to have d_{j} > 0 for some j ∉ J. Now assume that λ_{i} = λ′ so that λ = ^{σ}λ_{i} for some σ in the Weyl group of T_{i}. According to the proof of [Hum] 10.3 Lemma B we obtain λ from λ_{i} in the following way: Let σ_{j} be the reflection in the Weyl group corresponding to the simple root α_{j}. Write σ = σ_{j_{1}} … σ_{j_{t}} in reduced form. Then

λ_{i} − λ = ∑_{1≤s≤t} σ_{j_{s+1}} … σ_{j_{t}}(λ_{i})(α̌_{j_{s}})α_{j_{s}}

with all coefficients being nonnegative integers. Since the σ_{j} for j ∈ J fix λ_{i} we may assume that j_{t} ∉ J. Then the last term in the above sum is λ_{i}(α̌_{j_{t}})α_{j_{t}} whose coefficient is positive.

This claim means that ℓ_{i} is a full weight space of ρ_{i} ∘ ι_{(ν,b)}. But it follows from (+) also that the weight of ID on ℓ_{i} is maximal with respect to the natural order on the character group Q of ID among all weights of ID occurring in E_{i}. Hence ℓ_{i} must be the bottom step in the HN-filtration of the filtered K-isocrystal \underline{E}_{i,ν} := (E_{i}, ρ_{i}(b), F il_{ρ_{i}∘ν}^{·} E_{i}) for each ν ∈ Ξ_{i}. As such it carries the structure of a subobject \underline{ℓ}_{i,ν} ⊆ \underline{E}_{i,ν}. As noted already, due to ζ ∈ T′_{ξ,norm}, the HN-slope of \underline{E}_{i,ν} is zero. By the fundamental property of the HN-filtration (cf. [RZ] Prop. 1.4) the HN-slope of \underline{ℓ}_{i,ν} then must be strictly positive which means that

t_{H}(\underline{ℓ}_{i,ν}) > t_{N}^{L}(\underline{ℓ}_{i,ν}) .

Suppose that we find an 1 ≤ i ≤ m and a ν ∈ Ξ_{i} such that ℓ_{i} is transversal to the filtration F il_{ρ_{i}∘ν}^{·} E_{i}. Let (a_{1}, …, a_{r}), resp. (z_{1}, …, z_{r}), denote the filtration type (in the sense of section 5), resp. the slopes written in increasing order, of the corresponding \underline{E}_{i,ν}. The transversality means that t_{H}(\underline{ℓ}_{i,ν}) = a_{1}. On the other hand, since ℓ_{i} is a line we must have t_{N}^{L}(\underline{ℓ}_{i,ν}) = z_{j} ≥ z_{1}. But because of ζ ∈ T′_{ξ,norm}(K), once more [FR] Lemma 2.1, and Lemma 5.1 we have z_{1} ≥ a_{1} which leads to the contradictory inequality

t_{H}(\underline{ℓ}_{i,ν}) ≤ t_{N}^{L}(\underline{ℓ}_{i,ν}) .

It finally remains to justify our choice of ν. Since the filtration F il_{ρ_{i}∘ν}^{·} E_{i} is well defined for any ν ∈ Ξ (and not only ν ∈ Ξ_{i}) it suffices to establish the existence of some ν ∈ Ξ such that

ℓ_{i} is transversal to F il_{ρ_{i}∘ν}^{·} E_{i} for any 1 ≤ i ≤ m .

Let F_{i} ⊂ E_{i} denote the top step of the filtration F il_{ρ_{i}∘ξη_{L}}^{·} E_{i}. We have to find an element g ∈ G′(K) such that

ρ_{i}(g)(ℓ_{i}) ⊈ F_{i} for any 1 ≤ i ≤ m .

For each individual i the set U_{i} := {g ∈ G′ : ρ_{i}(g)(ℓ_{i}) ⊈ F_{i}} is Zariski open in G′. Since ρ_{i} is irreducible the set U_{i} is nonempty. The intersection U := U_{1} ∩…∩U_{m} therefore still is a nonempty Zariski open subset of G′. But G′(K) is Zariski dense in G′ (cf. [Hu1] §34.4). Hence U must contain a K-rational point g ∈ U(K). Then the cocharacter ν := g^{−1}(ξη_{L}) has the properties which we needed.

We summarize that, under the integrality assumption on η_{L}, any K-valued character of one of our Banach-Hecke algebras B(G, ρ_{ξ}|U_{0}) naturally gives rise to a nonempty set of Galois parameters Gal(\overline{L}/L) ⟶ G′(\overline{K}). The need to pass to the algebraic closure \overline{K} comes from two different sources: First the element ζ ∈ T′_{ξ,norm} giving rise to a K-valued character of B(G, ρ_{ξ}|U_{0}) in general is defined only over a finite extension of K; secondly, to make Steinberg’s theorem applicable we had to pass to the maximal unramified extension. In the spirit of our general speculation at the end of the last section we view this as an approximation to a general p-adic Langlands functoriality principle.

Without the integrality assumption on η_{L} one can proceed at least half way as follows. Let us fix, more generally, any natural number r ≥ 1. We introduce the category of r-filtered K-isocrystals F IC_{K,r} whose objects are triples \underline{D} = (D, φ, F il^{·}D) as before only that the filtration F il^{·}D is allowed to be indexed by r^{−1}ZZ (in particular, F IC_{K} = F IC_{K,1}). The invariants t_{H}(\underline{D}) and t_{N}^{L}(\underline{D}) as well as the notion of weak L-admissibility are defined literally in the same way leading to the full subcategory F IC_{K,r}^{L−adm} of F IC_{K,r}.

Proposition 6.4: F IC_{K,r}^{L−adm} is a K-linear neutral Tannakian category.

Proof: This follows by standard arguments from [Tot].

The tensor functor

\begin{matrix}I_{(ν,b)} : REP_{K}(G′) & ⟶ & F IC_{K,r} \\ (ρ, E) & ⟼ & (E, ρ(b), F il_{ρ∘ν}^{·} E) \end{matrix}

makes sense for any pair (ν, b) ∈ (X_{∗}(G′) ⊗ r^{−1}ZZ)(K) × G′(K) as does the notion of weak L-admissibility of such a pair. With these generalizations Prop. 6.1 continues to hold in complete generality (involving 2-filtered K-isocrystals) with literally the same proof. What is missing at present is the connection between the categories F IC_{K,r}^{L−adm} and Rep_{K}^{con}(Gal(\overline{L}/L)). This might involve a certain extension of the Galois group Gal(\overline{L}/L). We hope to come back to this problem in the future.

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Peter Schneider Mathematisches Institut Westfälische Wilhelms-Universität Münster Einsteinstr. 62 D-48149 Münster, Germany pschnei@math.uni-muenster.de http://www.uni-muenster.de/ math/u/schneider Jeremy Teitelbaum Department of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607, USA jeremy@uic.edu http://raphael.math.uic.edu/∼jeremy