homological algebra for schwartz algebras of reductive p-adic groups

HOMOLOGICAL ALGEBRA FOR SCHWARTZ ALGEBRAS OFREDUCTIVE P-ADIC GROUPS

RALF MEYER

Abstract. Let G be a reductive group over a non-Archimedean local field.Then the canonical functor from the derived category of smooth temperedrepresentations of G to the derived category of all smooth representations of G

is fully faithful. Here we consider representations on bornological vector spaces.As a consequence, if G is semi-simple, V and W are tempered irreduciblerepresentations of G, and V or W is square-integrable, then Extn

G(V, W ) ∼= 0

for all n ≥ 1. We use this to prove in full generality a formula for the formaldimension of square-integrable representations due to Schneider and Stuhler.

1. Introduction

Let G be a linear algebraic group over a non-Archimedean local field whoseconnected component of the identity element is reductive; we briefly call such groupsreductive p-adic groups. For the purposes of exposition, we assume throughout theintroduction that the connected centre of G is trivial, although we treat groupswith arbitrary centre in the main body of this article.

We are going to compare homological and cohomological computations for theHecke algebra H(G) and the Harish-Chandra Schwartz algebra S(G). Our mainresult asserts that the derived category of S(G) is a full subcategory of the derivedcategory of H(G). These derived categories incorporate a certain amount of func-tional analysis because S(G) is more than just an algebra. Before we discuss this,we sketch two purely algebraic applications of our main theorem.

Let Modalg(G) be the category of smooth representations of G on complex vec-tor spaces. We compute some extension spaces in this Abelian category. If bothV and W are irreducible tempered representations and one of them is square-integrable, then Extn

G(V,W ) = 0 for n ≥ 1. If the local field underlying G hascharacteristic 0, this is proven by very different means in [20]. We get a moretransparent proof that also works in prime characteristic.

The vanishing of ExtnG(V,W ) is almost trivial if V or W is supercuspidal because

then V or W is both projective and injective in Modalg(G). This is related tothe fact that supercuspidal representations are isolated points in the admissibledual. Square-integrable representations are isolated points in the tempered dual.Hence they are projective and injective in an appropriate category Mod

(S(G)

)of

tempered smooth representations of G. Both Mod(S(G)

)and Modalg(G) are full

subcategories in a larger category Mod(G). That is, we have fully faithful functors

Mod(S(G)

)→ Mod(G)← Modalg(G).

We will show that the induced functors between the derived categories,

Der(S(G)

)→ Der(G)← Deralg(G),

are still fully faithful. This contains the vanishing result for Ext as a special case.

2000 Mathematics Subject Classification. 20G05, 18E30.This research was supported by the EU-Network Quantum Spaces and Noncommutative Ge-

ometry (Contract HPRN-CT-2002-00280) and the Deutsche Forschungsgemeinschaft (SFB 478).1

2 RALF MEYER

Another application involves Euler characteristics for square-integrable repre-sentations. Let V be an irreducible square-integrable representation of G. By atheorem of Joseph Bernstein, any finitely generated smooth representation of G,such as V , has a finite type projective resolution P• → V . Its Euler characteristicis defined as

χ(V ) :=∑

(−1)n[Pn] ∈ K0

(H(G)

).

Since V is square-integrable, it is a projective S(G)-module and therefore definesa class [V ] ∈ K0

(S(G)

). We show that the map K0

(H(G)

)→ K0

(S(G)

)induced

by the embedding H(G)→ S(G) maps χ(V ) to [V ]. This is useful because in [20]Peter Schneider and Ulrich Stuhler construct very explicit finite type projectiveresolutions, so that we get a nice formula for [V ] ∈ K0

(S(G)

). This implies an

explicit formula for the formal dimension of V , which is proven in [20] if V issupercuspidal or if G has characteristic 0. One consequence of this formula is thatthe formal dimensions are quantised, that is, they are all multiples of some α > 0.This allows to estimate the number of irreducible square-integrable representationsthat contain a given representation of a compact open subgroup of G.

Although these applications can be stated purely algebraically, their proofs re-quire functional analysis. We may view S(G) just as an algebra and consider thecategory Modalg

(S(G)

)of modules over S(G) in the algebraic sense as in [21]. How-

ever, the functor Deralg(S(G)

)→ Deralg(G) fails to be fully faithful. This problem

already occurs for G = Z. The issue is that the tensor product of S(G) with itselfplays a crucial role. If we work in Modalg

(S(G)

), we have to deal with S(G)⊗S(G),

which appears quite intractable. In Mod(S(G)

)we meet instead the much simpler

completion S(G×G) of this space.Now it is time to explain briefly how we do analysis. I am an advocate of

bornologies as opposed to topologies. This means working with bounded subsetsand bounded maps instead of open subsets and continuous maps. General bornolog-ical vector spaces behave better than general topological vector spaces for purposesof representation theory and homological algebra (see [15,17]). The spaces that weshall use here carry both a bornology and a topology, and both structures deter-mine each other. Therefore, readers who are familiar with topological vector spacesmay be able to follow this article without learning much about bornologies. Weexplain some notions of bornological analysis along the way because they may beunfamiliar to many readers.

We let Mod(G) be the category of smooth representations of G on bornologicalvector spaces as in [15]. The algebras H(G) and S(G) are bornological algebras ina natural way. A smooth representation π : G → Aut(V ) on a bornological vectorspace V is called tempered if its integrated form extends to a bounded algebrahomomorphism S(G) → End(V ). We may identify Mod(G) with the category ofessential (or non-degenerate) bornological left modules over H(G) ([15]). As ournotation suggests, this identifies the subcategory Mod

(S(G)

)with the category

of essential bornological modules over S(G). We turn Mod(G) and Mod(S(G)

)into exact categories using the class of extensions with a bounded linear section.The exact category structure allows us to form the derived categories Der(G) andDer

(S(G)

)as in [11]. Actually, the passage to derived categories is rather easy in

both cases because our categories have enough projective and injective objects.Equipping a vector space with the finest possible bornology, we identify the

category of vector spaces with a full subcategory of the category of bornologicalvector spaces. Thus Modalg(G) becomes a full subcategory of Mod(G). Moreover,this embedding maps projective objects again to projective objects. Therefore, theinduced functor Deralg(G)→ Der(G) is still fully faithful. Our main theorem asserts

HOMOLOGICAL ALGEBRA FOR SCHWARTZ ALGEBRAS 3

that the canonical functor Der(S(G)

)→ Der(G) is fully faithful as well. The basic

technology for its proof is already contained in [16,17].In [17], I define the category Mod(A) of essential modules and its derived category

Der(A) for a “quasi-unital” bornological algebra A and extend some homologicalmachinery to this setting. A morphism A → B is called isocohomological if theinduced functor Der(B) → Der(A) is fully faithful. [17] gives several equivalentcharacterisations of isocohomological morphisms. The criterion that is most easyto verify is the following: let P• → A be a projective A-bimodule resolution of A;then A → B is isocohomological if and only if B ⊗A P• ⊗A B is a resolution of B(in both cases, resolution means that there is a bounded contracting homotopy).

The article [16] deals with the special case of the embedding C[G] → S1(G)for a finitely generated discrete group G and a certain Schwartz algebra S1(G),which is defined by `1-estimates. The chain complex whose contractibility decideswhether this embedding is isocohomological turns out to be a coarse geometricinvariant of G. That is, it depends only on the quasi-isometry class of a word-lengthfunction on G. If the group G admits a sufficiently nice combing, then I constructan explicit contracting homotopy of this chain complex. Thus C[G] → S1(G) isisocohomological for such groups.

The argument for Schwartz algebras of reductive p-adic groups follows the samepattern. Let H ⊆ G be some compact open subgroup and let X := G/H. Thisis a discrete space which inherits a canonical coarse geometric structure from G.Since G is reductive, it acts properly and cocompactly on a Euclidean building,namely, its affine Bruhat-Tits building. Such buildings are CAT(0) spaces andhence combable. Since X is coarsely equivalent to the building, it is combable aswell. Thus the geometric condition of [16] is easily fulfilled for all reductive p-adicgroups. However, we also have to check that the constructions in [16] are compatiblewith uniform smoothness of functions because G is no longer discrete. This forcesus to look more carefully at the geometry of the building.

2. Bornological analysis

Algebras like the Schwartz algebra S(G) of a reductive p-adic group carry anadditional structure that allows to do analysis in them. The homological algebrafor modules over such algebras simplifies if we take this additional structure intoaccount. One reason is that the complete tensor product S(G) ⊗ S(G) can beidentified with S(G2) (Lemma 2).

It is customary to describe this additional structure using a locally convex topol-ogy. We prefer to use bornologies instead. This means that we work with boundedsubsets and bounded operators instead of open subsets and continuous operators. Abasic reference on bornologies is [9]. We use bornologies because of their advantagesin connection with homological algebra (see [17]).

We mainly need bornological vector spaces that are complete and convex. There-fore, we drop these adjectives and tacitly require all bornologies to be complete andconvex. When we use incomplete bornologies, we explicitly say so.

We need two classes of examples: fine bornologies and von Neumann bornologies.Let V be a vector space over C. The fine bornology Fine(V ) is the finest possiblebornology on V . A subset T ⊆ V is bounded in Fine(V ) if and only if there is afinite-dimensional subspace VT ⊆ V such that T is a bounded subset of VT

∼= Rn

in the usual sense. We also write Fine(V ) for V equipped with the fine bornology.Any linear map Fine(V )→W is bounded. This means that Fine is a fully faithful

functor from the category of vector spaces to the category of bornological vectorspaces that is left-adjoint to the forgetful functor in the opposite direction.

4 RALF MEYER

Let V be a (quasi)complete locally convex topological vector space. A subsetT ⊆ V is called von Neumann bounded if it is absorbed by all neighbourhoodsof zero. These subsets form a bornology on V called the von Neumann bornology(following [9]). We write vN(V ) for V equipped with this bornology.

This defines a functor vN from topological to bornological vector spaces. Its re-striction to the full subcategory of Fréchet spaces or, more generally, of LF-spaces,is fully faithful. That is, a linear map between such spaces is bounded if and onlyif it is continuous. A crucial advantage of bornologies is that joint boundednessis much weaker than joint continuity for multilinear maps: if V1, . . . , Vn,W are(quasi)complete locally convex topological vector spaces, then any separately con-tinuous n-linear map V1 × · · · × Vn → W is (jointly) bounded. The converse alsoholds under mild hypotheses.

LetG be a reductive p-adic group. We carefully explain how the Schwartz algebraS(G) looks like as a bornological algebra. The most convenient definition for ourpurposes is due to Marie-France Vignéras ([25]). Let σ : G→ N be the usual scaleon G. It can be defined using a representation of G. Let L2(G) be the Hilbertspace of square-integrable functions with respect to some Haar measure on G. Let

Lσ2 (G) := f : G→ C | f · σk ∈ L2(G) for all k ∈ N.

A subset T ⊆ Lσ2 (G) is bounded if for all k ∈ N there exists a constant Ck ∈ R+

such that ‖f ·σk‖L2(G) ≤ Ck for all f ∈ T . This is the von Neumann bornology withrespect to the Fréchet topology on Lσ

2 (G) defined by the sequence of semi-norms

‖f‖k2 := ‖f · σk‖L2(G).

Let CO(G) be the set of compact open subgroups of G, ordered by inclusion. ForU ∈ CO(G), let S(GU) = Lσ

2 (GU) be the subspace of U -bi-invariant functionsin Lσ

2 (G). We give S(GU) the subspace bornology, that is, a subset is bounded ifand only if it is bounded in Lσ

2 (G). Finally, we let

S(G) := lim−→S(GU),

where U runs through CO(G). We equip S(G) with the direct-limit bornology. Thatis, a subset of S(G) is bounded if and only if it is a bounded subset of S(GU) forsome U ∈ CO(G). We may also characterise this bornology as the von Neumannbornology with respect to the direct-limit topology on S(G), using the well-knowndescription of bounded subsets in LF-spaces (see [24, Proposition 14.6]).

Lemma 1 ([25]). The definition of the Schwartz algebra above agrees with the oneof Harish-Chandra in [22, 26].

Proof. The first crucial point is that the space of double cosets GU—as opposed tothe group G itself—has polynomial growth with respect to the scale σ. It suffices tocheck this for a good maximal compact subgroup K because the map GU → GKis finite-to-one. By the Iwasawa decomposition, the double cosets in GK can beparametrised by points in a maximal split torus. The scale on G restricts to astandard word-length function on this torus, so that we get the desired polynomialgrowth. As a result, there exists d > 0 such that

∑x∈GU σ

−d(x) is bounded.Moreover, we need the following relationship between the growth of the double

cosets UxU and the Harish-Chandra spherical function Ξ: there are constantsC, r > 0 such that

vol(UxU) ≤ Cσ(x)r · Ξ(x)−2, Ξ(x)−2 ≤ Cσ(x)r · vol(UxU).


This follows from Equation I.1.(5) and Lemma II.1.1 in [26]. Hence∫G

|f(x)|2σ(x)s dx =∑

x∈GU

|f(UxU)|2σ(x)s vol(UxU)

≤∑

x∈GU

|f(UxU)|2Ξ(x)−2Cσ(x)r+s

≤ maxx∈G|f(x)|2Ξ(x)−2σ(x)r+s+d

∑y∈GU

Cσ−d(y).

A similar computation shows∫G

|f(x)|2σ(x)s dx ≥ maxx∈G|f(x)|2Ξ(x)−2C−1σ(x)s−r.

Therefore, the sequences of semi-norms ‖fσs‖2 and ‖fΞ−1σs‖∞ for s ∈ N areequivalent and define the same function space S(GU).

Convolution defines a continuous bilinear map S(GU) × S(GU) → S(GU)for any U ∈ CO(G) by [26, Lemme III.6.1]. Since S(GU) is a Fréchet space,boundedness and continuity of the convolution are equivalent. Since any boundedsubset of S(G) is already contained in S(GU) for some U , the convolution is abounded bilinear map on S(G), so that S(G) is a bornological algebra. In contrast,the convolution on S(G) is only separately continuous.

Now we return to the general theory and define the Hom functor and the tensorproduct. Let Hom(V,W ) be the vector space of bounded linear maps V → W .A subset T of Hom(V,W ) is bounded if and only if it is equibounded, that is,f(v) | f ∈ T, v ∈ S is bounded for any bounded subset S ⊆ V . This bornologyis automatically complete if W is.

The complete projective bornological tensor product ⊗ is defined in [8] by theexpected universal property: it is a bornological vector space V ⊗W together witha bounded bilinear map b : V ×W → V ⊗W such that l 7→ lb is a bijection betweenbounded linear maps V ⊗W → X and bounded bilinear maps V ×W → X. Thistensor product enjoys many useful properties. It is commutative, associative, andcommutes with direct limits. It satisfies the adjoint associativity relation

(1) Hom(V ⊗W,X) ∼= Hom(V,Hom(W,X)

).

Therefore, a bornological module over a bornological algebra A can be defined inthree equivalent ways, using a bounded linear map A→ End(V ), a bounded bilinearmap A× V → V , or a bounded linear map A ⊗ V → V .

Let ⊗ be the usual tensor product of vector spaces. The fine bornology functoris compatible with tensor products; that is, the obvious map V ⊗W → V ⊗W isa bornological isomorphism

(2) Fine(V ⊗W ) ∼= Fine(V ) ⊗ Fine(W )

for any two vector spaces V and W . More generally, if W is any bornologicalvector space, then the underlying vector space of Fine(V ) ⊗ W is equal to thepurely algebraic tensor product V ⊗W . A subset T ⊆ V ⊗W is bounded if andonly if there is a finite-dimensional subspace VT ⊆ V such that T is containedin and bounded in VT ⊗W ∼= Rn ⊗W ∼= Wn. Here Wn carries the direct-sumbornology. The reason for this is that ⊗ commutes with direct limits.

If V1 and V2 are Fréchet-Montel spaces, then we have a natural isomorphism

(3) vN(V1 ⊗π V2) ∼= vN(V1) ⊗ vN(V2),

where ⊗π denotes the complete projective topological tensor product (see [7, 24]).This isomorphism is proven in [13, Appendix A.1.4], based on results of Alexander

6 RALF MEYER

Grothendieck. The Montel condition means that all von Neumann bounded subsetsare precompact (equivalently, relatively compact).

Lemma 2. Let G be a reductive p-adic group. Then S(G) ⊗ S(G) ∼= S(G×G).

Proof. It is shown in [25] that S(GU) is a nuclear Fréchet space for all U ∈ CO(G);in fact, this follows easily from the proof of Lemma 1. Equip G2 with the scaleσ(a, b) := σ(a)σ(b) for all a, b ∈ G. By definition, S(G2) ∼= lim−→S(G2U2). Since ⊗commutes with direct limits,

S(G) ⊗ S(G) ∼= lim−→S(GU) ⊗ S(GU)

as well. It remains to prove S(GU)⊗2 ∼= S(G2U2). Since these Fréchet spacesare nuclear, they are Montel spaces. Hence (3) allows us to replace ⊗ by ⊗π. Nowwe merely have to recall the definition of nuclearity (see [7, 24]).

Let V and W be Fréchet spaces. The natural map V ⊗ W → Hom(V ′,W )defines another topology on V ⊗ W , which may be weaker than the projectivetensor product topology. A Fréchet space is nuclear if and only if this topologycoincides with the projective tensor product topology. Equivalently, there is onlyone topology on V ⊗ W for which the canonical maps V × W → V ⊗ W andV ⊗W → Hom(V ′,W ) are continuous. It is clear that the subspace topology fromS(G2U2) on S(GU)⊗S(GU) has these two properties. Hence it agrees with theprojective tensor product topology. Now the assertion follows because S(GU)⊗2

is dense in S(G2U2).

3. Basic homological algebra over the Hecke algebra

Throughout this section, G denotes a totally disconnected, locally compactgroup, H denotes a fixed compact open subgroup of G, and X := G/H.

Let V be a bornological vector space and let π : G→ Aut(V ) be a representationof G by bounded linear operators. The representation π is called smooth if for anybounded subset T ⊆ V there exists an open subgroup U ⊆ G such that π(g, v) = vfor all g ∈ U , v ∈ T (see [15]). For example, the left and right regular representa-tions of G on S(G) are smooth. If V carries the fine bornology, the definition aboveis equivalent to the usual notion of a smooth representation on a vector space.

Let Mod(G) be the category of smooth representations of G on bornologicalvector spaces; its morphisms are the G-equivariant bounded linear maps. LetModalg(G) be the category of smooth representations of G on C-vector spaces.The fine bornology functor identifies Modalg(G) with a full subcategory of Mod(G).

Let H(G) be the Hecke algebra of G; its elements are the locally constant,compactly supported functions on G. The convolution is defined by

f1 ∗ f2(g) =∫

G

f1(x)f2(x−1g) dx

for some left-invariant Haar measure dx; we normalise it so that vol(H) = 1. Weequip H(G) with the fine bornology, so that H(G) ∈ Modalg(G) ⊆ Mod(G). Moregenerally, given any bornological vector space V , we let H(G,V ) := H(G) ⊗ V .The underlying vector space of H(G,V ) is just H(G) ⊗ V because H(G) carriesthe fine bornology. Hence H(G,V ) is the space of locally constant, compactlysupported functions G→ V . The left regular representation λ and the right regularrepresentation ρ of G on H(G,V ) are defined by

λgf(x) := f(g−1x), ρgf(x) := f(xg)

as usual. They are both smooth.


Any continuous representation π : G→ Aut(V ) on a bornological vector space Vcan be integrated to a bounded algebra homomorphism H(G) → End(V ), whichwe again denote by π. By adjoint associativity, this corresponds to a map

π∗ : H(G,V ) = H(G) ⊗ V → V, f 7→∫

G

π(g, f(g)

)dg.

The map π∗ is G-equivariant if G acts on H(G,V ) by λ. By [15, Proposition 4.7],the representation π is smooth if and only if π∗ is a bornological quotient map,that is, any bounded subset of V is of the form π∗(T ) for some bounded subsetT ⊆ H(G,V ). Even more, if π is smooth, then π∗ has a bounded linear section.Namely, we can use

(4) σH : V → H(G,V ), σHv(g) = π(g−1, v)1H(g),

where 1H denotes the characteristic function of H. Thus the category Mod(G)becomes isomorphic to the category Mod

(H(G)

)of essential modules over H(G)

(see [15, Theorem 4.8]). The term “essential” is a synonym for “non-degenerate”that is not as widely used for other purposes.

Let Ext be the class of all extensions in Mod(G) that have a bounded linearsection. This turns Mod(G) into an exact category in the sense of Daniel Quillen.Hence the usual machinery of homological algebra applies to Mod(G): we can forma derived category Der(G) and derived functors (see [11, 17]). The exact categoryMod(G) has enough projective and injective objects, so that the usual recipes forcomputing derived functors apply. We shall use the following standard projectiveresolution in Mod(G), which already occurs in [16].

The homogeneous space X := G/H is discrete because H is open in G. Let

(5) Xn := (x0, . . . , xn) ∈ Xn+1 | x0 6= x1, . . . , xn−1 6= xn.

We equip C[Xn] with the fine bornology. We let G act diagonally on Xn and equipC[Xn] with the induced representation

g · f(x0, . . . , xn) := f(g−1x0, . . . , g−1xn).

The stabilisers of points

Stab(x0, . . . , xn) =n⋂

j=0

xjHx−1j

are compact open subgroups of G for each (x0, . . . , xn) ∈ Xn. Let X ′n ⊆ Xn be a

subset that contains exactly one representative from each orbit. We get

(6) Xn =∐

ξ∈X′n

G/ Stab(ξ), C[Xn] =⊕

ξ∈X′n

C[G/Stab(ξ)].

If U ∈ CO(G), then we have a natural isomorphism

HomG(C[G/U ], V )∼=→ Fix(U, V ), f 7→ f(1U ).

Since U is compact, this is an exact functor of V , so that C[G/U ] is a projectiveobject of Mod(G). Therefore, C[Xn] is projective by (6).

In the following, we view Xn as a subset of C[Xn] in the usual way. We let(x0, . . . , xn) = 0 if xj = xj+1 for some j ∈ 0, . . . , n−1. Thus (x0, . . . , xn) ∈ C[Xn]is defined for all (x0, . . . , xn) ∈ Xn+1.

We define the boundary map δ = δn : C[Xn+1]→ C[Xn] for n ∈ N by

δ((x0, . . . , xn+1)

):=

n+1∑j=0

(−1)j · (x0, . . . , xj , . . . , xn+1),

8 RALF MEYER

where xj means that xj is omitted. In terms of functions, we can write

(7) δφ(x0, . . . , xn) =n+1∑j=0

(−1)j∑y∈X

φ(x0, . . . , xj−1, y, xj , . . . , xn).

The operators δn are G-equivariant for all n ∈ N. We define the augmentation mapα : C[X0]→ C by α(x) = 1 for all x ∈ X0 = X. It is G-equivariant with respect tothe trivial representation of G on C. It is easy to see that δ2 = 0 and α δ0 = 0.Hence we get a chain complex

C•(X) := (C[Xn], δn)n∈N

over C. We also form the reduced complex C•(X), which has C[Xn] in degree n ≥ 1and ker(α : C[X]→ C) in degree 0. The complex C•(X) is exact. Thus C•(X)→ Cis a projective resolution of C in Modalg(G).

Next we define bivariant co-invariant spaces. For V,W ∈ Mod(G), let V ⊗G Wbe the quotient of V ⊗W by the closed linear span of v ⊗ w − gv ⊗ gw for v ∈ V ,w ∈ W , g ∈ G. Thus V ⊗G W is again a complete bornological vector space. Bydefinition, we have

V ⊗G W ∼= W ⊗G V ∼= (V ⊗W ) ⊗G C,where we equip V ⊗ W with the diagonal representation and C with the trivialrepresentation of G. If X is another bornological vector space, then we may identifyHom(V ⊗G W,X) with the space of bounded bilinear maps f : V ×W → X thatsatisfy f(gv, gw) = f(v, w) for all g ∈ G, v ∈ V , w ∈ W . This universal propertycharacterises V ⊗G W uniquely. It follows from the defining property of ⊗.

There is an alternative description of V ⊗G W in terms of H(G)-modules. TurnV into a right and W into a left bornological H(G)-module by

v ∗ f :=∫

G

f(g) g−1v dg, f ∗ w :=∫

G

f(g) gw dg

for all f ∈ H(G), v ∈ V , w ∈W . Let V ⊗H(G) W be the quotient of V ⊗W by theclosed linear span of v ∗ f ⊗ w − v ⊗ f ∗ w for v ∈ V , f ∈ H(G), w ∈W .

Lemma 3. V ⊗G W = V ⊗H(G) W , that is, the elements gv ⊗ gw − v ⊗ w andv ∗ f ⊗ w − v ⊗ f ∗ w generate the same closed linear subspace.

Proof. We have to show Hom(V ⊗G W,X) = Hom(V ⊗H(G) W,X) for all borno-logical vector spaces X. By definition, Hom(V ⊗G W,X) is the space of boundedbilinear maps l : V × W → X that satisfy l(g−1v, w) = l(v, gw) for all v ∈ V ,w ∈ W , g ∈ G. This implies l(v ∗ f, w) = l(v, f ∗ w) for all v ∈ V , w ∈ W ,f ∈ H(G). Conversely, suppose l(v ∗ f, w) = l(v, f ∗ w). Then

l(gv, g · (f ∗ w)

)= l(gv, (δg ∗ f) ∗ w)

= l(gv ∗ (δg ∗ f), w) = l((g−1 · gv) ∗ f, w

)= l(v, f ∗ w)

for all v ∈ V , w ∈ W , g ∈ G, f ∈ H(G). This implies l(gv, gw) = l(v, w) forall v ∈ V , w ∈ W , g ∈ G because any w ∈ W is fixed by some U ∈ CO(G) andtherefore of the form µU ∗ w, where µU is the normalised Haar measure of U .

Since ⊗G is functorial in both variables, we can apply it to chain complexes.Especially, we get a chain complex of bornological vector spaces V ⊗GC•(X). SinceC•(X) is a projective resolution of the trivial representation, we denote the chainhomotopy type of V ⊗GC•(X) by V ⊗L

G C. The homology vector spaces of V ⊗LG C

may be denoted TorGn (V,C) or TorH(G)

n (V,C). However, this passage to homologyforgets an important part of the structure, namely, the bornology. Therefore, it isbetter to work with V ⊗L

G C instead.


If V and W are just vector spaces, we can identify V ⊗G W with a purelyalgebraic construction. Let V ⊗G W be the quotient of V ⊗W by the linear spanof v ⊗ w − gv ⊗ gw for v ∈ V , w ∈W , g ∈ G. Then

Fine(V ⊗G W ) ∼= Fine(V ) ⊗G Fine(W ).

This follows from (2) and the fact that any linear subspace of a fine bornologicalvector space is closed. Therefore, Fine(V ) ⊗G C•(X) ∼= Fine

(V ⊗G C•(X)

), and

TorG∗ (V,C) is the homology of the chain complex V ⊗GC•(X). In this case, passage

to homology is harmless because V ⊗LG C carries the fine bornology; this implies

that it is quasi-isomorphic to its homology viewed as a complex with vanishingboundary map.

Our next goal is to describe V ⊗G C•(X) (Proposition 6). This requires somegeometric preparations.

Definition 4. Given a finite subset F ⊆ X, we define the relation ∼F on X by

(8) x ∼F y ⇐⇒ (x, y) ∈⋃g∈G

g · (H × F ) ⇐⇒ x−1y ∈ HFH.

Here we view x−1y ∈ GH and HFH ⊆ GH.A subset S ⊆ Xn is controlled by F if xi ∼F xj for all (x0, . . . , xn) ∈ S and all

i, j ∈ 0, . . . , n. We call S ⊆ Xn controlled if it is controlled by some finite F .Roughly speaking, this means that all entries of S are uniformly close.

A subset S ⊆ Xn is controlled if and only if S is G-finite, that is, there is a finitesubset F ⊆ Xn such that S ⊆ G · F . This alternative characterisation will be usedfrequently. Definition 4 emphasises a crucial link between the controlled supportcondition and geometric group theory.

A coarse (geometric) structure on a locally compact space such as X is a familyof relations on X satisfying some natural axioms due to John Roe (see also [6]).The subrelations of the relations ∼F above define a coarse geometric structure on Xin this sense. Since it is generated by G-invariant relations, it renders the actionof G on X isometric. This property already characterises the coarse structureuniquely: whenever a locally compact group acts properly and cocompactly on alocally compact space, there is a unique coarse structure for which this action isisometric (see [6, Example 6]). Moreover, with this coarse structure, the space Xis coarsely equivalent to G.

By definition, the notion of a controlled subset of Xn depends only on the coarsegeometric structure of X. Thus the space of functions on Xn of controlled supportonly depends on the large scale geometry of X.

Although our main examples, reductive groups, are unimodular, we want to treatgroups with non-trivial modular function as well. Therefore, we have to decorateseveral formulas with modular functions. We define the modular homomorphism∆G : G→ R>0 by ∆G(g)d(g−1) = dg and d(gh) = ∆G(h) dg for all h ∈ G.

Definition 5. Let C(Xn, V )∆ be the space of all maps φ : Xn → V that havecontrolled support and satisfy the covariance condition

(9) φ(gξ) = ∆G(g)−1π(g, φ(ξ)

)for all ξ ∈ Xn, g ∈ G. A subset T ⊆ C(Xn, V )∆ is bounded if φ(ξ) | φ ∈ T isbounded in V for all ξ ∈ Xn and the supports of all φ ∈ T are controlled by thesame finite subset F ⊆ X.

Proposition 6. For any V ∈ Mod(G), there is a natural bornological isomorphism

V ⊗G C[Xn] ∼= C(Xn, V )∆.

10 RALF MEYER

The induced boundary map on V ⊗G C[Xn] corresponds to the boundary map

δ = δn : C(Xn+1, V )∆ → C(Xn, V )∆

defined by (7).

We denote the resulting chain complex (C(Xn, V )∆, δn)n∈N by C(X•, V )∆.

Proof. The bifunctor ⊗G commutes with direct limits and in particular with directsums. Hence (6) yields

(10) V ⊗G C[Xn] ∼=⊕

ξ∈X′n

V ⊗G C[Gξ] ∼=⊕

ξ∈X′n

V ⊗G C[G/ Stab(ξ)].

Fix ξ ∈ X ′n and let Map(G · ξ, V )∆ be the space of all maps from G · ξ to V that

satisfy the covariance condition (9). We equip Map(G · ξ, V )∆ with the productbornology as in Definition 5. We claim that the map

I : V ⊗ C[Gξ]→ Map(Gξ, V ), v ⊗ φ 7→ [η 7→∫

G

π(h−1, v) · φ(hη) dh],

yields a bornological isomorphism

V ⊗G C[Gξ] ∼= MapG(Gξ, V )∆.

We check that I descends to V ⊗G C[Gξ] and maps into Map(Gξ, V )∆:

I(gv ⊗ gφ)(η) =∫

G

π(h−1, gv) · gφ(hη) dh =∫

G

π(h−1g, v) · φ(g−1hη) dh

=∫

G

π(h−1, v) · φ(hη) dh = I(v ⊗ φ),

I(v ⊗ φ)(gη) =∫

G

π(h−1, v) · φ(hgη) dh

=∫

G

π(gh−1, v)φ(hη) d(hg−1) = ∆G(g−1)π(g, I(v ⊗ φ)(η)

).

Thus we get a well-defined map V ⊗G C[Gξ] → Map(Gξ, V )∆. Evaluation at ξdefines a bornological isomorphism Map(Gξ, V )∆ ∼= Fix(Stab(ξ), V ). We claimthat the latter is isomorphic to V ⊗G C[G/ Stab(ξ)]. Since Stab(ξ) is compact andopen, the Haar measure µStab(ξ) of Stab(ξ) is an element of H(G). Convolution onthe right with µStab(ξ) is an idempotent left module homomorphism onH(G), whoserange is C[G/ Stab(ξ)]. Since V ⊗G H(G) ∼= V for all V , additivity implies thatV ⊗GC[G/ Stab(ξ)] is equal to the range of µStab(ξ) on V , that is, to Fix(Stab(ξ), V ).Thus we obtain an isomorphism V ⊗G C[Gξ] ∼= MapG(Gξ, V )∆, which can easilybe identified with the map I.

Recall that a subset of Xn is controlled if and only if it meets only finitely manyG-orbits. Therefore, we get the counterpart C(Xn, V )∆ ∼=

⊕ξ∈X′

nMap(Gξ, V )∆

to (10). We can piece our isomorphisms on orbits together to an isomorphism

I : V ⊗G C[Xn]→ C(Xn, V )∆, v ⊗ φ 7→ [ξ 7→∫

G

π(g−1, v)φ(gξ) dg].

A straightforward computation yields δ I(v ⊗ φ) = I(v ⊗ δφ) for all v ∈ V ,φ ∈ C[Xn] with δ as in (7). Therefore, I intertwines id ⊗G δ and δ.

Now let C•(X,V ) := C•(X)⊗V , equipped with the diagonal representation of G.Since C•(X) carries the fine bornology, the underlying vector space of C[Xn] ⊗ Vmay be identified with the space of functions Xn → V with finite support.

Lemma 7. The chain complex C•(X,V ) is a projective resolution of V in Mod(G).


Proof. The complex C•(X,V ) is exact because ⊗ is exact on extensions with abounded linear section. We have HomG(C[G/U ] ⊗ V,W ) ∼= HomU (V,W ) for anyU ∈ CO(G) and any smooth representation W . Since this is an exact functor of W ,C[G/U ] ⊗ V is projective. Equation (6) shows that C[Xn] ⊗ V is a direct sum ofsuch representations and therefore projective as well.

We view H(G) as a bimodule over itself in the usual way, by convolution onthe left and right. Since right convolution commutes with the left regular repre-sentation, the complex C•

(X,H(G)

)is a complex of H(G)-bimodules. The same

reasoning as in the proof of Lemma 7 shows that it is a projective H(G)-bimoduleresolution of H(G).

For V,W ∈ Mod(G), we let HomG(V,W ) be the space of bounded G-equivariantlinear maps V →W , equipped with the equibounded bornology. It agrees with thespace HomH(G)(V,W ) of bounded linear H(G)-module homomorphisms. We alsoapply the bifunctor HomG to chain complexes. In particular, we can plug in theprojective resolution C•(X,V ) of Lemma 7. The homotopy type of the resultingcochain complex of bornological vector spaces HomG(C•(X,V ),W ) is denoted byR HomG(V,W ). Its nth cohomology vector space is Extn

G(V,W ). As with V ⊗LG C,

it is preferable to retain the cochain complex itself.If V and W carry the fine bornology, then C•(X,V ) = C•(X) ⊗ V with the

fine bornology. Therefore, R HomG(V,W ) is equal to the space of all G-equivariantlinear maps C•(X) ⊗ V → W . Hence the Ext spaces above agree with the purelyalgebraic Ext spaces. In more fancy language, the embedding Modalg(G)→ Mod(G)induces a fully faithful functor between the derived categories Deralg(G)→ Der(G).This allows us to apply results proven using analysis in a purely algebraic context.

4. Isocohomological smooth convolution algebras

We introduce a class of convolution algebras on totally disconnected, locallycompact groups G. These have the technical properties that allow us to formulatethe problem. Then we examine the notion of an isocohomological embedding andformulate a necessary and sufficient condition for H(G)→ T (G) to be isocohomo-logical. This criterion involves the contractibility of a certain bornological chaincomplex, which is quite close to the one that arises in [16].

4.1. Unconditional smooth convolution algebras with rapid decay. Let Gbe a totally disconnected, locally compact group. Let σ : G→ R≥1 be a scale withthe following properties: σ(ab) ≤ σ(a)σ(b) and σ(a) = σ(a−1); σ is U -bi-invariantfor some U ∈ CO(G); the map σ is proper, that is, the subsets

(11) BR(G) := g ∈ G | σ(g) ≤ Rare compact for all R ≥ 1. The usual scale on a reductive p-adic groups has theseproperties. If the group G is finitely generated and discrete, then σ = 1 + ` or 2`

for a word-length function ` are good, inequivalent choices.Let U ∈ CO(G). Given sets S, S′ of functions GU → C we say that S′ domi-

nates S if for any φ ∈ S there exists φ′ ∈ S′ with |φ′(g)| ≥ |φ(g)| for all g ∈ G.

Definition 8. Let T (G) be a bornological vector space of functions φ : G→ C. Wecall T (G) an unconditional smooth convolution algebra of rapid decay if it satisfiesthe following conditions:8.1. T (G) contains H(G);8.2. H(G) is dense in T (G);8.3. the convolution extends to a bounded bilinear map T (G)× T (G)→ T (G);8.4. T (G) = lim−→T (GU) as bornological vector spaces, where U runs through

CO(G) and T (GU) is the space of U -bi-invariant functions in T (G);

12 RALF MEYER

8.5. if a set of functions GU → C is dominated by a bounded subset of T (GU),then it is itself a bounded subset of T (GU);

8.6. Mσ is a bounded linear operator on T (G).The first four conditions define a smooth convolution algebra, the fifth conditionmeans that the convolution algebra is unconditional, the last one means that it hasrapid decay.

An example of such a convolution algebra is the Schwartz algebra of a reductivep-adic group.

Let T (G) be an unconditional smooth convolution algebra of rapid decay. Arepresentation π : G → Aut(V ) is called T (G)-tempered if its integrated form ex-tends to a bounded algebra homomorphism T (G)→ End(V ) or, equivalently, to abounded bilinear map T (G)× V → V . The density of H(G) in T (G) implies thatthis extension is unique once it exists. Furthermore, G-equivariant maps are T (G)-module homomorphisms. Since the subalgebras T (GU) are unital, the algebraT (G) is “quasi-unital” in the notation of [17], so that the category Mod

(T (G)

)of

essential bornological left T (G)-modules is defined. This category is naturally iso-morphic to the category of T (G)-tempered smooth representations of G (see [17]).Thus Mod

(T (G)

)is a full subcategory of Mod(G).

The following lemmas prove some technical properties of T (G) that are obviousin most examples, anyway. Define PR : T (G) → H(G) by PRφ(x) = φ(x) forx ∈ BR(G) and PRφ(x) = 0 otherwise, with BR(G) as in (11).

Lemma 9. limR→∞ PR(φ) = φ uniformly for φ in a bounded subset of T (G).

Proof. If T ⊆ T (G) is bounded, then T ⊆ T (GU) for some U ∈ CO(G). Shrink-ing U further, we achieve that the scale σ is U -bi-invariant. We may further assumethat φ′ ∈ T whenever φ′ : GU → C is dominated by some φ ∈ T because T (G) isunconditional. Since Mσ is bounded, the subset Mσ(T ) ⊆ T (GU) is bounded aswell. For any φ ∈ T , we have |φ−PRφ| ≤ R−1|Mσφ|, so that φ−PRφ ∈ R−1Mσ(T ).This implies uniform convergence PR(φ)→ φ for φ ∈ T .

In the following, we briefly write

T (G2) := T (G) ⊗ T (G).

Lemma 2 justifies this notation for Schwartz algebras of reductive groups. In gen-eral, consider the bilinear maps

T (G)× T (G)→ C, (φ1, φ2) 7→ φ1(x)φ2(y)

for (x, y) ∈ G2. They extend to bounded linear functionals on T (G2) and hencemap T (G2) to a space of smooth functions on G2.

Lemma 10. This representation of T (G2) by functions on G2 is faithful, that is,φ ∈ T (G2) vanishes once φ(x, y) = 0 for all x, y ∈ G.

Proof. The claim follows easily from Lemma 9 (this is a well-known argumentin connection with Grothendieck’s Approximation Property). If φ(x, y) = 0 forall x, y ∈ G, then also (PR ⊗ PR)φ(x, y) = 0 for all R ∈ N, x, y ∈ G. SincePR ⊗PR(φ) ∈ H(G2), this implies PR ⊗PR(φ) = 0 for all R ∈ N. Lemma 9 impliesthat PR ⊗PR converges towards the identity operator on T (G) ⊗T (G). This yieldsφ = 0 as desired.

Hence we may view T (G2) as a space of functions on G2. It is easy to seethat T (G2) is again a smooth convolution algebra on G2. Equip G2 with the scaleσ2(a, b) := σ(a)σ(b) for a, b ∈ G. Then the operator Mσ2 = Mσ ⊗Mσ is bounded,that is, T (G2) also satisfies the rapid decay condition. However, T (G2) need not


be unconditional. We assume T (G2) to be unconditional in the following. This isneeded for the proof of our main theorem.

Let Tc(G) be H(G) equipped with the subspace bornology from T (G). Thisbornology is incomplete, of course. Similarly, we let Tc(G2) be H(G2) equippedwith the subspace bornology from T (G2).

Lemma 11. The completions of Tc(G) and Tc(G2) are naturally isomorphic toT (G) and T (G2).

Proof. It suffices to prove this for Tc(G). We verify by hand that T (G) satisfiesthe universal property that defines the completion of Tc(G). Alternatively, wecould use general characterisations of completions in [14, Section 4]. We must showthat any bounded linear map f : Tc(G) → W into a complete bornological vectorspace W extends uniquely to a bounded linear map on T (G). By Lemma 9, thesequence of operators PR : T (G)→ Tc(G) converges uniformly on bounded subsetstowards the identity map on T (G). Hence any bounded extension f of f satisfiesf(φ) = limR→∞ f PR(φ) for all φ ∈ T (G). Conversely, this prescription defines abounded linear extension of f .

4.2. Isocohomological convolution algebras. Let A be a quasi-unital algebrasuch as H(G) or T (G). In [17] I define the exact category Mod(A) of essentialbornological left A-modules and its derived category Der(A). A bounded algebrahomomorphism f : A → B between two quasi-unital bornological algebras inducesfunctors f∗ : Mod(B) → Mod(A) and f∗ : Der(B) → Der(A). Trivially, if f hasdense range then f∗ : Mod(B) → Mod(A) is fully faithful. We call f isocohomo-logical if f∗ : Der(B) → Der(A) is fully faithful as well ([17]). We are interested inthe embedding H(G) → T (G). If it is isocohomological, we briefly say that T (G)is isocohomological. The following conditions are proven in [17, Theorem 35] to beequivalent to T (G) being isocohomological:

• V ⊗LT (G)W

∼= V ⊗LGW for all V,W ∈ Der

(T (G)

)(recall that ⊗L

H(G)∼= ⊗L

G);• R HomT (G)(V,W ) ∼= R HomG(V,W ) for all V,W ∈ Der

(T (G)

);

• the functor f∗ : Der(T (G)

)→ Der(G) is fully faithful;

• T (G) ⊗LG V ∼= V for all V ∈ Mod

(T (G)

);

• T (G) ⊗LG T (G) ∼= T (G).

The last condition tends to be the easiest one to verify in practice. We will formulateit more concretely below. The signs “∼=” in these statements mean isomorphism inthe homotopy category of chain complexes of bornological vector spaces. This isstronger than an isomorphism of homology groups. As a consequence, we haveTorG

n (V,W ) ∼= TorT (G)n (V,W ) and Extn

G(V,W ) ∼= ExtnT (G)(V,W ) for all V,W ∈

Mod(T (G)

)if T (G) is isocohomological.

Notions equivalent to that of an isocohomological embedding have been definedindependently by several authors, as kindly pointed out to me by A. Yu. Pirkovskii(see [18] and the references given there). We warn the reader that in categories oftopological algebras some of the conditions above are no longer equivalent. Namely,the cohomological conditions in terms of the derived category and R Hom are weakerthan the homological conditions involving ⊗L.

We have seen in Section 3 that V ⊗GW ∼= (V ⊗W ) ⊗G C for all V,W ∈ Mod(G),where V ⊗W is equipped with the diagonal representation of G. Since V ⊗W isprojective if V or W is projective, this implies an isomorphism

V ⊗LG W ∼= (V ⊗W ) ⊗L

G C

14 RALF MEYER

for all V,W ∈ Der(G). Thus T (G) is isocohomological if and only if(T (G) ⊗ T (G)

)⊗L

G C = T (G2) ⊗LG C ∼= T (G).

Here we equip T (G2) with the diagonal representation of G, which is given by

g · f(x, y) := ∆G(g)f(xg, g−1y)

for all g ∈ G, f ∈ T (G2), x, y ∈ G because the left and right H(G)-modulestructures on T (G) are the integrated forms of the left regular representation λand the twisted right regular representation ρ ·∆G. The convolution map

T (G2) = T (G) ⊗ T (G) ∗→ T (G)

descends to a bounded linear map T (G2) ⊗G C → T (G). The latter map is abornological isomorphism because A ⊗A A ∼= A for any quasi-unital bornologicalalgebra by [17, Proposition 16]. Moreover, the convolution map T (G2)→ T (G) hasa bounded linear section, namely, the map T (G) → H

(G, T (G)

)⊆ T (G2) defined

in (4).We may use the projective resolution C•(X) → C to compute T (G2) ⊗L

G C.Proposition 6 identifies T (G2) ⊗G C•(X) with C(X•, T (G2))∆. We augment thischain complex by the map

(12) α : T (G2) ⊗G C[X0]id⊗Gα−−−−→ T (G2) ⊗G C ∗−→∼= T (G)

We let C(X•, T (G2))∆ be the subcomplex of C(X•, T (G2))∆ that we get if wereplace C(X0, T (G2))∆ by C(X0, T (G2))∆ := kerα.

Proposition 12. T (G) is isocohomological if and only if C(X•, T (G2))∆ has abounded contracting homotopy.

Proof. Our discussion of the convolution map implies that the augmentation mapin (12) is a surjection with a bounded linear section. Hence it is a chain homo-topy equivalence T (G2) ⊗L

G C → T (G) if and only if its kernel C(X0, T (G2))∆ iscontractible.

To give the reader an idea why the various characterisations of isocohomolog-ical embeddings listed above are equivalent, we explain how the contractibilityof C(X•, T (G2))∆ yields isomorphisms R HomT (G)(V,W ) ∼= R HomG(V,W ) forV,W ∈ Mod

(T (G)

). Almost the same argument yields V ⊗L

T (G) W∼= V ⊗L

G W .The extension to objects of the derived categories is a mere formality.

The space T (G2) carries a T (G)-bimodule structure via f1 ∗ (f2 ⊗ f3) ∗ f4 =(f1 ∗ f2) ⊗ (f3 ∗ f4). This structure commutes with the inner conjugation action,so that P• := T (G2) ⊗G C•(X) becomes a chain complex of bornological T (G)-bimodules over T (G). As above, we can compute these spaces explicitly:

T (G2) ⊗G C[Xn] ∼=⊕

ξ∈X′n

Fix(Stab ξ, T (G2)

).

It is not hard to see that T (G2) is a projective object of Mod(T (G2)

). That is,

T (G2) is a projective bimodule. Since the summands of T (G2) ⊗G C[Xn] are allretracts of T (G2), we conclude that T (G2)⊗GC[Xn] is a projective T (G)-bimodule.

Suppose now that P• is a resolution of T (G). Then it is a projective T (G2)-bimodule resolution. Since T (G) is projective as a right module, the contractinghomotopy of P• can be improved to consist of bounded right T (G)-module homo-morphisms. Therefore, P• ⊗T (G) V is again a resolution of V . Explicitly,

Pn ⊗T (G) V ∼=⊕

ξ∈X′n

Fix(Stab(ξ), T (G) ⊗ V )


because T (G)⊗T (G)V ∼= V . The summands are retracts of the projective left T (G)-module T (G) ⊗ V . Hence P• ⊗T (G) V is a projective left T (G)-module resolutionof V . We use it to compute

R HomT (G)(V,W ) = HomT (G)(P• ⊗T (G) V,W ).

Let U ∈ CO(G) act on T (G) ⊗V by ∆G|U ·ρ⊗π = ρ⊗π. If f : V →W is boundedand U -equivariant, then φ ⊗ v 7→ φ ∗ f(v) defines a bounded G-equivariant linearmap Fix(U, T (G) ⊗ V ) → W . One can show that this establishes a bornologicalisomorphism

HomT (G)(Fix(U, T (G) ⊗ V ),W ) ∼= HomU (V,W ).

This yields a natural isomorphism

HomT (G)(Pn ⊗T (G) V,W ) ∼=⊕

ξ∈X′n

HomStab(ξ)(V,W ).

The right hand side no longer depends on T (G)! Thus HomG(C•(X,V ),W ) is iso-morphic to the same complex, and R HomT (G)(V,W ) ∼= R HomG(V,W ) as asserted.

Next we simplify the chain complex C(X•, T (G2))∆. To φ ∈ C(Xn, T (G2))∆ weassociate a function φ∗ : G×G×Xn → C by φ∗(g, h, ξ) := φ(ξ)(g, h). This identifiesC(Xn, T (G2))∆ with a space of functions on G2×Xn by Lemma 10. More precisely,we get the space of functions φ : G2 ×Xn → C with the following properties:

• suppφ ⊆ G2 × S for some controlled subset S ⊆ Xn;• the function (a, b) 7→ φ(a, b, ξ) belongs to T (G2) for all ξ ∈ Xn;• φ(ag, g−1b, g−1ξ) = φ(a, b, ξ) for all ξ ∈ Xn, g, a, b ∈ G (the two modular

functions cancel).The last condition means that φ is determined by its restriction to 1 × G ×Xn

by φ(a, b, ξ) = φ(1, ab, aξ). Thus we identify C(Xn, T (G2))∆ with the followingfunction space on G×Xn:

Definition 13. Let C(G × Xn, T ) be the space of all functions φ : G × Xn → Cwith the following properties:13.1. suppφ ⊆ G× S for some controlled subset S ⊆ Xn;13.2. the function (a, b) 7→ φ(ab, aξ) belongs to T (G2) for all ξ ∈ Xn.A subset T ⊆ C(G×Xn, T ) is bounded if there is a controlled subset S ⊆ G suchthat suppφ ⊆ G × S for all φ ∈ T and if for any ξ ∈ Xn, the set of functions(a, b) 7→ φ(ab, aξ) for φ ∈ T is bounded in T (G2).

The boundary map δ on C(Xn, T (G2))∆ corresponds to the boundary map

δ : C(G×Xn+1, T )→ C(G×Xn, T ),

δφ(g, x0, . . . , xn) =n+1∑j=0

(−1)j∑y∈X

φ(g, x0, . . . , xj−1, y, xj , . . . , xn).

The augmentation map C(X, T (G2))∆ → T (G) corresponds to

(13) α : C(G×X, T )→ T (G), αφ(g) =∑x∈X

φ(g, x).

The proofs are easy computations, which we omit. Let C(G×X0, T ) ⊆ C(G×X0, T )be the kernel of α and let C(G×X•, T ) be the bornological chain complex that weget if we replace C(G×X0, T ) by C(G×X0, T ). Thus

C(G×X•, T ) ∼= C(X•, T (G2))∆.

16 RALF MEYER

Smooth functions of compact support automatically satisfy both conditions inDefinition 13, so that H(G)⊗C[Xn] ⊆ C(G×Xn, T ). These embeddings are com-patible with the boundary and augmentation maps. Thus H(G)⊗ C•(X) becomesa subcomplex of C(G×X•, T ). We write C(G×X•, Tc) and C(G×X•, Tc) for thechain complexes H(G)⊗ C•(X) and H(G)⊗ C•(X) equipped with the incompletesubspace bornologies from C(G×X•, T ). The complex C(G×X•, Tc) is contractiblebecause C•(X) is. However, the obvious contracting homotopy is unbounded.

Lemma 14. Suppose that there is a contracting homotopy D for C•(X) such thatidH(G) ⊗D is bounded on C(G×X•, Tc). Then T (G) is isocohomological.

Proof. We claim that C(G × X•, T ) is the completion of C(G × X•, Tc). ThenC(G×X•, T ) ∼= C(X•, T (G2))∆ inherits a bounded contracting homotopy becausecompletion is functorial. Proposition 12 yields that T (G) is isocohomological. Itremains to prove the claim. We do this by reducing the assertion to Lemma 11.Since C(G×X•, T ) is a direct summand in C(G×X•, T ), it suffices to prove thatC(G×X•, T ) is the completion of C(G×X•, Tc). Recall that X ′

• denotes a subsetof X• containing one point from each G-orbit. The decomposition of X• intoG-orbits yields a direct-sum decomposition

(14) C(G×X•, Tc) ∼=⊕

ξ∈X′•

Fix(Stab ξ, Tc(G2)

),

and a similar decomposition for C(G×X•, T ). Here direct sums are equipped withthe canonical bornology: a subset is bounded if it is contained in and bounded in afinite sub-sum. The reason for (14) is that a subset of Xn is controlled if and only ifit meets only finitely many G-orbits. Since completion commutes with direct sums,the assertion now follows from Lemma 11.

5. Contracting homotopies constructed from combings

In order to apply Lemma 14, we have to construct contracting homotopies ofC•(X). For this we use the geometric recipes of [16]. The only ingredient is asequence of maps pk : X → X with certain properties. We first explain how sucha sequence of maps gives rise to a contracting homotopy D of C•(X). Then weformulate conditions on (pk) and prove that they imply boundedness of D.

5.1. A recipe for contracting homotopies. The construction of C•(X) andC•(X) is natural: a map f : X → X induces a chain map f∗ : C•(X)→ C•(X) byf∗

((x0, . . . , xn)

):= (f(x0), . . . , f(xn)

)or, equivalently,

(15) f∗φ(x0, . . . , xn) =∑

yj : f(yj)=xj

φ(y0, . . . , yn).

Since α f∗ = α, this restricts to a chain map on C•(X). We have id∗ = id and(fg)∗ = f∗g∗. Let p0 be be the constant map x 7→ H for all x ∈ X. We claimthat (p0)∗ = 0 on C•(X). On C[Xn] for n ≥ 1 this is due to our convention that(x0, . . . , xn) = 0 if xi = xi+1 for some i. For φ ∈ C[X0], we get (p0)∗φ = α(φ) · (H),where (H) ∈ C[X] is the characteristic function of H ∈ X. This implies the claim.

Given maps f, f ′ : X → X, we define operators Dj(f, f ′) : C[Xn]→ C[Xn+1] forj ∈ 0, . . . , n by

(16) Dj(f, f ′)((x0, . . . , xn)

):=

(f(x0), . . . , f(xj), f ′(xj), . . . , f ′(xn)

)and let D(f, f ′) :=

∑nj=0(−1)jDj(f, f ′). It is checked in [16] that

[δ,D(f, f ′)] := δ D(f, f ′) +D(f, f ′) δ = f ′∗ − f∗.


Thus the chain maps f∗ on C•(X) for f : X → X are all chain homotopic. Inparticular, D(id, p0) is a contracting homotopy of C•(X) because (p0)∗ = 0.

However, this trivial contracting homotopy does not work for Lemma 14. Instead,we use a sequence of maps (pk)k∈N with p0 as above and limk→∞ pk = id, that is,for each x ∈ X there is k0 ∈ N such that pk(x) = x for all k ≥ k0. We let

Dk := D(pk, pk+1), Djk := Dj(pk, pk+1).

Observe thatDjk vanishes on the basis vector (x0, . . . , xn) unless pk(xj) 6= pk+1(xj).Therefore, all but finitely many summands of

D :=∞∑

k=0

Dk

vanish on any given basis vector. Thus D is a well-defined operator on C•(X).The operator D is a contracting homotopy of C•(X) because

[D, δ] =∞∑

k=0

[D(pk, pk+1), δ] =∞∑

k=0

(pk+1)∗ − (pk)∗ = limk→∞

(pk)∗ − (p0)∗ = id.

To verify this computation, plug in a basis vector and use that all but finitely manyterms vanish. This is the operator we want to use in Lemma 14.

5.2. Sufficient conditions for boundedness. Construct D as above and let D′ := idH(G) ⊗ D. We want this to be a bounded operator on C(G × X•, Tc). Forthis, we impose three further conditions on (pk). First, (pk) should be a combing.This notion comes from geometric group theory and is already used in [16]. Itallows us to control the support of D′φ for φ with controlled support. Secondly,the combing (pk) should be smooth. This allows us to control the smoothness ofthe functions (a, b) 7→ D′φ(ab, aξ) on G2 for ξ ∈ Xn. Only the third conditioninvolves the convolution algebra T (G). It asks for a certain sequence of operatorsTc(G)→ Tc(G2) to be equibounded.

The smoothness condition is vacuous for discrete groups. The third condition isalmost vacuous for `1-Schwartz algebras of discrete groups. Hence these two con-ditions are not needed in [16]. In our application to reductive groups, we constructthe operators (pk) using the retraction of the affine Bruhat-Tits building of thegroup along geodesic paths. This is a combing because Euclidean buildings areCAT(0) spaces. Its smoothness amounts to the existence of congruence subgroups.The third condition follows easily from Lemma 2.

We now formulate the above conditions on (pk) in detail and state the mainresult. We use the relation ∼F for a finite subset F ⊆ X = G/H defined in (8) byx ∼F y ⇐⇒ x−1y ∈ HFH.

Definition 15. A sequence of maps (pk)k∈N as above is called a combing of X ifit has the following additional two properties:15.1. there is a finite subset F ⊆ X such that pk(x) ∼F pk+1(x) for all k ∈ N,

x ∈ X;15.2. for any finite subset F ⊆ X there is a finite subset F ⊆ X such that pk(x) ∼F

pk(y) for all k ∈ N and x, y ∈ X with x ∼F y.We say that the combing has polynomial growth (with respect to the scale σ) if theleast k0 such that pk(gH) = gH for all k ≥ k0 grows at most polynomially in σ(g).(This definition of growth differs slightly from the one in [16].)

We may view the sequence(pk(x)

)as a path from H to x. The conditions on a

combing mean that these paths do not jump too far in each step and that nearbyelements have nearby paths

(pk(x)

).

18 RALF MEYER

Definition 16. A combing (pk)k∈G of G/H is called smooth if it has the followingtwo properties:

16.1. all maps pk are H-equivariant for some open subgroup H ⊆ H;16.2. for any U ∈ CO(G), there exists V ∈ CO(G) such that aV b ⊆ UabU for all

a, b ∈ G with pk(abH) = aH.

Definition 17. Let (pk) be a smooth combing of polynomial growth. Define

Rk : Tc(G)→ Tc(G2), Rkφ(a, b) :=

φ(ab) if pk(abH) = aH;0 otherwise.

We say that (pk) is compatible with T (G) if the sequence of operators (Rk) isequibounded.

Theorem 18. Let G be a totally disconnected, locally compact group and let T (G)be an unconditional smooth convolution algebra of rapid decay on G. Suppose alsothat the function space T (G2) on G2 is unconditional. If G/H for some com-pact open subgroup H ⊆ G admits a smooth combing of polynomial growth that iscompatible with T (G), then T (G) is isocohomological.

5.3. Proof of Theorem 18.

Lemma 19. Suppose that (pk) is a combing. Then for any controlled subset S ⊆Xn there is a controlled subset S ⊆ Xn+1 such that suppφ ⊆ S implies suppD(φ) ⊆S for all φ ∈ C[Xn].

Proof. Since S is controlled, there is a finite subset F ⊆ X such that xi ∼F xj forall i, j ∈ 0, . . . , n, (xi) ∈ S. Since (pk) is a combing, we can find a finite subsetF ⊆ X such that pk(xi) ∼F pk+1(xi) and pk(xi) ∼F pk(xj) for all k ∈ N and all(xj) ∈ S. Let F ′ := HFHFH ⊆ G/H. If x ∼F y ∼F z, then x ∼F ′ z. Hence(pk(x0), . . . , pk(xj), pk+1(xj), . . . , pk+1(xn)

)is controlled by F ′ for all (xi) ∈ S.

This means that all summands in D(x0, . . . , xn) are controlled by F ′.

Hence D′ := idH(G)⊗D preserves controlled supports in C(G×X•, Tc). We haveseen in (14) that

C(G×X•, Tc) ∼=⊕

ξ∈X′•

Fix(Stab ξ, Tc(G2)

).

The isomorphism sends φ : G×Xn → C to the family of functions (φξ)ξ∈X′n

definedby φξ(a, b) := φ(ab, aξ). Thus we may describe any operator on C(G×X•, Tc) by ablock matrix. In particular, we get D′ = (D′ξη)ξ,η∈X′

•with certain operators

D′ξη : Fix(Stab η, Tc(G2)

)→ Fix

(Stab ξ, Tc(G2)

).

The fact that D′ preserves controlled supports means that for fixed η we haveD′ξη = 0 for all but finitely many ξ. Thus the whole operator D′ is bounded if andonly if all its matrix entries D′ξη are bounded.

For j, n ∈ N, n ≥ j, define pjk : Xn → Xn+2 by

pjk

((x0, . . . , xn)

):=

(pk(x0), . . . , pk(xj), pk+1(xj), . . . , pk+1(xn)

).

Then the operator Djk : C[Xn]→ C[Xn+1] is given by

Djkφ(ξ) =∑

η ∈ p−1jk (ξ)

φ(η).


Let D′jk = idH(G)⊗Djk and let D′jk,ξη be the matrix entries of D′jk with respect tothe decomposition (14). Thus D′ξη =

∑k∈N

∑nj=0(−1)jD′jk,ξη. Writing φξ(a, b) =

φ(ab, aξ) and φη(ag, g−1b) = φ(ab, agη), we get

(17) D′jk,ξηψ(a, b) =∑

g ∈ G/ Stab(η) | pjk(agη) = aξψ(ag, g−1b)

= vol(Stab η)−1

∫g ∈ G | pjk(agη) = aξ

ψ(ag, g−1b) dg

for ψ ∈ Fix(Stab η, Tc(G2)

). The right hand side of (17) makes sense for arbitrary

ψ ∈ Tc(G2) and extends D′jk,ξη to an operator on Tc(G2). Now we fix ξ, η untilfurther notice and sometimes omit them from our notation.

Let U ⊆ Stab(η) be an open subgroup and φ ∈ Tc(GU). Let µU ∈ H(G) bethe normalised Haar measure of U , that is, suppµU = U and µU (g) = vol(U)−1 forg ∈ U . Equation (17) yields

vol(Stab η)D′jk,ξη(φ⊗ µU )(a, b)

= vol(U)−1

∫g ∈ bU | pjk(agη) = aξ

φ(ag) dg =

φ(ab) if pjk(abη) = aξ;0 otherwise.

Let χj,ξη(a, b) be the number of k ∈ N with pjk(abη) = aξ and let

χ(a, b) = χξη(a, b) :=n∑

j=0

(−1)jχj,ξη(a, b).

These numbers are finite for any a, b ∈ G for the same reason that guarantees thatthe sum defining D is finite on each basis vector. Define

A = Aξη : Tc(G)→ Tc(G2), Aφ(a, b) = χ(a, b) · φ(ab).

Our computation shows that Aφ = D′ξη(φ⊗µU )·vol(Stab η) if U is an open subgroupof Stab η and φ ∈ Tc(GU).

Lemma 20. The operator D′ξη is bounded if and only if Aξη is bounded.

Proof. The boundedness of D′ξη implies that A is bounded on Tc(GU) for suffi-ciently small U and hence on all of Tc(G). Suppose conversely that A is bounded.We turn Tc(G2) into an (incomplete) bornological right Tc(G)-module by

(f1 ⊗ f2) ∗ f3 := f1 ⊗ (f2 ∗ f3).This bilinear map Tc(G2)×Tc(G)→ Tc(G2) is bounded because the convolution inT (G) is bounded. The operators D′jk,ξη and hence D′ξη are Tc(G)-module homo-morphisms by (17). Let U ⊆ Stab(η) be open and φ1, φ2 ∈ Tc(GU). Then

D′ξη(φ1 ⊗ φ2) = D′ξη(φ1 ⊗ µU ) ∗ φ2 = vol(Stab η)−1A(φ1) ∗ φ2.

This implies the boundedness of D′ξη because U is arbitrarily small and A and theconvolution Tc(G2)→ Tc(G) are bounded.

Lemma 21. If the combing is smooth, then for any U ∈ CO(G) there is V ∈ CO(G)such that A maps Tc(GU) into Tc(G2V 2).

Proof. Let a, b ∈ G. Clearly, χ(a, by) = χ(a, b) for y ∈ Stab(η). Since pk isH-equivariant, so is pjk. Hence χ(ha, b) = χ(a, b) for h ∈ H. Therefore, Aφ(ua, b) =Aφ(a, b) = Aφ(a, bu) for φ ∈ Tc(GU) provided U ⊆ H ∩ Stab(η). Moreover, wehave Aφ(a, xb) = Aφ(ax, b) if x ∈ Stab(ξ).

We may assume that the zeroth components η0 and ξ0 are H: any G-orbiton Xn has such a representative. Then pjk(abη) = aξ implies pk(abH) = aH. By

20 RALF MEYER

the definition of a smooth combing, there is V ∈ CO(G) such that aV b ⊆ UabUwhenever pjk(abη) = aξ for some k ∈ N. Hence φ(avb) = φ(ab) for (a, b) ∈ suppχand v ∈ V . We may shrink V such that V ⊆ Stab(ξ)∩U . Then χ(a, vb) = χ(a, b) =χ(av, b) as well, so that A maps Tc(GU) to Tc(G2V 2).

Definition 17 requires the following sequence of operators to be equibounded:

Rk : Tc(G)→ Tc(G2), Rkφ(a, b) :=

φ(ab) if pk(abH) = aH;0 otherwise.

Hence R :=∑

k∈N(k+1)−2Rk is bounded. We have Rφ(a, b) = φ(ab)χ′(a, b), where

χ′(a, b) =∑

k ∈ N | pk(abH) = aH(k + 1)−2.

Now let S ⊆ Tc(G) be bounded. Then S ⊆ Tc(GU) for some U ∈ CO(G). Wehave already found V ∈ CO(G) such that Aφ ∈ Tc(G2V 2) for all φ ∈ Tc(GU).Since pointwise multiplication by the scale σ is bounded on Tc(GV ), it followsthat the set of functions σ(a)Nσ(b)NR(S) is bounded in Tc(G2V 2). We claimthat σ(a)Nσ(b)NR(S) dominates A(S) for sufficiently large N . Since T (G2) isunconditional, this implies the boundedness of A.

Since A and R are multiplication operators, the claim follows if χ′σN dominatesχj,ξη for any fixed j, ξ, η. This is what we are going to prove. Let k0(ab) be theleast k0 such that pk(abηj) = abηj for all k ≥ k0. The polynomial growth ofthe combing implies that k0(ab) is dominated by Cσ(ab)N ≤ Cσ(a)Nσ(b)N forsufficiently large C,N . Since ξj 6= ξj+1, we have pjk(abη) /∈ Gξ for k ≥ k0. Hence,if pjk(abη) = aξ, then k < k0. We choose the set of representatives X ′

• such thatξ0 = H for all ξ ∈ X ′

•. Then pjk(abη) = aξ implies pk(abH) = aH. Therefore, foreach summand 1 in χj,ξη(a, b) there is a summand 1/(k+1)2 ≥ 1/k2

0 in χ′(a, b). Thisyields the desired estimate χj,ξη(a, b) ≤ k0(a, b)2χ′(a, b) ≤ C2σ(a)2Nσ(b)2Nχ′(a, b).Thus the operators Aξη are bounded for all ξ, η. This implies boundedness of D′ξη

by Lemma 20. By Lemma 19, it follows that D′ is bounded. Finally, Lemma 14yields that T (G) is isocohomological. This finishes the proof of Theorem 18.

6. A smooth combing for reductive p-adic groups

The following theorem is the main goal of this section. In addition, we prove avariant (Theorem 28) that deals with the subcategories of χ-homogeneous repre-sentations for a character χ : C(G)→ U(1) of the connected centre of G.

Theorem 22. The Schwartz algebra S(G) of a reductive p-adic group G is isoco-homological.

Proof. We are going to apply Theorem 18. Let σ be the standard scale as in thedefinition of the Schwartz algebra and let T (G) := S(G). The space T (G2) isdefined as T (G) ⊗ T (G). This notation is permitted because of Lemma 2, whichidentifies S(G) ⊗ S(G) with the Schwartz algebra of G2 (which is again a reductivep-adic group). Clearly, the Schwartz algebras S(G) and S(G2) are unconditionalsmooth convolution algebras of rapid decay.

We let BT = BT (G) be the affine Bruhat-Tits building of G, as defined in[5, 23]. This is a Euclidean building on which G acts isometrically, properly, andcocompactly. Let C(G) be the connected centre of G, so that the quotient G/C(G)is semi-simple. In Section 7, we will also use the variant BT

(G/C(G)

)of BT (G),

which we call the semi-simple affine Bruhat-Tits building of G.Let G be the connected component of G as an algebraic group. Thus G is a

reductive group and G is a finite extension of G. Inspection of the definition in [23]


shows that the buildings for G and G are equal. We remark that it is not hard toreduce the case of general reductive p-adic groups to the special case of connectedsemi-simple groups, or even connected simple groups. At first I followed this routemyself. Eventually, it turned out that this intermediate step is unnecessary becauseall arguments work directly in the generality we need.

Let ξ0 ∈ BT , H := Stab(ξ0), and X := G/H. We have H ∈ CO(G), and X maybe identified with the discrete subset Gξ0 ⊆ BT . We need a combing of X. Asa preparation, we construct a combing of BT , using that Euclidean buildings areCAT(0) spaces, that is, have “non-positive curvature” (see [4,5]). In particular, anytwo points in BT are joined by a unique geodesic. For ξ ∈ BT , let

p(ξ) : [0, d(ξ, ξ0)]→ BT , t 7→ pt(ξ),

be the unit speed geodesic segment from ξ0 to ξ; extend this by pt(ξ) := ξ fort > d(ξ, ξ0). Restricting to t ∈ N, we get a sequence of maps pk : BT → BT .

Lemma 23. The maps pk for k ∈ N form a combing of linear growth of BT .

This means d(pk(ξ), pk(η)

)≤ R · d(ξ, η) + R and d

(pk(ξ), pk+1(ξ)

)≤ R for all

ξ, η ∈ BT , k ∈ N, for some R > 0. Linear growth means that the least k0 such thatpk(ξ) is constant for k ≥ k0 grows at most linearly in d(ξ, ξ0).

Proof. By construction, d(ps(ξ), pt(ξ)

)≤ |t−s| for all s, t ∈ R+, ξ ∈ BT , and ps(ξ)

is constant for s ≥ d(ξ, ξ0). The lemma follows if we prove the following claim:d(pt(ξ), pt(η)

)≤ d(ξ, η) for all ξ, η ∈ BT , t ∈ R+.

Fix ξ, η ∈ BT and t ∈ R+. We may assume d(ξ, ξ0) ≥ d(η, ξ0) (otherwiseexchange ξ and η) and d(ξ, ξ0) ≥ t (otherwise pt(ξ) = ξ and pt(η) = η). Let d∗ bethe usual flat Euclidean metric on R2. Let ξ∗ and η∗ be points in R2 with

d∗(ξ∗, 0) = d(ξ, ξ0), d∗(η∗, 0) = d(η, ξ0), d∗(ξ∗, η∗) = d(ξ, η).

The CAT(0) condition means that distances between points on the boundary of thegeodesic triangle (ξ, η, ξ0) are dominated by the distances between the correspond-ing points in the comparison triangle (ξ∗, η∗, 0). Here the point pt(ξ) correspondsto the point pt(ξ)∗ on [0, ξ∗] of distance t from the origin. The point pt(η) cor-responds to the point pt(η)∗ of distance mint, d(η, ξ0) from the origin. An easy

@@

@@

@@

@@@0 ξ∗

η∗

BB

BBB

BBBBB

BBBBBB

BBBBBBBB

BBBBBBBBB

JJ

JJ

JJ

JJJ

pt(ξ)∗

pt(η)∗

Figure 1. A comparison triangle

computation or a glance at Figure 1 shows d∗(pt(ξ)∗, pt(η)∗) ≤ d∗(ξ∗, η∗). By theCAT(0) condition, this implies d

(pt(ξ), pt(η)

)≤ d(ξ, η).

We identify G/H with the orbit Gξ0 ⊆ BT . Since the group action is cocompact,there is some R > 0 such that for any ξ ∈ BT there exists ξ′ ∈ Gξ0 with d(ξ, ξ′) < R.We let p′k(ξ) for ξ ∈ BT be a point in Gξ0 with d

(p′k(ξ), pk(ξ)

)< R. We claim that

any such choice defines a combing of G/H.If we equip Gξ0 with the metric d from BT , the maps p′k on Gξ0 still form a

combing in the metric sense because they are “close” to the combing (pk). The

22 RALF MEYER

subspace metric from BT and the relations ∼F in Definition 15 generate the samecoarse geometric structure on X. That is, for any R > 0 there is a finite subsetF ⊆ X such that d(x, y) ≤ R implies x ∼F y, and for any finite subset F ⊆ Xthere is R > 0 such that x ∼F y implies d(x, y) ≤ R. This is easy to verify byhand. Alternatively, it follows from the uniqueness of coarse structures mentionedafter Definition 15. Hence (p′k) is a combing of G/H in the sense we need.

We also need the combing to be smooth. To get this, we must choose the basepoint ξ0 and the approximations p′k(ξ) more carefully. This requires some geometricfacts about the apartments in the building. Let K be the non-Archimedean localfield over which G is defined. Let S ⊆ G be a maximal K-split torus of G. Wedo not distinguish in our notation between the algebraic groups S and G and theirlocally compact groups of K-rational points. Let X∗(S) and X∗(S) be the groupsof algebraic characters and cocharacters of S, respectively. The R-vector spaceA := X∗(S)⊗ R is the basic apartment of (G,S).

Let Φ ⊆ X∗(S) be the set of roots of G relative to S. Choose a simple systemof roots ∆ ⊆ Φ and let A+ ⊆ A be the corresponding closed Weyl chamber:

A+ := x ∈ A | α(x) ≥ 0 for all α ∈ ∆.

Let W be the Weyl group of the root system Φ. It is the Coxeter group generatedby orthogonal reflections in the hyperplanes α(x) = 0 for α ∈ ∆. The positivecone A+ is a fundamental domain for this action, that is, W (A+) = A (see [10]).

Let Z ⊆ G be the centraliser of S. There is a canonical homomorphism ν : Z → A(see [23, (1.2)]). Its kernel is compact and its range is a lattice Λ ⊆ A; that is, Λis a discrete and cocompact subgroup of A. Moreover, let Λ+ := Λ ∩ A+. Since Λis free Abelian, we can lift it to a subgroup of Z and view Λ ⊆ Z ⊆ G. Let Φaf

be the set of affine roots as in [23, (1.6)]. These are affine functions α : A → R ofthe form α(x) = α0(x) + γ with α0 ∈ Φ and certain γ ∈ R. Recall that Φ ⊆ Φaf

and that Φaf is invariant under translation by Λ. The subsets of A of the formx ∈ A | α(x) = 0 and x ∈ A | α(x) ≥ 0 for α ∈ Φaf are called walls andhalf-apartments, respectively.

We define the closure cl(Ω) ⊆ A of a non-empty subset Ω ⊆ A as the intersectionof all closed half-apartments containing Ω (see [5, (7.1.2)]). We claim that

(18) cl(0, ξ) = A+ ∩ (ξ −A+)

for all ξ ∈ Λ+. Let B := A+ ∩ (ξ −A+). This is an intersection of half-apartmentscontaining 0 and ξ because Φ ∪ (ξ − Φ) ⊆ Φaf . Hence cl(0, ξ) ⊆ B. It remainsto show that any half-apartment C containing 0 and ξ also contains B. Let C bedefined by the equation α ≥ 0 for some affine root α with linear part α0 ∈ Φ. Wedistinguish the cases α0 > 0 and α0 < 0. If α0 > 0, then α(x) = α0(x) + α(0) ≥α0(x) is non-negative on A+; if α0 < 0, then α(ξ − x) = −α0(x) + α(ξ) ≥ −α0(x)is non-negative on A+, so that α is non-negative on ξ −A+. Thus B ⊆ C in eithercase. This finishes the proof that cl(0, ξ) = B.

Lemma 24. There is R > 0 such that for all ξ ∈ Λ+ and all η ∈ cl(0, ξ), thereis η′ ∈ Λ ∩ cl(0, ξ) with d(η, η′) ≤ R.

Proof. Let ∆ = α1, . . . , αs be the system of simple roots that determines A+.If G is semi-simple, these roots form a basis of A. In general, they are linearlyindependent, so that we can extend them to a basis by certain αj for s < j ≤ r.Define a vector space isomorphism γ : A → Rr by γ(η)j = αj(η) for j = 1, . . . , r.This identifies A+ with the set of (xj) ∈ Rr with xj ≥ 0 for 1 ≤ j ≤ s. Equation (18)identifies B := cl(0, ξ) with

γ(B) = (xj) ∈ Rr | 0 ≤ xj ≤ γ(ξ)j , j = 1, . . . , s.


We may assume αj ∈ X∗(S) ⊗ Q ⊆ A, so that γ(Λ) ⊆ Qr. Replacing γ byn−1γ for some n ∈ N∗, we can achieve Zr ⊆ γ(Λ). Hence if (xj) ∈ γ(B), then thetruncated vector b(xj)c := (bxjc) belongs to γ(B ∩ Λ). It satisfies |xj − bxjc| < 1for all j. Since the norm ‖γ(η)‖∞ is equivalent to the Euclidean norm on A, wehave d(γ−1bγ(η)c, η) < R for all η ∈ B for some R > 0.

The building BT can be defined as the quotient of G×A by a certain equivalencerelation. We may view gA for g ∈ G as a subspace of BT ; these are the apartmentsof BT . We now choose ξ0 to be the origin in A ⊆ BT . Recall that H ⊆ G denotesthe stabiliser of ξ0. We have the Cartan decomposition G = HΛ+H by [23, (3.3.3)],so that Gξ0 = HΛ+ξ0. Let G ⊆ G be the connected component of the identity (asan algebraic variety) and let H := H ∩ G. These are open normal subgroups offinite index in G and H, respectively, and BT (G) is isomorphic to BT (G) equippedwith a canonical action of G.

Choose Λ′+ ⊆ Λ+ to contain one representative for each H-orbit in Gξ0. Fixξ ∈ Λ′+ and k ∈ N. We further decompose Hξ as a disjoint union of finitely manyH-orbits Hhjξ = hjHξ for suitable h1, . . . , hN ∈ H. We let p′k(ξ) be some pointin Λ ∩ cl(0, ξ) ⊆ A ⊆ BT that has minimal distance from pk(ξ).

Proposition 25. Let Ω ⊆ A ⊆, Ω 6= ∅. If g ∈ G satisfies gx = x for all x ∈ Ω,then gx = x for all x ∈ cl(Ω). (This may fail if we allow g ∈ G.)

Proof. The proof requires some facts about stabilisers of points in BT , which areconveniently summarised in [20, Section I.1]. The subgroups PΩ ⊆ G defined theremanifestly satisfy PΩ = Pcl(Ω). This implies our claim because

PΩ = g ∈ G | gx = x ∀x ∈ Ω.

Proposition 25 allows us to define p′k(hjhξ) := hjhp′k(ξ) for all h ∈ H. Let-

ting ξ, j vary, we get a map p′k : Gξ0 → Gξ0. It is H-equivariant because His normal in H. Since pk(ξ) ∈ cl(0, ξ), Lemma 24 yields R > 0 such thatd(pk(ξ), p′k(ξ)

)≤ R for all k ∈ N, ξ ∈ Λ′+. The same holds for ξ ∈ Gξ0 be-

cause pk is H-equivariant and G acts isometrically on BT . Moreover, p′0(ξ) = ξ0and p′k(ξ) = ξ for k ≥ d(ξ, ξ0). Thus (p′k) is a combing of G/H of linear growth.

Lemma 26. The combing (p′k) is smooth.

Proof. There is a decreasing sequence (Un)n∈N in CO(G) such that each Un isnormal in H and can be written as U+

n · U−n = U−n · U+n with

(19) λU+n ⊆ U+

n λ, U−n λ ⊆ λU−nfor all λ ∈ Λ+ (see [20, Section I.2]). Let a, b ∈ G satisfy p′k(abH) = aH. Writeab = h1λh2 with h1, h2 ∈ H, λ ∈ Λ′+, and h1 chosen as carefully as above if G isdisconnected. Then a = h1p

′k(λ)h3 with the same h1 and some h3 ∈ H. Hence

b = h−13 p′k(λ)−1λh2. Since p′k(λ) ∈ cl(0, λ), equation (18) yields p′k(λ) ∈ Λ+ and

p′k(λ)−1λ ∈ Λ+. Using (19) and that Un is normal in H, we get

aUnb = h1p′k(λ)U+

n U−n p

′k(λ)−1λh2 ⊆ h1U

+n p

′k(λ)p′k(λ)−1λU−n h2

⊆ h1UnλUnh2 = Unh1λh2Un = UnabUn.

Thus the combing (p′k) is smooth in the sense of Definition 16.

Finally, the compatibility condition of Definition 17 is easy to check using theexplicit description of S(G2) in Lemma 2. In order to cover also the Schwartzalgebras for discrete groups, which are defined by `1-estimates, we define spaces

Lσp (G) := f : G→ C | f · σk ∈ Lp(G) ∀k ∈ N

24 RALF MEYER

for all 1 ≤ p <∞, and equip them with the evident bornology: a subset T ⊆ Lσp (G)

is bounded if and only if for any k ∈ N there is Ck such that ‖f ·σk‖Lp(G) ≤ Ck forall f ∈ T . Let Lσ

p (GU) be the subspace of U -bi-invariant functions in Lσp (G).

For p = 2, this agrees with the previous definition, so that Lemma 2 yields

S(G) = lim−→Lσ2 (GU), S(G2) = lim−→Lσ

2 (G2U2).

Lemma 27. Let (pk) be a combing of polynomial growth on G/H. Then thesequence of operators (Rk) used in Definition 17 is uniformly bounded as operatorsLσ

p (G)→ Lσp (G2). Here we use the scale σ(a, b) := σ(a)σ(b) on G2.

Proof. The operator Wφ(x, y) := φ(x, x−1y) is an isometry of Lp(G2). It is alsosufficiently compatible with the scale on G2 for W and its inverse to be boundedlinear operators on Lσ

p (G2). We have Rkφ(a, b) = φ(ab)1pk(abH)(a), where 1pk(abH)

denotes the characteristic function of pk(abH) ⊆ G. Hence

WRkφ(x, y) = φ(y) · 1pk(yH)(x).

Since the combing (pk) has polynomial growth, σ(pk(yH)

)is controlled by a poly-

nomial in σ(y). The boundedness of W Rk is now immediate because all cosetsxH have volume 1. This implies the boundedness of Rk.

Since the combing (p′k) is smooth, for any U ∈ CO(G) there exists V ∈ CO(G)such that Rk maps Tc(GU) into Tc(G2V 2). Together with Lemma 27 for p = 2,this yields that the combing (p′k) is compatible with S(G) in the sense of Defini-tion 17. We have now verified all the hypotheses of Theorem 18. Thus S(G) isisocohomological.

6.1. Decomposition with respect to the centre of G. As before, we let Gbe a reductive p-adic group. Let C(G) ⊆ G be the connected centre of G andlet χ : C(G) → U(1) be a unitary character on C(G). Let Modχ(G) be the fullsubcategory of Mod(G) whose objects are the representations π : G→ Aut(V ) thatsatisfy π(z) = χ(z)idV for all z ∈ C(G). Let

Modχ

(S(G)

):= Mod

(S(G)

)∩Modχ(G)

be the subcategory of tempered representations in Modχ(G). The class of exten-sions with a bounded linear section turns Modχ(G) and Modχ

(S(G)

)into exact

categories, so that we can form the derived categories Derχ(G) and Derχ(S(G)

).

Let Gss := G/C(G), this is again a reductive p-adic group. If χ = 1, thenModχ(G) = Mod(Gss) and Modχ

(S(G)

)= Mod

(S(Gss)

). In general, there are

quasi-unital bornological algebras Hχ(G) and Sχ(G) such that

Modχ(G) ∼= Mod(Hχ(G)

), Modχ

(S(G)

) ∼= Mod(Sχ(G)

).

We briefly recall their well-known definitions. A C(G)-invariant subset of G iscalled C(G)-compact if its image in Gss is compact. Let Hχ(G) be the spaceof locally constant functions f : G → C with C(G)-compact support such thatf(z−1g) = χ(z)f(g) for all g ∈ G, z ∈ C(G). If f1, f2 ∈ Hχ(G), then the functionh 7→ f1(h)f2(h−1g) is C(G)-invariant, so that

f1 ∗ f2(g) :=∫

Gss

f1(h)f2(h−1g) dh

makes sense; here dh denotes the Haar measure on Gss. This turns Hχ(G) into analgebra, which we equip with the fine bornology.

Let Lσ2 (G)χ be the space of functions f : G→ C that satisfy f(z−1g) = χ(z)f(g)

for all g ∈ G, z ∈ C(G), and such that zg 7→ |f(g)| is an element of Lσ2 (Gss). Let

Sχ(G) := lim−→Lσ2 (GU)χ


where U runs through the set of compact open subgroups with U ∩ C(G) ⊆ kerχ.The same estimates as for S(Gss) show that the convolution on Hχ(G) extends toa bounded multiplication on Sχ(G).

Consider the map

ρ : H(G)→ Hχ(G), ρf(g) :=∫

C(G)

χ(z)f(zg) dz.

For appropriately normalised Haar measures, this is a surjective, bounded algebrahomomorphism; that is, Hχ(G) is a quotient algebra of H(G). Using ρ, we can pullback Hχ(G)-modules to H(G)-modules. This construction maps essential modulesagain to essential modules (ρ is a proper morphism in the notation of [17]). Thuswe have got a functor ρ∗ : Mod

(Hχ(G)

)→ Mod

(H(G)

). Since ρ is surjective, ρ∗ is

fully faithful. Thus Mod(Hχ(G)

)becomes a full subcategory of Mod(G). It is easy

to identify this subcategory with Modχ(G). If (V, π) ∈ Modχ(G), then V becomesan essential Hχ(G)-module by

π(f) :=∫

Gss

f(g)π(g) dg.

This is well-defined because f(gz)π(gz) = f(g)π(g) for all g ∈ G, z ∈ C(G).We can extend ρ to a bounded algebra homomorphism ρS : S(G) → Sχ(G).

The map ρS has a bounded linear section and its kernel is the closure of ker ρ ⊆H(G). Therefore, bounded algebra homomorphisms Sχ(G) → End(V ) correspondto bounded algebra homomorphisms S(G) → End(V ) whose restriction to H(G)vanishes on ker ρ. Equivalently, Mod

(Sχ(G)

) ∼= Mod(S(G)

)∩ Modχ(G). Thus

Hχ(G) and Sχ(G) have the required properties.

Theorem 28. The embedding Hχ(G)→ Sχ(G) is isocohomological.

Proof. Let Hχ(G)op be the opposite algebra of Hχ(G), so that Mod(Hχ(G)op) isthe category of right Hχ(G)-modules. Since Hχ(G)op ∼= Hχ−1(G), we have anisomorphism of categories Mod(Hχ(G)op) ∼= Modχ−1(G).

Equip X ∈ Mod(Hχ(G)op), V ∈ Mod(Hχ(G)

)with the associated representa-

tions of G. We equip X ⊗ V with the diagonal representation. Since χ and χ−1

cancel, C(G) acts trivially on X ⊗ V . Thus we obtain a bifunctor

(20) Mod(Hχ(G)op)×Mod(Hχ(G)

)→ Mod(Gss), (X,V ) 7→ X ⊗ V.

This functor is evidently exact for extensions with a bounded linear section. More-over, we claim that X ⊗V is projective if X or V are projective. It suffices to treatthe case where X is projective. We may even assume that X is a free essentialmodule X0 ⊗ Hχ(G). The diagonal representation on X0 ⊗ Hχ(G) ⊗ Y is isomor-phic to the regular representation ρg⊗ 1⊗ 1 on H(Gss) ⊗X0 ⊗Y . The intertwiningoperator is given by Φ(x⊗ f ⊗ y)(g) := f(g)x⊗ gy for all g ∈ G; this function onlydepends on the class of g in Gss.

Let X ∈ Mod(Hχ(G)op), V ∈ Mod(Hχ(G)

). Then X ⊗Hχ(G) V is defined as the

quotient of X ⊗ V by the closed linear span of x ∗ f ⊗ v − x ⊗ f ∗ v for x ∈ X,f ∈ Hχ(G), v ∈ V . Since ρ : H(G) → Hχ(G) is surjective, this is the same asX ⊗H(G) V , which we have identified with X ⊗G V in Section 3. Thus

X ⊗Hχ(G) V ∼= C ⊗Gss (X ⊗ V ).

The same assertion holds for the total derived functors because the bifunctor in (20)is exact and preserves projectives. Especially, we get

Sχ(G) ⊗LHχ(G) Sχ(G) ∼= C ⊗L

Gss

(Sχ(G) ⊗ Sχ(G)

).

26 RALF MEYER

Here Sχ(G) ⊗ Sχ(G) is equipped with the inner conjugation action of Gss. Weidentify Sχ(G) ⊗Sχ(G) ∼= Sχ×χ(G×G) as in Lemma 2. By [17, Theorem 35.2], theembedding Hχ(G)→ Sχ(G) is isocohomological if and only if Sχ(G) ⊗L

Hχ(G) Sχ(G)is a resolution of Sχ(G). Thus the assertion that we have to prove is equivalent to

C ⊗LGssSχ×χ(G×G) ∼= Sχ(G).

We already know C ⊗LGssS(Gss×Gss) ∼= S(Gss) because S(Gss) is isocohomological

(Theorem 22) and this condition is equivalent to S(Gss) being isocohomological.Now we choose a continuous section s : Gss → G; this is possible because G is

totally disconnected. It yields bornological isomorphisms

Ψ′ : Sχ(G)→ S(Gss), Ψ′f(g) := f s(g),Ψ: Sχ×χ(G×G)→ S(Gss ×Gss), Ψf(g, h) := f

(s(g), s(g)−1s(gh)

).

The isomorphism Ψ intertwines the inner conjugation actions of Gss on Sχ×χ(G×G)and S(Gss ×Gss). Thus we get isomorphisms

(21) C ⊗LGssSχ×χ(G×G) ∼= C ⊗L

GssS(Gss ×Gss) ∼= S(Gss) ∼= Sχ(G).

It is easy to see that the composite isomorphism is induced by the convolution mapin Sχ(G).

7. Applications to representation theory

Let G be a reductive p-adic group, let C(G) be its centre, and let Gss := G/C(G).So far we have used very large projective H(G)-module resolutions, which offergreat flexibility for writing down contracting homotopies. Now we consider muchsmaller projective resolutions, which are useful for explicit calculations. We writeMod(χ)(G) if it makes no difference whether we work in Mod(G) or Modχ(G) forsome character χ : C(G) → U(1). Similarly, we write H(χ)(G) and S(χ)(G). Theactual applications of our main theorem are contained in Sections 7.1, 7.7 and 7.8.The other subsections contain small variations on known results. Our presentationdiffers somewhat from the accounts in [20, 25] because we want to exhibit connec-tions with K-theory and assembly maps. I would like to thank Peter Schneider forexplaining to me some of the results of [20].

7.1. Cohomological dimension. Let rkG = dimBT (G) be the rank of G.

Theorem 29. The cohomological dimensions of the exact categories Mod(G) andMod

(S(G)

)are (at most) rkG; that is, any object has a projective resolution of

length rkG. Similarly, the cohomological dimensions of Modχ(G) and Modχ

(S(G)

)for a character χ : C(G)→ U(1) are at most rkGss.

Proof. The assertions are well-known for Mod(G) and Modχ(G). For the proof,equip BT = BT (G) with a CW-complex structure for whichG acts by cellular maps.Then the cellular chain complex C•(BT ) is a projective H(G)-module resolutionof the trivial representation of length rkG. The chain complex C•(BT ) ⊗ V withthe diagonal representation of G is a projective H(G)-module resolution of V forarbitrary V ∈ Mod(G). If V ∈ Modχ(G), we use the building BT (Gss) instead;C•

(BT (Gss)

)⊗ V is a projective resolution of V in Modχ(G).

What is new is that we get the same assertions for Mod(S(G)

)and Mod

(Sχ(G)

).

Since the argument is the same in both cases, we only write it down for S(G). LetV ∈ Mod

(S(G)

)and let P• → V be a projective resolution in Mod(G) of length

rkG. Then S(G) ⊗G P• has the homotopy type of

(22) S(G) ⊗LG V ∼= S(G) ⊗L

S(G) V∼= V


because S(G) is isocohomological (Theorem 22); here we use one of the equiva-lent characterisations of isocohomological embeddings listed in Section 4.2. Equa-tion (22) means that S(G) ⊗G P• is a resolution of V . This resolution is projectiveand has length rkG.

Conversely, there is V ∈ Mod(G) with Extrk GG (V, V ) 6= 0. Hence the cohomo-

logical dimension of Mod(G) is equal to rkG. We can even take V tempered andirreducible. Hence Extrk G

S(G)(V, V ) 6= 0 as well because S(G) is isocohomological.Thus Mod

(S(G)

)also has cohomological dimension equal to rkG. Similarly, the

cohomological dimension of Modχ(G) and Modχ

(S(G)

)is equal to rkGss.

7.2. Finite projective resolutions. We use a result of Joseph Bernstein to attachan Euler characteristic Eul(V ) in K0

(H(χ)(G)

)to a finitely generated representation

V ∈ Mod(χ)(G).

Definition 30. A smooth representation V is called finitely generated if there existfinitely many elements v1, . . . , vn such that the map

H(G)n → V, (f1, . . . , fn) 7→n∑

j=1

fj ∗ vj

is a bornological quotient map.

An admissible representation is finitely generated if and only if it has finite length,that is, it has a Jordan-Hölder series of finite length.

SinceH(G)n carries the fine bornology, the same is true for its quotients. Hence afinitely generated representation necessarily belongs to Modalg(G). In the situationof Definition 30, there exists U ∈ CO(G) fixing vj for all j ∈ 1, . . . , n. Thus weget a bornological quotient map H(G/U)n V . Conversely, H(G/U)n is finitelygenerated and projective. Thus a smooth representation is finitely generated if andonly if it is a quotient of H(G/U)n for some U ∈ CO(G). If V ∈ Modχ(G), then wemay replace H(G/U)n by Hχ(G/U)n for some U ∈ CO(G) with χ|U∩C(G) = 1.

Definition 31. An object of Mod(χ)(G) has type (FP) if it admits a resolution offinite length by finitely generated projective objects of Mod(χ)(G). Such a resolutionis called a finite projective resolution.

Theorem 32 (Joseph Bernstein). An object of Mod(χ)(G) has type (FP) if andonly if it is finitely generated.

Proof. It is trivial that representations of type (FP) are finitely generated. Con-versely, if V is finitely generated, then V is a quotient of a finitely generated projec-tive representation, say, ∂0 : H(χ)(G/U)n V . By [3, Remark 3.12], subrepresenta-tions of finitely generated representations are again finitely generated. Especially,ker ∂0 is finitely generated. By induction, we get a resolution (Pn, ∂n) of V byfinitely generated projective objects. By Theorem 29, the kernel of ∂n : Pn → Pn−1

is projective for sufficiently large n. Hence

0→ ker ∂n → Pn → . . .→ P0 → V

is a finite projective resolution.

The algebraic K-theory K0

(H(χ)(G)

)is the Grothendieck group of the monoid

of finitely generated projective H(χ)(G)-modules. This is so because H(χ)(G) is aunion of unital subalgebras.

Definition 33. Let V ∈ Mod(χ)(G) be finitely generated. Then V is of type (FP)by Bernstein’s Theorem 32. Choose a finite projective resolution

0→ Pn → . . .→ P0 → V → 0.

28 RALF MEYER

The Euler characteristic of V is defined by

Eul(V ) :=n∑

j=0

(−1)j [Pj ] ∈ K0

(H(χ)(G)

).

We check that this does not depend on the resolution (see also [19, Section 1.7]).Define the Euler characteristic Eul(P•) for finite projective complexes in the obviousfashion. Let P• and P ′• be two finite projective resolutions of V . The identity mapon V lifts to a chain homotopy equivalence f : P• → P ′•. Hence the mapping cone Cf

of f is contractible. The Euler characteristic vanishes for contractible complexes.Hence Eul(Cf ) = 0. This is equivalent to Eul(P•) = Eul(P ′•).

Definition 34. Let

HH0

(H(χ)(G)

):= H(χ)(G)/[H(χ)(G),H(χ)(G)].

The universal trace is a map

truniv : K0

(H(χ)(G)

)→ HH0

(H(χ)(G)

).

If (pij) ∈ Mn

(H(χ)(G)

)is an idempotent with H(χ)(G)n · (pij) ∼= V , then we have

truniv[V ] =[∑

pii

].

The above definitions are inspired by constructions of Hyman Bass in [2], wheretruniv Eul(V ) is constructed for modules of type (FP) over unital algebras.

7.3. Traces from admissible representations. Let W ∈ Mod(χ)(G) be an ad-missible representation. Its integrated form is an algebra homomorphism ρ fromH(χ)(G) to the algebra Endfin(W ) := W ⊗W of smooth finite rank operators on W .Here W denotes the contragradient representation and Endfin(W ) carries the finebornology. Composing ρ with the standard trace on Endfin(W ), we get a tracetrW : HH0

(H(χ)(G)

)→ C and a functional

τW : K0

(H(χ)(G)

)→ Z.

The following computation of τW is a variant of [2, Proposition 4.2].

Proposition 35. Let V,W ∈ Mod(χ)(G), let V be finitely generated projectiveand let W be admissible. Then τW [V ] = dim HomG(V,W ) and HomG(V,W ) isfinite-dimensional.

Proof. The functoriality of K0 for the homomorphism H(χ)(G)→ Endfin(W ) maps[V ] to the class of the finitely generated projective module

V ′ := Endfin(W ) ⊗H(χ)(G) V ∼= W ⊗ (W ⊗G V ) ∼= W dim W⊗GV

over Endfin(W ). Thus τW [V ] = dim W ⊗G V . By adjoint associativity,

Hom(W ⊗G V,C) ∼= HomG

(V,Hom(W ,C)

) ∼= HomG

(V, ˜W

).

We have W ∼= ˜W because W is admissible. Thus τW [V ] = dim HomG(V,W ).

Let V,W ∈ Mod(χ)(G), let W be admissible and let V be finitely generated. ByBernstein’s Theorem 32, there is a finite projective resolution P• → V in Mod(χ)(G).By Proposition 35, HomG(P•,W ) is a chain complex of finite-dimensional vectorspaces. Hence its homology Extn

H(χ)(G)(V,W ) is finite-dimensional as well and

(23)∞∑

n=0

(−1)n dim ExtnH(χ)(G)(V,W ) =

∞∑n=0

(−1)n dim HomG(Pn,W )

=∞∑

n=0

(−1)nτW [Pn] = τW(Eul(V )

).


We call this the Euler-Poincaré characteristic EP(χ)(V,W ) of V and W (compare[20, page 135]).

7.4. Formal dimensions. Evaluation at 1 ∈ G is a trace on H(χ)(G), that is, alinear functional τ1 : HH0

(H(χ)(G)

)→ C. The functional

dim := τ1 truniv : K0

(H(χ)(G)

)→ C

computes the formal dimension for finitely generated projective H(χ)(G)-modules.Recall that an irreducible representation in Modχ(G) is projective if and only if itis supercuspidal. Unless C(G) is compact, Mod(G) has no irreducible projectiveobjects.

We can also define the formal dimension for representations that are square-integrable (see [22]). Let (V, π) ∈ Modχ(G) be irreducible and square-integrable(or, more precisely, square-integrable modulo the centre C(G)). Since irreduciblerepresentations are admissible, V carries the fine bornology. Moreover, square-integrable representations are tempered. Thus the integrated form of π extendsto a bounded homomorphism π : Sχ(G) → Endfin(V ). Since V is irreducible, thishomomorphism is surjective. The crucial property of irreducible square-integrablerepresentation is that there is an ideal I ⊆ Sχ(G) such that I ⊕ kerπ ∼= Sχ(G).Thus π|I : I → Endfin(V ) is an algebra isomorphism. It is necessarily a bornologicalisomorphism because it is bounded and Endfin(V ) carries the fine bornology.

Proposition 36. Let V ∈ Modχ(G) be irreducible and square-integrable. Then Vis both projective and injective as an object of Modχ

(S(G)

).

Proof. The direct-sum decomposition Sχ(G) ∼= kerπ ⊕ Endfin(V ) gives rise to anequivalence of exact categories

Modχ

(S(G)

) ∼= Mod(kerπ)×Mod(Endfin(V )

).

The representation V belongs to the second factor. The algebra Endfin(V ) is canon-ically Morita equivalent to C, so that Mod

(Endfin(V )

)and Mod(C) are equivalent

exact categories. The easiest way to get this Morita equivalence uses a basis in Vto identify Endfin(V ) ∼=

⋃∞n=1Mn(C). Since any extension in Mod(C) splits, any

object of Mod(C) is both injective and projective.

We can also define K0

(S(χ)(G)

), HH0

(S(χ)(G)

), and

truniv : K0

(S(χ)(G)

)→ HH0

(S(χ)(G)

).

It is irrelevant for the following whether we divide by the linear or closed lin-ear span of the commutators in the definition of HH0

(S(χ)(G)

). The trace τ1

extends to a bounded trace τS1 : HH0

(S(χ)(G)

)→ C. This induces a functional

dimS : K0

(S(χ)(G)

)→ Z. An irreducible square-integrable representation V de-

fines a class [V ] ∈ K0

(S(χ)(G)

)by Proposition 36; we define its formal dimension

by dimS V := dimS [V ].The embedding H(χ)(G)→ S(χ)(G) induces natural maps

ι : K0

(H(χ)(G)

)→ K0

(S(χ)(G)

), [V ] 7→ [S(χ)(V ) ⊗H(χ)(G) V ],

ι : HH0

(H(χ)(G)

)→ HH0

(S(χ)(G)

). [f ] 7→ [f ].

These maps are compatible with the universal traces and satisfy τS1 ι = τ1 anddimS ι = dim.

It is shown in [25] that S(χ)(G) is closed under holomorphic functional calculusin the C∗-algebra C∗red,(χ)(G). Hence

K0

(S(χ)(G)

) ∼= K0

(C∗red,(χ)(G)

).

30 RALF MEYER

It follows also that any finitely generated projective module V over S(χ)(G) is therange of a self-adjoint idempotent element in Mn

(S(χ)(G)

)for some n ∈ N. Since

the trace τS1 is positive, we get dimS V > 0 unless V = 0.Yet another notion of formal dimension comes from the theory of von Neumann

algebras. The (χ-twisted) group von Neumann algebra ofG is the closureN(χ)(G) ofH(χ)(G) or S(χ)(G) in the weak operator topology on L2(G)(χ). We may extend τS1to a positive unbounded trace τN

1 on N(χ)(G). Any normal ∗-representation ρ ofN(χ)(G) on a separable Hilbert space is isomorphic to the left regular representationon the Hilbert space

(L2(G)(χ) ⊗ `2(N)

)· pρ for some projection pρ ∈ N(χ)(G) ⊗

B(`2N) where ⊗ denotes spatial tensor products of Hilbert spaces and von Neumannalgebras, respectively; the projection pρ is unique up to unitary equivalence. Wedefine the formal dimension dimN (ρ) to be τN

1 (pρ) ∈ [0,∞]; this does not dependon the choice of pρ.

By definition, we have dimN (L2(G)n(χ) ·e) = dimS [e] if e ∈Mn

(S(χ)(G)

)is a self-

adjoint idempotent. Since S(χ)(G) is closed under holomorphic functional calculusin the reduced group C∗-algebra C∗red,(χ)(G), any idempotent element of S(χ)(G) issimilar to a self-adjoint idempotent. Hence dimN (L2(G)n

(χ) · e) = dimS [e] holds forany idempotent e ∈Mn

(S(χ)(G)

). We have

L2(G)(χ) ⊗S(χ)(G) S(χ)(G)n · e ∼= L2(G)n(χ) · e.

Therefore, if V is a finitely generated projective left S(χ)(G)-module V , then wemay view L2(G)(χ) ⊗S(χ)(G) V as a Hilbert space equipped with a faithful normal∗-representation of N(χ)(G); the resulting representation is uniquely determinedup to unitary equivalence because any two self-adjoint idempotents realising V areunitarily equivalent in S(χ)(G). The formal dimensions from the Schwartz algebraand the von Neumann algebra are compatible in the following sense:

dimN (L2(G)(χ) ⊗S(χ)(G) V ) ∼= dimS(V ).

Thus dimS V only depends on the unitary equivalence class of the associated unitaryrepresentation L2(G)(χ) ⊗S(χ)(G) V .

7.5. Compactly induced representations. Equip U ∈ CO(G) with the restric-tion of the Haar measure from G. The map iGU : H(U) → H(G) that extendsfunctions by 0 outside U is an algebra homomorphism. Hence it induces a map

(iGU )! : Rep(U) ∼= K0

(H(U)

)→ K0

(H(G)

), [V ] 7→ [H(G)⊗H(U) V ].

This is the standard functoriality of K-theory. We denote it by (iGU )! because thisnotation is used in [17]. We call representations of the form (iGU )!(V ) compactlyinduced because H(G)⊗H(U) V ∼= c-IndG

U (V ) (see [15]).Let U, V ∈ CO(G) and suppose that gUg−1 ⊆ V for some g ∈ G. Then we have

iGU = γ−1g iGV iVgUg−1 γg, where γg denotes conjugation by g. One checks that γg

acts trivially on K0

(H(G)

). Hence (iGU )! is the composite of (iGV )! and the map

Rep(U)∼=→ Rep(gUg−1)→ Rep(V ) that is associated to the group homomorphism

U → V , x 7→ gxg−1. Let Sub(G) be the category whose objects are the compactopen subgroups ofG and whose morphisms are these special group homomorphisms.We have exhibited that U 7→ Rep(U) is a module over this category. The variousmaps (iGU )! combine to a natural map

lim−→Sub(G)

Rep(U)→ K0

(H(G)

).


We call this map the assembly map for K0

(H(G)

)because it is a variant of the

Farrell-Jones assembly map for discrete groups (see [12, Conjecture 3.3]), which isin turn closely related to the Baum-Connes assembly map.

The above definitions carry over to Hχ(G) in a straightforward fashion. LetCO

(G;C(G)

)be the set of C(G)-compact open subgroups of G containing C(G).

The projection to Gss identifies CO(G;C(G)

)with CO(Gss). As above, we get

algebra homomorphisms iGU : Hχ(U) → Hχ(G) for U ∈ CO(G;C(G)

). There is an

analogue of the Peter-Weyl theorem for Hχ(U); that is, Hχ(U) is a direct sum ofmatrix algebras. Therefore, finitely generated projective modules over Hχ(U) arethe same as finite-dimensional representations in Modχ(U). This justifies definingRepχ(U) := K0

(Hχ(U)

). As above, we can factor (iGU )! through (iGV )! if U is

subconjugate to V . The relevant category organising these subconjugations is thecategory Sub

(G;C(G)

)whose set of objects is CO

(G;C(G)

)and whose morphisms

are the group homomorphisms U → V of the form x 7→ gxg−1 for some g ∈ G.Thus we get an assembly map

(24) lim−→Sub

(G;C(G)

) Repχ(U)→ K0

(Hχ(G)

).

Let Γ(Gss, dg) ⊆ R be the subgroup generated by vol(U)−1 for U ∈ CO(Gss).Since vol(U)/ vol(V ) ∈ N for V ⊆ U , this group is already generated by vol(U)−1

for maximal compact subgroups U ⊆ Gss. We have Γ(Gss, dg) = αZ for someα > 0 because there are only finitely many maximal compact subgroups andvol(U)/ vol(V ) ∈ Q for all U, V ∈ CO(Gss). The number α depends on thechoice of the Haar measure, of course. We let size(U) := α vol

(U/C(G)

), so that

size(U)−1 ∈ N for all U ∈ CO(G;C(G)

).

Lemma 37. Let U ∈ CO(G;C(G)

), and let W ∈ Modχ(U) be finite-dimensional.

Let cW : U → C be the character of W . Then truniv(iGU )![W ] ∈ HH0

(Hχ(G)

)is

represented by the function

(25) cGU,W (g) :=

α size(U)−1cW (g) for g ∈ U ,0 for g /∈ U .

Moreover, dim(iGU )![W ] = α size(U)−1 dim(W ). Thus dimx ∈ αZ for all x in therange of the assembly map (24).

A similar result holds for compactly induced projective objects of Mod(G).

Proof. Since the universal trace is compatible with the functoriality of K0 and HH0,the first assertion follows if truniv[W ] = vol(U/C(G))−1cW (g) in HH0

(Hχ(U)

). We

briefly recall how this well-known identity is proved. We may assume that W isirreducible. Hence there is an idempotent pW ∈ Hχ(U) with W ∼= Hχ(U)pW .Thus truniv[W ] = [pW ]. We can compute cW (g) for g ∈ G as the trace of the finiterank operator f 7→ λ(g)f ∗ pW on Hχ(U). This operator has the integral kernel(x, y) 7→ pW (y−1g−1x), so that

cW (g) = cW (g−1) =∫

U/C(G)

pW (x−1gx) dx.

This implies [cW ] =∫

U/C(G)[W ] dx = vol

(U/C(G)

)[W ] because conjugation does

not change the class in HH0

(Hχ(U)

). We get the formula for formal dimensions

because dimx = truniv(x)(1) and cW (1) = dimW . This lies in αZ by constructionof α. By additivity, we get dimx ∈ αZ for all x in the range of the assemblymap (24).

32 RALF MEYER

7.6. Explicit finite projective resolutions. Let V ∈ Modχ(G) be of finitelength. Peter Schneider and Ulrich Stuhler construct an explicit finite projective res-olution for such V in [20]. We only sketch the construction very briefly. Let BT (Gss)be the affine Bruhat-Tits building of Gss. One defines a coefficient system γe(V ) onBT (Gss), which depends on an auxiliary parameter e ∈ N (see [20, Section II.2]);its value on a facet F of BT (Gss) is the—finite-dimensional—space Fix(Ue

F , V ) forcertain Ue

F ∈ CO(G). The cellular chain complex C•(BT (Gss), γe(V )

)with values

in γe(V ) is a resolution of V for sufficiently large e ([20, Theorem II.3.1]). It is afinite projective resolution of V in Modχ(G) because the stabilisers of facets belongto CO

(G;C(G)

)and the set of facets is Gss-finite.

Proposition 38. If V ∈ Modχ(G) has finite length, then Eul(V ) belongs to therange of the assembly map (24). Hence dim

(Eul(V )

)∈ αZ.

Define the Euler-Poincaré function fVEP ∈ Hχ(G) of V as in [20, page 135].

Then [fVEP] = truniv Eul(V ) ∈ HH0

(Hχ(G)

). Thus dim

(Eul(V )

)= fEP(1) and

EPχ(V,W ) = trW (fVEP) for all admissible W ∈ Modχ(G).

See also [20, Proposition III.4.22] and [20, Proposition III.4.1].

Proof. The finite projective resolution C•(BT (Gss), γe(V )

)is explicitly built out

of compactly induced representations. Hence Eul(V ) belongs to the range of theassembly map. Lemma 37 yields dim

(Eul(V )

)∈ αZ and allows us to compute

truniv Eul(V ). Inspection shows that this is exactly [fVEP].

Proposition 38 yields dimV = dim Eul(V ) ∈ αZ if V is irreducible supercuspidal.This rationality result is due to Marie-France Vignéras ([25]).

7.7. Euler characteristics and formal dimensions for square-integrablerepresentations.

Theorem 39. Let V ∈ Modχ(G) be irreducible and square-integrable. Let

ι : K0

(Hχ(G)

)→ K0

(Sχ(G)

)be induced by the embedding Hχ(G) → Sχ(G). Then ι

(Eul(V )

)= [V ]. Hence [V ]

lies in the range of the assembly map

lim−→Sub

(G;C(G)

) Repχ(U)→ K0

(Hχ(G)

)→ K0

(Sχ(G)

)and fV

EP(1) = dim(Eul(V )

)= dimS(V ). This number belongs to α ·N≥1 with α as

in Lemma 37.

Proof. Choose e ∈ N large enough such that C•(BT (Gss), γe(V )

)is a projective

Hχ(G)-module resolution of V . Then

ιEul(V ) =∞∑

n=0

(−1)n[Sχ(G) ⊗Hχ(G) Cn

(BT (Gss), γe(V )

)].

Since Sχ(G) is isocohomological (Theorem 28), Sχ(G) ⊗LHχ(G) V

∼= V . There-fore, Sχ(G) ⊗Hχ(G) C•

(BT (Gss), γe(V )

)is still a projective Sχ(G)-module resolu-

tion of V . Since V is projective as well (Proposition 36), this resolution splitsby bounded Sχ(G)-module homomorphisms. This implies [V ] = ιEul(V ). Theremaining assertions now follow from Proposition 38 and dimS(V ) > 0.

Theorem 40. Let U ∈ CO(G;C(G)

)and let W ∈ Modχ(U) be finite-dimensional.

Then there are at most dim(W ) · size(U)−1 different irreducible square-integrablerepresentations whose restriction to U contains the representation W .


Proof. Let V1, . . . , VN be pairwise non-isomorphic irreducible square-integrable rep-resentations whose restriction to U contains W . Let X := Sχ(G) ⊗Hχ(U)W . Thereare natural adjoint associativity isomorphisms

HomHχ(U)(W,Vj) ∼= HomSχ(G)(X,Vj)

for all j (see [17]). Thus we get non-zero maps X → Vj . They are surjectiveand admit bounded linear sections because the representations Vj are irreducibleand carry the fine bornology; since the representations Vj are projective (Proposi-tion 36), they even admit G-equivariant bounded linear sections. Thus V1, . . . , VN

are direct summands of X. Since they are not isomorphic,⊕Vj is a direct sum-

mand of X as well. Therefore,∑N

j=1 dimS(Vj) ≤ dimS X. Lemma 37 and The-orem 39 yield dimS X = α dim(W ) size(U)−1 and dimS(Vj) ≥ α for all j. HenceN ≤ dim(W ) size(U)−1.

An irreducible square-integrable representation that is not supercuspidal is asubquotient of a representation that we get by Jacquet induction from a properLevi subgroup. It is desirable in this situation to compute the formal dimension(and other invariants) of V from its cuspidal data. This gives rise to some ratherintricate computations; these are carried out in [1] for representations of Glm(D)for a division algebra D.

7.8. Some vanishing results.

Theorem 41. Let V,W ∈ Modχ(G) be irreducible and tempered. If V or W issquare-integrable, then Extn

G(V,W ) = 0 for all n ≥ 1 and

EPχ(V,W ) =

1 if V ∼= W ;0 otherwise.

Proof. Since V and W are tempered and Sχ(G) is isocohomological, we have

ExtnHχ(G)(V,W ) ∼= Extn

Sχ(G)(V,W ).

The latter vanishes for n ≥ 1 by Proposition 36. For n = 0 we are dealing withHomG(V,W ), which is computed by Schur’s Lemma.

Theorem 42. If V is irreducible and tempered but not square-integrable, thenfVEP(1) = dim Eul(V ) = 0.

Proof. This follows from the abstract Plancherel Theorem and Theorem 41 asin the proof of [20, Corollary III.4.7]. We merely outline the proof. The ab-stract Plancherel theorem applied to the type I C∗-algebra C∗red,χ(G) yields thatfVEP(1) is the integral of its Fourier transform W 7→ trW (fV

EP) with respect tosome measure µ, which is called the Plancherel measure. Here W runs throughthe tempered irreducible representations in Modχ(G). Proposition 38 asserts thattrW (fV

EP) = EPχ(V,W ).We have Extn

Hχ(G)(V,W ) = 0 for all n ∈ N and hence EPχ(V,W ) = 0 unlessV and W have the same infinitesimal character. Since the infinitesimal characteris finite-to-one, the support of the function W 7→ EPχ(V,W ) is finite. Hence onlyatoms of the Plancherel measure µ contribute to the integral

fVEP(1) =

∫EPχ(V,W ) dµ(W ).

These atoms are exactly the square-integrable representations. Now Theorem 41yields fV

EP(1) = 0 unless V is square-integrable. In addition, this computationshows that fV

EP(1) = dimS(V ) if V is square-integrable (compare Theorem 39).

34 RALF MEYER

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E-mail address: [email protected]

Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einstein-str. 62, 48149 Münster, Germany

homological algebra for schwartz algebras of reductive p-adic groups

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