Multiplicative Weightings of Lie Groupoids

• The paper’s main contribution is the formulation of multiplicative weightings for Lie groupoids, extending manifold weighting to a setting that harmonizes with groupoid operations.
• It introduces a rigorous filtration on the algebra of smooth functions, ensuring compatibility with source, target, multiplication, and associated graded structures.
• These weightings enable advanced applications such as pseudodifferential calculus, index theory, and the treatment of singularities in geometric analysis.

A multiplicative weighting for Lie groupoids is a filtration-theoretic structure that equips a Lie groupoid $G \rightrightarrows M$—and a subgroupoid $H \rightrightarrows N$—with a compatible filtration of the algebra of smooth functions, extending the concept of a weighting of manifolds along submanifolds to a setting where groupoid morphisms and operations interact compatibly with the filtration. This formalism links graded geometric structures, filtered manifolds, and groupoid-based analytic machinery such as deformation groupoids and the pseudodifferential calculus on filtered spaces. The theory arises in the foundational work of Meinrenken, Loizides, and Hudson, establishing a framework for analytic and cohomological generalizations in geometric analysis and index theory (Meinrenken, 15 Jan 2026, Loizides et al., 2020, Hudson, 14 Aug 2025).

1. Weightings Along Subgroupoids

Let $G \rightrightarrows M$ be a Lie groupoid with source and target maps $s, t: G \to M$, and multiplication $\mathrm{Mult}_G: G^{(2)} \to G$. An order-$r$ weighting of $G$ along a closed embedded subgroupoid $H \subset G$ (with units $N \subset M$) consists of a decreasing filtration

$$C^\infty(G) = C^\infty(G)_{(0)} \supseteq C^\infty(G)_{(1)} \supseteq \cdots \supseteq C^\infty(G)_{(r)} \supseteq \cdots$$

satisfying:

• $C^\infty(G)_{(1)}$ is the vanishing ideal of $H$, i.e., $I_H = \{f \in C^\infty(G) \mid f|_H = 0\}$;
• The induced filtration on $T^*G|_H$ by $C^\infty(G)_{(i)}$ is by subbundles, terminating at zero at degree $r$; sections of $(T^*G|_H)_{(i)}$ are $C^\infty(H)$-linear spans of $d f|_H$ for $f \in C^\infty(G)_{(i)}$;
• The filtration satisfies the Leibniz rule: $$C^\infty(G)_{(i)} \cap I_H^2 = \sum_{0 < j < i} C^\infty(G)_{(j)} \cdot C^\infty(G)_{(i-j)}$$ for $2 \leq i \leq r$.

In local coordinates $(x_1, ..., x_{\dim G})$ of weights $w_a$, $f \in C^\infty(G)_{(i)}$ if and only if $f = \sum_s a_s x^s$ with $\sum_a s_a w_a \geq i$ (Meinrenken, 15 Jan 2026, Hudson, 14 Aug 2025).

2. Multiplicativity: Compatibility and Characterizations

A weighting as above is multiplicative if it is compatible with the Lie groupoid structure. The following conditions—each equivalent—capture this compatibility (Meinrenken, 15 Jan 2026, Loizides et al., 2020, Hudson, 14 Aug 2025):

• Structure maps: The source and target maps $$s, t: (G, \{ C^\infty(G)_{(i)} \}) \rightarrow (M, \{ C^\infty(M)_{(i)} \})$$ are weighted submersions; the unit inclusion $u: M \hookrightarrow G$ is a weighted embedding with image $H$; the multiplication $\mathrm{Mult}_G$ is a weighted submersion on the appropriately weighted product of $G$ with itself.
• Tangent and cotangent flags: For each $i$, the filtered subbundle $(TG|_H)_{(-i)} \subset TG|_H$ is a Lie–subgroupoid of $TG \rightrightarrows TM$ (i.e., closed under groupoid operations).
• Graph condition: The graph of the groupoid multiplication $$\mathrm{Gr}(\mathrm{Mult}_G) \subset G \times G \times G$$ is a weighted submanifold.
• Weighted deformation space: The weighted deformation groupoid $\delta_W(G, H)$ over $\mathbb{R}$ is a groupoid whose fiber at $0$ is the weighted normal bundle (the osculating groupoid), and at $c \neq 0$ is $G$ itself.

The necessity of these properties is justified by considering the effect of the weighted structure on the filtrations of the groupoid tangent spaces and the requirement that these be preserved by groupoid operations. Conversely, given that the tangent flags are subgroupoids, weighted local coordinates adapted to the groupoid structure can be constructed (Meinrenken, 15 Jan 2026, Loizides et al., 2020).

3. Infinitesimal (Lie Algebroid) Aspects and Integration

The infinitesimal counterpart of a multiplicative weighting is an infintesimally multiplicative (IM) weighting on the Lie algebroid $A \to M$ of $G$, along the Lie subalgebroid $B \to N$ of $H$. This is a filtration

$$\cdots \supseteq \Gamma(A)_{(i)} \supseteq \Gamma(A)_{(i+1)} \supseteq \cdots$$

such that

• the anchor map $a: (A)_{(i)} \to \mathfrak{X}(M)_{(i)}$ preserves the filtration,
• the bracket satisfies $$[\Gamma(A)_{(i)}, \Gamma(A)_{(j)}] \subseteq \Gamma(A)_{(i+j)}$$.

These IM-weightings are in one-to-one correspondence with multiplicative weightings on source-simply connected groupoids, via the integration of graded Lie subalgebroids of $T_rA|_B$ to Lie subgroupoids of $T_rG|_H$. The process mirrors the classical Lie III theorem for integrating algebroid data to global groupoid structures (Loizides et al., 2020, Hudson, 14 Aug 2025).

4. Normal and Deformation Groupoids

Given a multiplicative weighting, several groupoid constructions acquire compatible weighted structures:

• The weighted normal groupoid $\nu_W(G, H) \rightrightarrows \nu_W(M, N)$ (“osculating groupoid”) is a graded Lie groupoid over the weighted normal bundle, constructed from the associated graded algebra of the filtration.
• The weighted deformation groupoid $\delta_W(G, H) \rightrightarrows \delta_W(M, N)$ is a Lie groupoid over $\mathbb{R}$, specializing to $\nu_W(G, H)$ at $0$ and $G$ elsewhere (the Rees deformation).
• The weighted blow-up groupoid $$\mathrm{Bl}^w(G, H) \rightrightarrows \mathrm{Bl}^w(M, N)$$ is obtained by removing the zero locus in $\delta_W(G, H)$ and quotienting by the $\mathbb{R}_{>0}$ action.

All these constructions inherit compatible multiplicative (or infinitesimally multiplicative) structures, allowing for analytic and geometric operations mirroring those of the underlying groupoid, and are central to applications in analysis and microlocal geometry (Loizides et al., 2020, Meinrenken, 15 Jan 2026).

5. Explicit Examples

Broad classes of groupoids admit natural multiplicative weightings:

Example Type	$G$	$H$
Pair groupoid	$M \times M \rightrightarrows M$	$N \times N$
Action groupoid	$K \ltimes M \rightrightarrows M$	$K \ltimes N$
Gauge groupoid	$(P \times P)/K \rightrightarrows M$	$(P|_N \times P|_N)/K$
Vector bundle groupoid	$E \rightrightarrows M$	$W \rightarrow N$

• For the pair groupoid, weighting $M \times M$ along the diagonal recovers the tangent groupoid of Connes and the classical deformation to the normal cone (Meinrenken, 15 Jan 2026, Loizides et al., 2020).
• For action and gauge groupoids, weights are pulled back via the groupoid structure; normal bundles are given by the associated groupoid action on linearized data.
• For vector bundle groupoids, linear weightings respect the vector bundle structure and are compatible with the groupoid operations (Hudson, 14 Aug 2025, Loizides et al., 2020).

6. Analytic, Cohomological, and Geometric Applications

Multiplicative weightings enable significant analytical and geometric constructions:

• Filtration of cochain complexes: Introducing a multiplicative weighting yields a filtration of the differentiable cochain complex of $G$, with the associated graded complex controlling characteristic classes of the osculating groupoid (Meinrenken, 15 Jan 2026).
• Pseudodifferential analysis: The framework generalizes van Erp–Yuncken’s pseudodifferential calculus on filtered manifolds to groupoids with multiplicative weighting—symbol maps, parametrices, and index theory for anisotropic hypoelliptic operators can be formulated using the deformation groupoid and its zoom action (Meinrenken, 15 Jan 2026).
• Singular geometric structures: Constructions such as the weighted blow-up groupoid provide tools for handling geometric singularities (e.g., in sub-Riemannian geometry or fibred boundary analysis), extending the reach of groupoid-based analysis to spaces with corners or singular stratifications (Loizides et al., 2020, Hudson, 14 Aug 2025).

7. Connections and Further Developments

The theory of multiplicative weightings dovetails with multiple advanced topics and research initiatives:

• The correspondence between multiplicative and IM-weightings provides a bridge from filtered differential geometry to global groupoid theory, opening avenues for the study of filtered groupoids, hypoelliptic operators, and index theory on singular spaces (Hudson, 14 Aug 2025, Loizides et al., 2020).
• Weighted VB-groupoids, higher tangent prolongations, and their applications in singularity theory and quantization are under active investigation.
• Applications to deformation quantization, weighted Morita equivalence, C*-algebraic deformation theory, and analytic structures on groupoids are anticipated (cf. van Erp–Yuncken, Melrose’s $b$-calculus) (Meinrenken, 15 Jan 2026, Hudson, 14 Aug 2025).

The development of multiplicative weightings for Lie groupoids situates these structures as foundational to the modern analysis and geometry of filtered and singular spaces, providing intrinsic, coordinate-free tools for the unified study of deformations, characteristic classes, and the microlocal analysis of groupoid and algebroid structures.