New York Journal of Mathematics

New York J. Math. 8 (2002) 241–256.

Gaussian Weighted Unreduced L_{2} Cohomology of Locally Symmetric Spaces

Stephen S. Bullock

Abstract. Let (M, g) be a complete, noncompact Riemannian manifold of finite volume. For w : M → (0, ∞) a weighting function, the w weighted unreduced L_{2} cohomology is defined as the usual unreduced L_{2} cohomology except that dvol is replaced by the measure w dvol. This paper proves that in the case M = Γ\G/K is a locally symmetric space of nonpositive sectional curvature and arbitrary rank whereupon w(m) = e^{−dist(m,p_{0})^{2}} is the Gaussian relative to some basepoint p_{0}, the w weighted unreduced L_{2} cohomology is isomorphic to the usual de Rham cohomology. This isomorphism extends to the standard coefficient bundles.

Note that weights for the de Rham cohomology of exponential decay have already been constructed; see [Bor83], [Bor90] and [Fra98]. The Gaussian weight behaves differently in terms of coefficient bundles.

Contents

0. Introduction 242 1. Sheaves on M 244

1.1. Topology of M 244 1.2. L_{2} sheaves and metrics 246

2. Proof of quasiisomorphism 246 2.1. Contraction onto U invariant forms 246 2.2. Contraction on 𝔞^{>0} 250

3. Appendices 254 3.1. Appendix: Regularity for locally symmetric spaces 254 3.2. Appendix: Extensions to nonarithmetic lattices 255

References 255

Received June 28, 2002. Mathematics Subject Classification. 53C, 22E, 14F. Key words and phrases. weighted L_{2} cohomology, locally symmetric space, Gaussian weight. This research supported by the U.M. VIGRE grant.

241^{ISSN 1076-9803/02}

0. Introduction

Let (M, g) be a locally symmetric finite volume manifold of nonpositive sectional curvature. Thus, the universal cover X = G/K is a globally symmetric space of noncompact type. For convenience, say X is Riemannian irreducible, so G is ℝ semisimple. In particular, the center of G contains no nonidentity ℚ split tori. Let K ⊂ G be a maximal compact subgroup. Finally, suppose a rational structure G_{ℚ} ⊂ G with arithmetic torsionfree Γ ⊂ G_{ℚ} the fundamental group of M. Nonarithmetic lattices occur in rank_{ℝ}G = 1 and present no difficulties per §3.2.

Let E be a finite dimensional representation of G_{ℂ}. There is as described in [MM63] a coefficient bundle 𝔼 = E ×_{Γ}X ≅ (Γ\G)×_{K} E. Any choice of orthonormal basis of E induces (local) orthonormal sections of a Hermitian metric on 𝔼, parallel to a canonical flat connection. This bundle metric on 𝔼 may be explicitly described as follows. Fix ⟨−, −⟩_{0} a metric on the base vector space E; this should be averaged over K so as to be K invariant. Let x_{0} ∈ X be stabilized by K. Then for v, w elements of E, v ×_{Γ} (g · x_{0}) is an element of the fiber of E ×_{Γ} X at (Γg) · x_{0}. We define ⟨v ×_{Γ} (g · x_{0}), w ×_{Γ} (g · x_{0})⟩_{𝔼} = ⟨g^{−1} · v, g^{−1} · w⟩_{0}.

Remark 0.1. 1) ℂ × M the untwisted coefficient bundle arises from the trivial

representation.

2) On Riemannian (M, g), there is no reason to both add a weight in front of dvol and allow the choice of arbitrary bundle metrics on ℂ × M. However, in the locally symmetric case the bundle metrics on ℂ may not be standard.

Franke constructed [Fra98, p. 187] weighting functions w_{λ} ∈ O(a^{λ}) on each Siegel set in the almost fundamental domain of the fundamental theorem of reduction theory of [BHC62]. In our notation:

• P = UMA is the Langlands decomposition of a minimal rational parabolic, and 𝔞 is the Lie algebra of A with Killing dual 𝔞̌. Also, take {q_{i}}_{i=1}^{c} a set of representatives for Γ\G_{ℚ}/P_{ℚ} indexing the cusps of M.

• H(−) is the associated height function:

(H = log ∘ π_{A}) : G = (UMA)K → 𝔞 ≅ ℝ^{ℓ}

for ℓ = rank_{ℚ}G.

• ⟨−, −⟩ denotes the pairing between 𝔞 and 𝔞̌.

• χ : ℝ → ℝ is a cutoff function which is 1 on [T, ∞) and 0 on (−∞, T − 1] with bounded gradient.

• Breaking with [Fra98], Σ^{+}(𝔤, 𝔞) denotes the positive restricted ℚ roots of 𝔤 according to P, and Σ^{++}(𝔤, 𝔞) denotes the simple restricted ℚ roots.

Each λ ∈ 𝔞̌ produces a weight function on M of bounded gradient [Fra98, p. 187]:

(1) w_{λ}(ΓgK) =

∑_{i=1}^{c} ∑_{[γ]∈(Γ∩UM)\Γ} {exp(⟨λ, H(γq_{i}^{−1}g)⟩) ∏_{α∈Σ^{++}(𝔤,𝔞)} χ(⟨α, H(γq_{i}^{−1}g)⟩)} .

Remark 0.2. Some care must be taken in choosing T above. The details are omitted to avoid a lengthy review of reduction theory [Fra98, §2.1].

Gaussian Weighted Unreduced L_{2} Cohomology 243

The present argument replaces the linear term λ by a negative definite quadratic form −Q ∈ 𝔞̌ ⊗ 𝔞̌:

(2) w_{−Q}(ΓgK) =

∑_{i=1}^{c} ∑_{[γ]∈(Γ∩UM)\Γ} {exp(−⟨Q, H(γq_{i}^{−1}g)⟩) ∏_{α∈Σ^{++}(𝔤,𝔞)} χ(⟨α, H(γq_{i}^{−1}g)⟩)} .

Here, if Q = ∑_{i,j} λ_{i} ⊗ λ_{j}, then ⟨Q, H⟩ = ∑_{i,j} λ_{i}(H)λ_{j}(H) for H ∈ 𝔞.

Note that there exists Q > 0 which causes the weight above to be O(e^{−r^{2}}). Indeed, the Killing form is the restriction of the metric of M to the flat sector corresponding to 𝔞^{≥0}, and one may choose Q realizing the Killing form. Thus the fundamental theorem of reduction theory will imply w_{−Q} ∈ O(e^{−r^{2}}).

Remark 0.3. Note exp(⟨λ, H(−)⟩) in these weighting functions may instead be thought of as the exponential of a Busemann function of X. This appears in the literature as a scholium of Lemma 1.3 [Leu95, p. 395].

These weighting functions will be used in the following context. Define

(3) Ω_{(2),w}^{•}(M, 𝔼) = {ω ∈ Ω^{•}(M, 𝔼) | ∫_{M} |η|_{𝔼}^{2}w dvol < ∞ for η = ω, dω} .

This is a cochain complex using the exterior derivative for forms with coefficients in 𝔼. The resulting cohomology is denoted H_{(2),w}^{•}(M, 𝔼). In case w = w_{λ} or w = w_{−Q}, the subscript of the weight may replace the weight itself in all notations. Finally, w is omitted if w ≡ 1.

These conventions allow for a restatement of a restatement of prior work. In [Fra98, p. 90], Franke proves a result which rephrases Borel’s results [Bor83] on forms of uniform moderate growth. In the present notation,

lim_{λ→(∞,...,∞)}H_{(2),−λ}^{•}(M, 𝔼) ≅ H^{•}(M, 𝔼). (4)

The present result is similar. It asserts H_{(2),−Q}^{•}(M, 𝔼) ≅ H^{•}(M, 𝔼). But there is one novelty. Namely, the present result is uniform in 𝔼, meaning for fixed Q_{0} the current result holds for all 𝔼. In contrast, any fixed λ_{0} has some 𝔼(λ_{0}) for which w_{−λ_{0}} produces a nontrivial boundary truncation.

The precise statement requires more notation. Let M be the reductive Borel-Serre compactification of M introduced in [Zuc82]. For the remainder, j : M ↪ M is open inclusion so that j_{∗}Ω^{•}(𝔼) is an incarnation of R^{•}j_{∗}𝔼, while A^{•}(E) is the presheaf of special forms on M of [GHM94, §14], Equation (11). The sheafification Sh(A^{•}(E)) is the weighted cohomology sheaf W^{−∞}C^{•}(E), whose hypercohomology is the usual de Rham cohomology with coefficients in 𝔼. Finally, L_{−Q}^{•}(E) is the sheafification of the presheaf L_{−Q}^{•}(E) of w_{−Q} L_{2} forms.

Lemma 0.4. Let M locally symmetric with 𝔼 a standard bundle induced by E a finite dimensional representation. 𝔼 carries a Hermitian metric in the standard quasiisometry class. Let Q > 0 in 𝔞̌ ⊗ 𝔞̌. Then there is a quasiisomorphism on M

j_{∗}Ω^{•}(𝔼) ← Sh(A^{•}(E)) → L_{−Q}^{•}(E). (5)

The following consequence is immediate, since quasiisomorphisms imply isomorphic hypercohomology.

Theorem 0.5. Suppose the hypotheses of Lemma 0.4. Then one has

H_{(2),−Q}^{•}(M, 𝔼) ≅ H^{•}(M, 𝔼).

Corollary 0.6 follows. In fact, there now exists a strong weighted Kodaira decomposition^{∗} given the finite dimensional answer above, while the Gaussian w = e^{−r^{2}} results via choice 𝔼 = ℂ × M for ℂ trivial and −Q realizing the Killing form.

Corollary 0.6. Suppose (M, g) is a locally symmetric nonpositively curved finite volume Riemannian manifold. Let r be distance to a basepoint and w = e^{−r^{2}} be the Gaussian weight on M. Then H_{(2),w}^{•}(M) ≅ H^{•}(M), where the latter is the usual de Rham cohomology.

The above should be compared to the following conjecture by Edward Bueler. To the author’s knowledge, this remains open even in the rank one finite volume locally symmetric case.

Bueler’s Conjecture (1.1 of [Bue99]). Suppose (M^{n}, g) is complete, Ric ≥ −c^{2}. Let w(m) = ρ_{t}(m, p_{0}) for p_{0} ∈ M fixed and t fixed, where ρ_{t}(x, y) is the heat kernel of the metric Laplacian on functions associated to g. Then for β^{p} = dim H^{p}(M) the usual Betti number of M,

dim ker Δ_{w} = β^{p}, 0 ≤ p ≤ n. (6)

Also, the complex dom d is Fredholm, implying a strong Kodaira decomposition and finite dimensional H_{(2),w}^{•}(M).

Finally, the result has certain apparently unexciting implications for Lie algebra cohomology. Nonetheless, errors might be made in deriving them from the literature. A short appendix is included.

Remark 0.7. The author thanks Gopal Prasad for comments on an early manuscript and the preprint and thanks the referee for enumerating several improvements and a few errata.

1. Sheaves on M

1.1. Topology of M. The introduction mentioned in passing the reductive Borel-Serre compactification of M, denoted here M. This compactification originated in [Zuc82] and at its creation was optimized for computing L_{2} cohomology sheaves. It also plays a central role in [GHM94], being the compactification on which the various incarnations of weighted cohomology are constructed. Besides [Zuc82], the compactification was also used in Theorem A of [Nai99]. The present argument is similar but less delicate due to rapid decay of the weight.

To facilitate computations of stalk cohomologies of differential graded sheaves, say at y ∈ Y, a sequence {U_{i}}_{i=1}^{∞} of open neighborhoods of y with {y} = ∩_{i=1}^{∞}U_{i} is useful. Such a system of neighborhoods will be called fundamental. The main issue with a compactification M is to build such fundamental neighborhoods about n ∈ ∂M rather than m ∈ M.

By definition, X is naturally a quotient of the Borel-Serre enlargement X̃ of [BS73], with Γ\X̃ = M̃ a manifold with corners. This is described in [Zuc82], pp.

∗See [BL92] for an explanation and references.

184 and 190, where the projections p_{P} : e′(P) → ê(P) of individual boundary strata patch to a global projection.

Thus, say P = UMA is the rational Langlands decomposition of the real points of an arbitrary rank fixed parabolic ℚ subgroup P. Here, U is the unipotent radical of P. Recall x_{0} ∈ X corresponding to K is a fixed choice of basepoint corresponding to a fixed Gaussian. Then A is identified with a maximal ℚ split torus in the center of P/U, so that A is moreover invariant under the Cartan involution corresponding to x_{0}. Then MA is similarly identified to P/U, while M is identified with intersection of the squares of all ℚ characters [GHM94, p. 150].

Since U is the exponential of a space of positive 𝔞 weights, it is a simply connected nilpotent group. On the other hand, M is reductive with the Cartan involution of G having a well-defined restriction to M, so that X_{M} = M/(K ∩ M) is a globally symmetric space of rank strictly less than X for P nontrivial. We finally fix notations Γ_{U} = Γ ∩ U and Γ_{M} for the projection of Γ onto M ≅ U\P/A. Then Γ_{U} is always a lattice subgroup in U given that P is rational, so that via a standard property of simply connected nilpotent groups Γ_{U}\U is compact. On the other hand, M_{M} = Γ_{M}\X_{M} is a lower rank finite volume locally symmetric space.

Now choose η_{1} × η_{2} ⊂ UM open with η_{1} so large as to include a fundamental domain for the nilpotent lattice Γ_{U} ⊂ U and η_{2} ⊂ M being K ∩ M invariant with compact closure. Let {H_{i}}_{i=1}^{r} be dual to {α_{i}}_{i=1}^{r} = Σ^{++}(𝔭, 𝔞) for r the rank of P. We define the group isomorphism ψ : (0, ∞)^{dimA} → A as ψ(t_{1}, …, t_{r}) = exp(∑_{i=1}^{r} t_{i}H_{i}). Note ψ is not an isometry for the usual flat metric of (0, ∞)^{r}. Henceforth, dt⃗^{2} denotes not ∑_{i=1}^{r} dt_{i}^{2} but rather the flat pullback metric from 𝔞^{>0}.

For T > 0, the Siegel set 𝔖(T, η_{1} × η_{2}) is η_{1}η_{2}ψ([T, ∞)^{dimA}) ⊂ G. The term also refers to 𝔖(T, η_{1} × η_{2}) · x_{0}. The fundamental neighborhoods of n ∈ ∂M are minor modifications of such Siegel sets.

Describing these requires a review of the construction of M. To do so, let X̃ denote the Borel-Serre enlargement of the globally symmetric space X = G/K and X be the reductive Borel-Serre enlargement of [Zuc82]. Then the added boundary strata e′(P) of M̃ = Γ\X̃ for P a ℚ parabolic admits a natural fibration

Γ_{U}\U ⟶ e′(P)⟶^{p_{P}}(Γ_{M}\X_{M} = ê(P)). (7)

Zucker verifies that these projections p_{P} patch in a well defined way given the incidence of e′(P) and e′(R) for P ⊂ R parabolic, so that M is the singular image under the patched projection of the manifold with corners M̃. Thus, each point in a boundary strata of M is associated to a point in some Γ_{M}\X_{M}.

We finally describe how to modify a Siegel set to form a fundamental neighborhood to n ∈ ∂M. By construction, there is a rational parabolic P = UMA with n ∈ ê(P). Choose η_{2}(i) for i = 1, 2, … a sequence of fundamental neighborhoods about n in Γ_{M}\X_{M}, and choose η_{1} ⊂ U as above. Then for π_{Γ} : X → M projection,

Ũ_{i} = π_{Γ}[𝔖(i, η_{1} × η_{2}(i))] ⊔ η_{2}(i) ⊂ Γ\X ⊔ ê(P) (8)

is the closure of the interior of π_{Γ}[𝔖(i, η_{1} × η_{2}(i))] in M and forms a fundamental system of neighborhoods about n ∈ ∂M as i → ∞.

1.2. L_{2} sheaves and metrics. We briefly recall the two sheaves on M which will be shown to be quasiisomorphic. The first j_{∗}Ω^{•}(𝔼) is

Γ(U, j_{∗}Ω^{•}(𝔼)) = Ω^{•}(U ∩ M, 𝔼). (9)

Here, Γ denotes sections rather than a group. For the second, consider this presheaf:

Γ(U, L_{−Q}^{•}(E)) = {ω ∈ Γ(j_{∗}Ω^{•}(𝔼)) | ∫_{U∩M}(|ω|^{2} + |dω|^{2})w_{−Q}dvol < ∞} . (10)

Then L_{−Q}^{•}(E) is the sheafification, i.e., the differential graded sheaf implementing the corresponding local integrability condition.

One may now check that the sections on the compact space M of L_{−Q}^{•}(E) are exactly the globally weighted L^{2} forms, given that sheafification replaces the global square integrability with the local version. These global sections also compute the hypercohomology, since the argument for fineness on p. 191 of [Zuc82] generalizes to the present weighted L_{2} sheaf.

Finally, the differential graded presheaf A^{•}(E) of special forms is constructed in [GHM94, §14]. It fits into the d.g.s. in the middle of the quasiisomorphism of Equation (5):

j_{∗}Ω^{•}(𝔼) ← Sh(A^{•}(E)) → L_{−Q}^{•}(E).

To describe these, let U_{i} correspond to a fundamental system about n ∈ ∂M.

(11) Γ(U_{i}, A^{•}(E)|_{ê(P)}) =

{∑ α_{j} ⊗ ω_{j} ⊗ ϕ_{j} | α_{j} ∈ ℂ, ω_{j} ∈ Ω^{•}(η_{2}(i), 𝔼), ϕ_{j} ∈ C^{•}(𝔲, E)} .

Here, C^{•}(𝔲, E) = ∧^{•}𝔲^{∗} ⊗ E is identified with Ω^{•}(U, 𝔼)^{U} ⊂ Ω^{•}((Γ ∩ U)\U, 𝔼), and η_{2}(i) ⊂ X_{M} is an open set with compact closure.

The arrow pointing left in Equation (5) is inclusion. It is true but not obvious that the arrow which points right is also inclusion. To see this, recall U_{i} ≅ (Γ_{U}\U)× η(i) × (0, ∞)^{r} via a diffeomorphic decomposition. The η(i) is a Riemannian factor, say with coordinates carrying a local metric dx_{M}^{2} with ∇R ≡ 0 on X_{M}. The Γ_{U}\U factor collapses; say β runs over Σ^{+}(𝔭, 𝔞_{P}) for 𝔲 = ⊕_{β}𝔲_{β} with du_{β}^{2} an inner product. Then taking ψ as our coordinates on the last factor, [Bor74, §4] shows that the pullback of the metric on M via the Langlands decomposition is

ds^{2} = dt⃗^{2} + dx_{M}^{2} + ∑_{β>0} exp(−2⟨β, H ∘ ψ(t⃗)⟩)du_{β}^{2}. (12)

Here, dt⃗^{2} is the pullback under ψ of the flat metric on 𝔞^{>0} rather than ∑_{i=1}^{r} dt_{i}^{2}. The above formula then demands every special form is square integrable versus the weights of Equation (2).

2. Proof of quasiisomorphism

2.1. Contraction onto U invariant forms. A direct limit of cochain homotopies is a cochain homotopy. Thus, the quasiisomorphism will be implied by checking for the appropriate presheaves that on each fundamental neighborhood U_{i} of a generic boundary point n ∈ ∂M we may construct cochain homotopy equivalences of each of Γ(U_{i}, L_{−Q}^{•}(E)) and Γ(U_{i}, j_{∗}Ω^{•}(𝔼)) onto the space of special forms Γ(U_{i}, A^{•}(E)).

The first step of the argument described in §2.1 is known. It reproves the Nomizu/van Est theorem of [Nom54] and [vE58] while checking L_{2} boundedness conditions on boundary neighborhoods. See [Zuc82, pp. 195–201]. However, the argument has already been cited multiple times without discussion by other authors. A mild variant reappears here.

2.1.1. Γ_{U}\U as a tower. Let β vary over Σ^{+}(𝔭, 𝔞_{P}) for 𝔲 = ⊕_{β}𝔲_{β}. Since Σ^{+}(𝔭, 𝔞_{P}) is a set of restricted roots, perhaps dim 𝔲_{β} > 1. Nonetheless, we may choose a basis {X_{i}}_{i=1}^{n−1} ⊂ 𝔲 so that each X_{i} ∈ 𝔲_{β(i)} for i = 1 … n − 1 and moreover i > j demands β(i) ≤ β(j). In particular, these conventions demand X_{1} ∈ 𝔷(𝔲).

Definition 2.1. For 0 ≤ k ≤ n−1, 𝔲_{(k)} = ⟨X_{1}, …, X_{k}⟩ with the brackets denoting the real span. U_{(k)} = exp𝔲_{k} and Γ_{U,(k)} = U_{(k)} ∩ Γ_{U}. As 𝔲_{(k)} is an ideal in 𝔲, set 𝔲^{(k)} = 𝔲/𝔲_{(k)} a Lie algebra with U^{(k)} = exp𝔲^{(k)} and Γ_{U}^{(k)} = Γ_{U}/Γ_{U,(k)}. Finally, put T^{(k)} = (Γ_{U}^{(k)}/Γ_{U}^{(k−1)})\(U^{(k)}/U^{(k−1)}).

Here, T^{(k)} is a torus since T^{(k)} ≅ {exp nℓX_{k}}_{n∈ℤ}\{exp tX_{k}}_{t∈ℝ} for some length ℓ. Thus, T^{(k)} ≅ SO(2). This allows Γ_{U}\U to be written as a tower of SO(2) principal bundles. The tower written horizontally is

\begin{matrix}T^{(1)} & & T^{(i)} & & T^{(i+1)} \\ ↓ & & ↓ & & ↓ \\ (Γ_{U}\U) & → ⋯ & Γ_{U}^{(i)}\U^{(i)} & → & Γ_{U}^{(i+1)}\U^{(i+1)} & → ⋯ → Γ_{U}^{(n−1)}\U^{(n−1)} = \{e\}. \end{matrix} (13)

Fix a fundamental neighborhood V = U_{k} for some k, i.e., V = (Γ_{U}\U) × η(k) × (0, ∞)^{r} for η(k) ⊂ X_{M} an open set with compact closure. For V^{(i)} = (Γ_{U}^{(i)}\U^{(i)}) × η(k)×(0, ∞)^{r}, extend the notation Π^{(i)} to also denote Π^{(i)} : V → V^{(i)} the composite projection onto V^{(i)}. Finally, define C^{∞}(V)^{i} = {f ∈ C^{∞}(V)|Xf ≡ 0 if X ∈ 𝔲_{(i)}}. The following lemma follows via an omitted induction.

Lemma 2.2. There is a bijective correspondence f ↔ f between C^{∞}(V^{(i)}) and C^{∞}(V)^{i} which respects Π^{(i)} in the sense that f ∘ Π^{(i)} = f.

2.1.2. Antiderivative operators of SO(2) actions. For the remainder of this section, we fix a principal SO(2) bundle SO(2) → E→^{Π}B. In the application, E = V^{(i)} while B = V^{(i+1)}. We use Z to denote the vector field which traverses the SO(2) fiber in time 2π. In the application, this is some renormalization of X_{i} dependent on the circumference of the circular fibers.

Definition 2.3. Let f : E → ℂ. Then a(f)(x) = \frac{1}{2π} ∫_{0}^{2π} f(xe^{2πsZ})ds. We put f_{0} = f − a(f).

Definition 2.4. Pick any constant c, and let N ⊂ E be an open set with a SO(2) principal bundle chart φ : N → W ×SO(2; ℝ). Then letting θ stand for exp(θ(E_{21}− E_{12})) in the argument of φ(w, −),

[sf]φ(w, θ) = ∫_{c}^{θ} f_{0}(φ(w, ψ))dψ − \frac{1}{2π} ∫_{0}^{2π} ∫_{c}^{θ} f_{0}(φ(w, ψ))dψ dθ (14)

is readily verified to be independent of c and moreover the local choice of N. The associated globally defined operator on E will be denoted s : C_{c}^{∞}(E) → C_{c}^{∞}(E).

These two operators on functions allow us to redefine Zucker’s cochain homotopy operators without recourse to d_{SO(2)}^{∗} the exterior derivative adjoint in an SO(2) direction. Certain boundedness properties and commutation properties of each are required in the computations that complete the argument. These are collected in the following lemma, which is stated without proof. Each part follows from either i) the definitions, ii) the Schwarz inequality, or iii) dominated convergence.

Lemma 2.5. i) Say μ is some measure on E which restricts to a product measure on each SO(2) principal bundle chart. Then a : C_{c}^{∞}(E) → C_{c}^{∞}(E) is w μ L_{2} bounded for any w > 0 constant on every SO(2) fiber.

ii) The same holds for s : C_{c}^{∞}(E) → C_{c}^{∞}(E). iii) Zaf = aZf ≡ 0. iv) Zsf = sZf = f_{0}.

v) Suppose that in every bundle chart φ(W × SO(2)), the vector field X has no component in the vertical SO(2) direction. Then [X, s] = 0 and [X, a] = 0.

2.1.3. Central case of cochain homotopy. Recall the conventions Σ^{++}(𝔭, 𝔞) = {α_{i}}_{i=1}^{r} and λ = ∑ c_{i}α_{i}. Let N the set of 𝔞 weights of C^{•}(𝔲, E) induced by 𝔲 = ⊕_{(𝔭,𝔞)}𝔲_{α}, and say 2ρ = ∑_{Σ^{+}(𝔭,𝔞)} dim 𝔲_{β}β is the exponential decay rate of the volume form. Change conventions slightly so that L_{(2)}^{•}((0, ∞)^{r}, k(t⃗)) may denote k(t⃗) weighted L^{2} forms on the sector. Finally, V is some fixed fundamental neighborhood U_{i}. Then we have the decomposition V ≅_{dif} (Γ_{U}\U) × η_{2} × (0, ∞)^{r}. Thus

(15) Γ(A^{•}(𝔼)) = ℂ ⊗ Ω^{•}(η_{2}, 𝔼) ⊗ C^{•}(𝔲, E)
Γ(L_{−Q}^{•}(E)) = ⊕_{ν∈N} {[L_{(2)}^{•}((0, ∞)^{r}, exp(−⟨Q, H ∘ ψ(t⃗)⟩ − 2⟨ν + ρ, H ∘ ψ(t⃗)⟩)
ˆ⊗L_{2}((Γ ∩ U)\U))] ˆ⊗L_{2}^{•}(η_{2}, 𝔼)⊗C^{•}(𝔲, E)_{ν}}.

Before describing the generic case of a cochain homotopy which projects down the tower of Equation (13), we begin with the case of Z = X_{1} central. Several simplifications arise.

Thus, say L_{Z} is the Lie derivative of Z. Since Z is central, in terms of the decompositions the Lie derivative is given by L_{Z}(f ⊗ ϕ_{1} ⊗ ϕ_{2}) = Zf ⊗ ϕ_{1} ⊗ ϕ_{2}. Moreover, a computation using Z central also verifies [d, s ⊗ 1 ⊗ 1] = 0 for s the antiderivative with respect to Z. Neither fact is true for noncentral directions.

Definition 2.6. The linear operators P, B are P = a⊗1⊗1 and B = (s⊗1⊗1)∘ι_{Z} according to Equation (15).

Using Cartan’s magic formula dι_{Z}+ι_{Z}d = L_{Z}, we see that (B, P) forms a cochain homotopy of Ω_{c}^{•}(V, 𝔼):

dB + Bd = s(dι_{Z} + ι_{Z}d) = s(L_{Z}) = 1 − P. (16)

Lemma 2.5 and Equation (12) will moreover imply that P and B are bounded via the weighted L_{2} conditions. Thus, (B, P) is a well-defined cochain homotopy on each of the three spaces of sections. Note that the boundedness check for B uses ι_{Z} bounded, which follows since |Z| → 0 as m → ∂M. As the forms in im P for each of the three d.g.s. are invariant under the SO(2) action given by integrating Z, they may now be viewed as forms on V^{(2)} = (Γ_{U}^{(2)}\U^{(2)}) × η(i) × (0, ∞)^{r} rather than on V = V^{(1)}. The next section describes the induction argument that continues this process until the output forms are invariant under 𝔲 and hence U.

2.1.4. Generic cochain homotopy. Recall that for 0 ≤ ℓ ≤ n − 1, C^{∞}(V)^{ℓ} = {f ∈ C^{∞}(V) | Xf ≡ 0 if X ∈ 𝔲_{(ℓ)}}. Via the Langlands decomposition, we may view \{\frac{∂}{∂t_{i}} \}_{i=1}^{r}⊔\{Y_{j}\}⊔\{X_{k}\} as a frame on V. Here the coordinate fields span (0, ∞)^{r}, Y_{j} span the patch η_{2} of X_{M}, and X_{k} ∈ 𝔲_{β(k)} so that k_{1} ≤ k_{2} demands β(k_{1}) ≥ β(k_{2}). Then using coefficient functions C^{∞}(V)^{ℓ} in the dual coframe produces a filtration Ω^{•}(V, 𝔼) ⊃ Ω^{•}(V, 𝔼)^{1} ⊃ … Ω^{•}(V, 𝔼)^{n−1}. Each space of forms of Equation (15) intersects the subspaces of this filtration nontrivially, and the central case of cochain homotopy of Subsection 2.1.3 contracts each onto its Ω^{•}(V, 𝔼)^{1} subspace. The special forms are Ω^{•}(V, 𝔼)^{n−1}.

Induction hypothesis: Each space of forms of Equation (15) has been contracted onto its (ℓ − 1)^{st} subspace.

Definition 2.7. Extend the definition of s(f) and a(f) to act on Ω^{•}(V, 𝔼) by acting on the coefficient functions of the above coframe. Fix throughout the subsection the convention that Z is the positive multiple of X_{ℓ} with flow traversing the SO(2) fiber of V^{(ℓ−1)} in time 2π. Then P = a and B̃ = s ∘ ι_{Z}.

(P, B̃) is not a cochain homotopy equivalence, but the argument may be salvaged. To begin, suppose throughout fτ ⊗η⊗ϕ is decomposed according to Equation (15), with η a form on η_{2} ⊂ X_{M}. Then for Θ_{j} : C^{•}(𝔲, E) → C^{•+1}(𝔲, E) left exterior multiplication by X_{j}, ι_{j} left interior multiplication, and Θ_{t,i}, ι_{t,i} similar for τ ∈ ∧^{•}[dt_{i}]_{i=1}^{r}, we have the following expression for the exterior derivative. It is intended that this formula extend linearly:

(17) d(fτ ⊗ η ⊗ ϕ) = ∑_{i=1}^{r} \frac{∂f}{∂t_{i}} Θ_{t,i}τ ⊗ η ⊗ ϕ + (−1)^{degτ∧η}(Zf)τ ⊗ η ⊗ Θ_{Z}ψ

+ ∑_{j≠ℓ}(−1)^{degτ∧η}X_{j}fτ ⊗ η ⊗ Θ_{j}ϕ + (−1)^{degη}fτ ⊗ dη ⊗ ϕ

+ (−1)^{degτ∧η}fτ ⊗ η ⊗ dϕ.

Recalling B̃(fτ ⊗ η ⊗ ϕ) = (−1)^{degτ∧η}sfτ ⊗ η ⊗ ϕ, we compute dB̃(fτ ⊗ η ⊗ ϕ) and B̃d(fτ ⊗ η ⊗ ϕ).

(18)  B̃d(f ⊗ η ⊗ ϕ) = (−1)^{degτ∧η} ∑_{i=1}^{r} (s\frac{∂}{∂t_{i}} ) Θ_{t,i}τ ⊗ η ⊗ ι_{Z}ϕ

+ s(Zf)τ ⊗ η ⊗ ι_{Z}Θ_{Z}ϕ + ∑_{j≠ℓ} s(X_{j}f)τ ⊗ η ⊗ ι_{Z}Θ_{j}ϕ

+ (−1)^{degτ+1}(sf)τ ⊗ dη ⊗ ι_{Z}ϕ + (sf)τ ⊗ η ⊗ ι_{Z}dϕ

dB̃(f ⊗ η ⊗ ϕ) = (−1)^{degτ∧η} ∑_{i=1}^{r} \frac{∂}{∂t_{i}} (sf)Θ_{t,i}τ ⊗ η ⊗ ι_{Z}ϕ

+ (Zs)fτ ⊗ η ⊗ Θ_{Z}ι_{Z}ϕ + ∑_{j≠ℓ}(X_{j}s)fτ ⊗ η ⊗ Θ_{j}ι_{Z}ϕ

+ (−1)^{degτ} (sf)τ ⊗ dη ⊗ ϕ + (sf)τ ⊗ η ⊗ dι_{Z}ϕ.

Thus using Θ_{Z}ι_{Z} +ι_{Z}Θ_{Z} = 1, Θ_{j}ι_{Z} +ι_{Z}Θ_{j} = 0, Zs = sZ = 1−a, and the obvious cancellations, one obtains

(19)  (dB̃ + B̃d)(fτ ⊗ η ⊗ ϕ) = (Zs)f ⊗ η ⊗ ϕ + (sf)τ ⊗ η ⊗ (ι_{Z}d + dι_{Z})ϕ
= (1 − P)(fτ ⊗ η ⊗ ϕ) − Q(fτ ⊗ η ⊗ ϕ)

for Q(fτ ⊗ η ⊗ ϕ) = −(sf)τ ⊗ η ⊗ (dι_{Z} + ι_{Z}d)ϕ and extended linearly. Some preliminary facts about Q must be argued before constructing the genuine cochain homotopy.

Lemma 2.8. i) Let −λ_{0} be the lowest 𝔞 weight of C^{•}(𝔲, E), while Z ∈ 𝔲_{β}. Then

for N so large that C^{•}(𝔲, E)_{−λ_{0}+nβ} = {0} for n ≥ N, one has Q^{N} = 0. ii) [d, P] = 0. iii) [d, Q] = 0. iv) QP = P Q = 0.

v) Q is bounded in the weighted L_{2} norm on forms.

Sketch. i) follows from the similar statement for ι_{Z}d + dι_{Z}, since d : C^{•}(𝔲, E) → C^{•}(𝔲, E) preserves the weight space decomposition. ii) is a computation, and iii) follows from ii) since [d, dB̃ + B̃d] = 0. iv) is a computation using saf = asf ≡ 0.

v) follows from the similar statement for s and the metric Equation (12). □

Now label B = B̃(∑_{i=0}^{N−1} Q^{i}) for N per the lemma. A computation checks that (B, P) is a cochain homotopy, using i)-iv) of the lemma. A check using v) shows that the cochain homotopy respects the complexes of weighted L_{2} forms and special forms. Thus each space of forms in decomposition Equation (15) may be contracted onto its U invariant subspace.

Next, apply the Poincaré lemma to η_{2} so that Y_{∗}f ≡ 0 for the coefficient functions in the coframe \{\frac{∂}{∂t_{i}} \}⊔\{Y_{j}\}⊔\{X_{k}\}. This is standard. The final cochain homotopies compute the weighted L_{2} cohomology of sectors (0, ∞)^{r}.

2.2. Contraction on 𝔞^{>0}.

2.2.1. Cochain homotopies on sectors. The goal of this subsection is to generalize a result first quoted in [Bul01] from half lines to sectors. The original result is that for k(t) a weight on (0, ∞) satisfying the differential inequality \frac{d}{dt}logk(t) < −ϵ^{2} < ∞, one has the following inequality:

∫_{0}^{∞} ∥∫_{0}^{t} f(s)ds∥^{2} k(t)dt < const ∫_{0}^{∞} ∥f(t)∥^{2}k(t)dt. (20)

The present argument is thus quite indebted to [Muc72] and [BH92].

The setting throughout is (0, ∞) · C, where C ⊂ S^{n−1} is a set of vectors on the sphere which is open with compact closure so that (0, ∞) · C is starlike about 0. The scalar multiplication will also be thought of as providing coordinates, so that x⃗ in the cone may also be written as (t, θ⃗) for t ∈ (0, ∞) and θ⃗ ∈ C. We also use t rather than the more typical r for the radial coordinate, given the previous paragraph.

Proposition 2.9. In the setting above, let k(x⃗) > 0 be a smooth weighting function on (0, ∞) · C so that raywise within the cone the following differential inequality holds:

(21) \frac{∂}{∂t}log k(t, θ⃗) ≤ −2ϵ < 0, uniformly over all θ⃗ ∈ C.

Abbreviate X = \frac{∂}{∂t}, and define the integral operator by

I(f_{I}dt dθ_{I} + g_{I}dθ_{I}) = (∫_{X} f_{I}) dt dθ_{I} + (∫_{X} g_{I}) dθ_{I}. (22)

Here, the dθ_{I} are wedges of duals of coordinate fields of S^{n−1} which expand linearly away from 0 when lifted to the cone, and (∫_{X} f)(x⃗) = ∫_{0}^{|x⃗ |} f(sx⃗/|x⃗ |)ds. Finally, define B_{•} = ι_{X} ∘ I for • ≥ 1 and B_{0} ≡ 0, where ι_{X} is contraction by X. Then each B_{•} is bounded in the k(x⃗) weighted L_{2} norm on compactly supported smooth forms on the cone, and thus each extends to the L_{2} space of forms L_{2}((0, ∞)·C, k(x⃗)dvol).

Remark 2.10. The most natural of these operators is B_{1}, which is just the line integral B_{1}ϕ = ∫ x⃗ ; 0 ϕ.

Corollary 2.11. Given the raywise differential inequality (21), the weighted L_{2} cohomology of the sector is

H_{(2)}^{•}((0, ∞) · C, k(x⃗)) ≅ ℂ ⊕ 0 ⋯ ⊕ 0. (23)

In fact, the cochain homotopy onto the right-hand side is given by taking P_{0}f = f(0 ), P_{•} ≡ 0 else, and B_{•} per Proposition 2.9.

Remark 2.12. This would imply that in Equation (15),

H_{(2)}^{•}((0, ∞)^{r}, exp(−⟨Q, H · ψ(t⃗)⟩ − 2⟨ν + ρ, H ∘ ψ(t⃗)⟩) ≅ ℂ ⊕ 0 ⊕ 0 · · · . (24)

That in turn ends the final cochain homotopy on the differential presheaf sections, proving Lemma 0.4 hence Theorem 0.5 hence 0.6.

Proof of Corollary 2.11. Via dominated convergence, one has that [d, I] ≡ 0. Now since the coordinate coframe in (22) was dual to a coordinate frame, the Lie derivative of the radial direction L_{X} in fact just differentiates coefficient functions in terms of this coframe. Thus, L_{X}Iϕ = ϕ. Now for ϕ a form of degree at least one, so that neither B term is a zero operator, Cartan’s formula shows

(dB + Bd)ϕ = dι_{X}Iϕ + ι_{X}Idϕ = (dι_{X} + ι_{X}d)Iϕ = L_{X}Iϕ = ϕ = (1 − P)ϕ.

On the other hand, for functions (dB_{0} + B_{1}d)f = ∫ x⃗ ; 0 df = f(x⃗) − f(0 ). Finally, the critical boundedness check for B_{•} follows from 2.9. □

Proof of Proposition 2.9. The proposition follows given that one can use the uniform raywise estimate (21) to show that for some finite constant C(I, 2ϵ) and any multiindex I fixed one has

(25) ∫_{(0,∞)·C} ∥∫_{X} f_{I} ∥^{2} t^{−2#I}k(x⃗) dvol ≤ C(I, 2ϵ) ∫_{(0,∞)·C} ∥f_{I}∥^{2}t^{−2#I}k(x⃗) dvol.

Note that dvol = t dt dθ⃗ simply adds another factor of t. The lemmas of the next subsection, namely 2.13, 2.14, and 2.15, prove the estimate (25) above. □

2.2.2. Lemmas and estimates. As stated in the last subsection and the introduction, the present argument follows [Muc72] and [BH92]. Also, the lemmas below check that the constant which bounds the growth rate of the exterior derivative is traced explicitly through the present computation and found to be 4/ϵ^{2} for 2ϵ per (21).

Lemma 2.13. Let n ∈ ℤ. Consider the inequality

∫_{C} ∫_{(0,∞)} ∥∫_{X} f∥^{2} t^{n}k(t, θ⃗) dt dθ⃗ ≤ C(λ) ∫_{C} ∫_{(0,∞)} ∥f∥^{2}t^{n}k(t, θ⃗) dt dθ⃗ (26)

for f varying over those square integrable functions of k(x⃗)t^{n} dt dθ⃗. If there exists a function λ(x⃗) so that

{k(s, θ⃗)^{−1}t^{−n}λ(s, θ⃗) ∫_{s}^{∞} k(t, θ⃗)t^{n} (∫_{0}^{t} λ(u, θ⃗)^{−1}du) dt} ≤ C_{λ} < ∞ (27)

uniformly over θ⃗ ∈ C and s > 0, then (26) holds with C(λ) = C_{λ} < ∞.

Lemma 2.14. Suppose we define a possibly infinite constant B as follows:

B = sup_{C}sup_{r>0}\sqrt{∫_{0}^{r} k(t, θ⃗)^{−1}t^{−n}dt ∫_{r}^{∞} k(t, θ⃗)t^{n}dt} . (28)

Then we may choose λ(t, θ⃗) = t^{n}k(t, θ⃗)\sqrt{∫_{0}^{t} s^{−n}k(t, θ⃗)ds} which guarantees the expression of (27) of Lemma 2.13 is bounded above by C_{λ} ≤ 4B^{2}.

Lemma 2.15. Suppose we label M = sup_{t>0}e^{2ϵt}(∫_{t}^{∞} e^{−ϵt})^{−2} = ϵ^{−2}. Then given the uniform raywise bound of (21), one has \sqrt{M} ≥ B for B the constant of (28) of Lemma 2.14. Thus (26) of Lemma 2.13 holds with

C_{λ} ≤ 4B^{2} ≤ 4M = 4ϵ^{−2} < ∞. (29)

Proof of Lemma 2.13. Using the Schwarz inequality for λ(s, θ⃗)ds,

∥∫_{X} f∥^{2} ≤ ∫_{0}^{t} ∥f(s, θ⃗)∥^{2}λ(s, θ⃗)ds (∫_{0}^{t} λ(u, θ⃗)^{−1}du) .

Now one substitutes into the weighted L_{2} integral and applies Fubini’s theorem.

∫_{C} ∫_{0}^{∞} ∥∫_{0}^{t} f(s, θ⃗)ds∥^{2} k(x⃗)t^{n}dt dθ⃗
≤ ∫_{C} ∫_{0}^{∞} ∫_{0}^{t} ∥f(s, θ⃗)∥^{2}λ(s, θ⃗)ds (∫_{0}^{t} λ(u, θ⃗)^{−1}du) k(t, θ⃗)t^{n}dt dθ⃗
= ∫_{C} ∫_{0}^{∞} ∫_{s}^{∞} ∥f∥^{2}λ(s, θ⃗) (∫_{0}^{t} λ(u, θ⃗)^{−1}du) k(t, θ⃗)t^{n}dt ds dθ⃗
= ∫_{C} ∫_{0}^{∞}{− − −}∥f∥^{2}k(s, θ⃗)s^{n}ds dθ⃗
≤ C_{λ} ∫_{C} ∫_{0}^{∞} ∥f∥^{2}k(s, θ⃗)s^{n}ds dθ⃗

with {− − −} per the statement of the lemma. The lemma follows. □

Proof of Lemma 2.14. Recall λ(s, θ⃗) = s^{n}k(s, θ⃗)\sqrt{∫_{0}^{s} u^{−n}k(u, θ⃗)^{−1}du}. For this

choice of λ, let us note as a preliminary that for each θ⃗

∫_{0}^{t} λ(u, θ⃗)^{−1}du = ∫_{0}^{t} 2\frac{d}{du}\sqrt{∫_{0}^{u} w^{−n}k(w, θ⃗)^{−1}dw} du = 2\sqrt{∫_{0}^{t} u^{−n}k(u, θ⃗)^{−1}du}.

Thus one obtains the following estimates, beginning by plugging λ(t, θ⃗) above into the expression of Lemma 2.13 to be bounded.

k(s,θ⃗)^{−1}s^{−n}s^{n}k(s, θ⃗) (∫_{0}^{s} u^{−n}k(u, θ⃗)^{−1}du)^{1/2} ∫_{s}^{∞} k(t, θ⃗) [∫_{0}^{t} λ(u, θ⃗)^{−1}du] dt

= (∫_{0}^{s} u^{−n}k(u, θ⃗)^{−1}du)^{1/2} ∫_{s}^{∞} k(t, θ⃗)t^{n} [2 (∫_{0}^{t} u^{−n}k(u, θ⃗)^{−1}du)^{1/2}] dt

≤ (2B) (∫_{0}^{s} u^{−n}k(u, θ⃗)^{−1}du)^{1/2} ∫_{s}^{∞} k(t, θ⃗)t^{n} (∫_{t}^{∞} u^{n}k(u, θ⃗)du)^{−1/2} dt

= (2B) (∫_{0}^{s} u^{−n}k(u, θ⃗)^{−1}du)^{1/2} 2 (∫_{s}^{∞} t^{n}k(t, θ⃗)dt)^{1/2}
≤ 4B^{2}.

This concludes the proof of the lemma. □

Proof of Lemma 2.15. For each fixed θ⃗_{0}, choose a decreasing function dθ⃗_{0} so that k(t, θ⃗_{0}) = dθ⃗_{0}(t)e^{−ϵt}. Now label M̃ as the following supremum:

M̃ = sup_{C}sup_{t} (∫_{t}^{∞} s^{n}k(s, θ⃗)ds)^{2} t^{−2n}k(t, θ⃗)^{−2}.

Then in fact since all the dθ⃗ decrease, one has

M̃ ≤ sup_{C}sup_{t} \frac{d_{θ⃗} (t)^{2}(∫_{t}^{∞} e^{−ϵt}dt)^{2}}{d_{θ⃗} (t)^{2}e^{−2ϵt}} = M < ∞.

Now flipping the definition about, k(s, θ⃗)^{−1}s^{−n} ≤ Mk(s, θ⃗)s^{n}(∫_{s}^{∞} t^{n}k(t, θ⃗)dt)^{−2}. Whence substituting within the integral,

(∫_{0}^{t} k(s, θ⃗)^{−1}s^{−n}ds)^{1/2}

≤ M^{1/2} [∫_{0}^{t} k(s, θ⃗)s^{n} (∫_{s}^{∞} u^{n}k(u, θ⃗)du)^{−2} ds]^{1/2}

= M^{1/2} [(∫_{t}^{∞} s^{n}k(s, θ⃗)ds)^{−1} − (∫_{0}^{∞} s^{n}k(s, θ⃗)ds)^{−1}]^{1/2}

≤ M^{1/2} (∫_{t}^{∞} s^{n}k(s, θ⃗)ds)^{−1/2} .

Thus M^{1/2} ≥ (∫_{t}^{∞} s^{n}k(s, θ⃗)ds)^{1/2}(∫_{0}^{t} k(s, θ⃗)^{−1}s^{−n}ds)^{1/2} uniformly over t and θ⃗ ∈ C, as we wished to prove. □

Remark 2.16. Reading through the argument, one notices that dlog t^{n} → 0 as t → ∞ for any n ∈ ℤ. As logarithmic derivatives are multiplicative, one notes after the proof that one might instead use the estimate 4ϵ^{−2} directly without the extra factors of t^{n} by renaming k̃(t, θ⃗) = t^{n}k(t, θ⃗).

3. Appendices

3.1. Appendix: Regularity for locally symmetric spaces. The decay of the weights given by formula (2) is extremely rapid. Informal discussions lead the present author to thus believe there are no applications of the (𝔤, K) cohomology of the right regular representation of w_{−Q} or the multiplicative inverse. Nonetheless, references are included here to make these well-defined concepts.

We begin by recalling two regularity results. (N, g) is any complete manifold. Then a regularization result, due to Cheeger [Che80] shows that the (unreduced) L_{2} cohomologies computed with either smooth square integrable forms or with the Hilbert complex of L_{2} forms coincide. The proof uses Friedrich’s mollifiers. Similar results arise as scholia in the weighted case; see [BL92]. This justifies Equation (3) stating that H_{(2),w}^{•}(N) is the cohomology of the complex

{ω | ∫_{M} |ω|^{2} w dvol < ∞ and ∫_{M} |dω|^{2} w dvol < ∞} .

In the locally symmetric case, a stronger regularity holds. Retaining M = Γ\G/K as in the body, let U(𝔤) denote the universal enveloping algebra of 𝔤, i.e., the algebra of left G invariant differential operators on G. We wish to express the w weighted cohomology as the Lie algebra cohomology of a U(𝔤) module. Stated simply, we want the same cohomology to be produced by an even smaller complex:

{ω | ∫_{Γ\X} |Dω|^{2}w dvol < ∞ ∀D ∈ U(𝔤)} . (30)

Borel proved this stronger regularity result in the unweighted case [Bor83]. Specifically, he proved a cochain homotopy from the cohomology of Equation (3) onto that of Equation (30). For the argument, mollification takes places via a convolution in G.

Definition 3.1 ([Fra98, p. 191]). A weight w is admissible if |∇log w| is bounded.

Franke proved the stronger regularity in the weighted case^{†} , given that w is admissible. Of course, Gaussians are inadmissible. Nonetheless, Borel’s argument generalizes immediately to the case at hand. The proof will not be repeated, but a new statement is in order.

Theorem 3.2 ([Bor83], 2.7, p. 618). Let w : Γ\X → (0, ∞) be a Borel measurable weighting function on a locally symmetric space, with w̃ the right K invariant lift. We do not suppose |dlogw| is bounded. Put L_{2,w̃}^{∞}(Γ\G) = V be the Harish-Chandra module of smooth vectors within the regular representation L_{2,w̃}(Γ\G). Then there is an isomorphism H_{(2),w}^{•}(Γ\X, 𝔼) ≅ H^{•}(𝔤, K, V ⊗ E).

†In fact, he proves both regularities directly with weighted Sobolev inequalities.

Remark 3.3. 1) For most w, w̃dg is not G invariant. This does not destroy the statement, but it should be read carefully. For the (𝔤, K) cohomology actually depends on V^{∞} = {f|Df ∈ L_{2,w̃}(Γ\G)∀D ∈ U(𝔤)}.

2) The argument runs roughly as follows. First, lift the question to C^{0}(Γ\G) ⊗ ∧^{•}𝔤^{∗}⊗E. Here, one may produce a cochain homotopy (B, P) whose projection operator P convolves with an approximate convolution identity supported on a neighborhood of the identity in G. That clearly is w̃dg bounded and maps the coefficient functions into C^{∞}, but unfortunately the associated B does not preserve the subspace ∩_{X∈𝔨}ker ι_{X} associated to the (𝔤, K) cohomology. The argument circumvents that problem via explicit analysis of an appropriate spectral sequence.

3.2. Appendix: Extensions to nonarithmetic lattices. There are no great difficulties in generalizing these results to the nonarithmetic lattices. Such lattices only exist in case rankG_{ℝ} = 1. Below appears a short description of the minor modifications of the previous argument which apply to the nonarithmetic case.

• There is strictly speaking no ℚ rank. However, since we suppose M noncompact, the case at hand mimics the case where both ranks are one. In particular, λ ∈ 𝔞̌^{>0} should be read as λ nonzero and positive according to a parabolic preserving some line. The correct definition of −Q < 0 follows.

• Rational parabolics are generally replaced by parabolics stabilizing a line in the language of [GR70], i.e., the stabilizer of the boundary point the lift of an embedded ray of M. Equivalently, any parabolic P with P = UA and (Γ ∩ U)\U compact may replace a rational parabolic in the body.

• Given these conventions, M is point-end.

• The special form presheaf is again just U invariant forms in some neighborhood of a boundary point of the point-end compactification. Weighted L_{2} cohomology is defined in the usual way [Zuc82, p. 175].

Then one can readily repeat all arguments, using Theorem 1.2 of [GR70] in place of the reduction theory of [BHC62].

References

[Bor74] Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4), 7 (1974), 235–272, (1975), MR 52 #8338, Zbl 0316.57026.

[Bor83] Armand Borel, Regularization theorems in Lie algebra cohomology. Applications, Duke Math. J., 50(3) (1983) 605–623, MR 84h:17009, Zbl 0528.22010.

[Bor90] Armand Borel, Correction and complement to the paper: “Regularization theorems in Lie algebra cohomology. Applications” [Duke Math. J. 50 (1983), no. 3, 605–623], Duke Math. J., 60(1) (1990), 299–301, MR 91b:17019, Zbl 0689.22007.

[BHC62] Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2), 75 (1962), 485–535, MR 26 #5081, Zbl 0107.14804.

[BS73] Armand Borel and Jean-Pierre Serre, Corners and arithmetic groups, Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault, Comment. Math. Helv., 48 (1973), 436–491, MR 52 #8337, Zbl 0274.22011.

[BH92] R. C. Brown and D. B. Hinton, A weighted Hardy’s inequality and nonoscillatory differential equations, Quaestiones Math., 15(2) (1992), 197–212, MR 94a:26039, Zbl 0766.26013.

[BL92] J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal., 108(1) (1992), 88–132, MR 93k:58208, Zbl 0826.46065.

[Bue99] Edward L. Bueler, The heat kernel weighted Hodge Laplacian on noncompact manifolds, Trans. Amer. Math. Soc., 351(2) (1999), 683–713, MR 99d:58164, Zbl 0920.58002.

256 Stephen S. Bullock

[Bul01] Stephen S. Bullock, Weighted L_{2} cohomology of asymptotically hyperbolic manifolds, New York J. Math., 7 (2001), 7–22, MR 2002f:58029.

[Che80] Jeff Cheeger, On the Hodge theory of Riemannian pseudomanifolds, in “Geometry of the Laplace operator” (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Amer. Math. Soc., Providence, R.I., 1980, 91–146, MR 83a:58081, Zbl 0461.58002.

[Fra98] Jens Franke, Harmonic analysis in weighted L_{2}-spaces, Ann. Sci. École Norm. Sup. (4), 31(2) (1998), 181–279, MR 2000f:11065, Zbl 0938.11026.

[GR70] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279–326, MR 42 #1943, Zbl 0206.03603.

[GHM94] M. Goresky, G. Harder, and R. MacPherson, Weighted cohomology, Invent. Math., 116(1-3) (1994), 139–213, MR 95c:11068, Zbl 0849.11047.

[Leu95] Enrico Leuzinger, An exhaustion of locally symmetric spaces by compact submanifolds with corners, Invent. Math., 121(2) (1995), 389–410, MR 97f:53085, Zbl 0844.53040.

[MM63] Yozô Matsushima and Shingo Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2), 78 (1963), 365–416, MR 27 #2997, Zbl 0125.10702.

[Muc72] Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math., 44 (1972) (Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I), 31–38, MR 47 #418, Zbl 0236.26015.

[Nai99] Arvind Nair, Weighted cohomology of arithmetic groups, Ann. of Math. (2), 150(1) (1999), 1–31, MR 2000i:11083.

[Nom54] Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2), 59 (1954), 531–538, MR 16,219c, Zbl 0058.02202.

[vE58] W. T. van Est, A generalization of the Cartan-Leray spectral sequence. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math., 20 (1958), 399–413, MR 21 #2236, Zbl 0084.39202.

[Zuc82] Steven Zucker, L_{2} cohomology of warped products and arithmetic groups, Invent. Math., 70(2) (1982), 169–218, MR 86j:32063, Zbl 0508.20020.

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