Technical Summary: Fredholm Criteria for G-Pseudodifferential Operators

Problem Statement

This paper investigates the Fredholm property of a specific class of operators acting on Sobolev spaces of sections of vector bundles over a closed manifold $M$. Let $G$ be a compact Lie group acting smoothly and isometrically on $M$. The authors consider $G$-pseudodifferential operators, defined as operators of the form:
$$(P + D)u = Pu + \int_G D_g(T_g u) \, dg$$
where $P$ is a classical pseudodifferential operator of order $m$, $D_g$ is a smooth family of pseudodifferential operators of order $m$, $T_g$ represents the shift operator defined by the group action ($T_g u(x) = g \cdot u(g^{-1}x)$), and $dg$ is the normalized Haar measure.

The central problem is to establish necessary and sufficient conditions for such an operator to be Fredholm (i.e., invertible modulo compact operators). While the ellipticity of $P$ guarantees the Fredholm property, the presence of the integral term involving the group action complicates the analysis, particularly when the action is not free or when the group is non-trivial.

Methodology

The authors employ a combination of $C^*$-algebra techniques, local principles, and pseudodifferential uniformization.

1. Reduction to Order Zero: Using the Atkinson theorem and properties of the Laplacian, the authors reduce the general problem of order $m$ to the case of operators of order 0 acting on $L^2(M, E)$.
2. Simonenko's Local Principle: The core methodological tool is a general Simonenko local principle derived in previous work by the first author [6]. This principle relates the global Fredholm property of an operator to its local invertibility on the quotient space $M/G$. The authors apply this to the $C^*$-algebra of $G$-pseudodifferential operators, treating $C(M/G)$ as a subalgebra of the Calkin algebra.
3. Decomposition of the Manifold: The authors identify a subset $P \subset M$ consisting of points where the conormal bundle to the group orbits is trivial (i.e., $(T^*_G M)_x = \{0\}$). They show that $P$ is a union of clopen orbits. The operator is decomposed based on the orthogonal projection onto $M \setminus P$ and $P$.
4. Uniformization for Finite Groups: In the specific case where the group $\Gamma$ is finite, the authors utilize a pseudodifferential uniformization principle. They construct a unitary isomorphism that maps $\Gamma$-pseudodifferential operators on $L^2(M, E)$ to $\Gamma$-invariant classical pseudodifferential operators on the larger space $L^2(M, E \otimes \mathbb{C}[\Gamma])$. This allows the application of known Fredholm criteria for invariant operators.

Key Contributions and Results

1. General Simonenko Principle for $G$-Operators

The paper establishes a general criterion for the Fredholm property of $G$-pseudodifferential operators when $M$ contains no clopen orbits (a condition satisfied, for instance, if $M$ is connected and not a single orbit, or if $\dim M > \dim G$).

Theorem (Simonenko's principle for $G$-operators):
Let $D$ be a $G$-pseudodifferential operator. Then $Id + D$ is Fredholm if and only if it is locally invertible on the quotient space $M/G$.

• Local Invertibility: An operator is locally invertible at a point in the quotient if it admits a local inverse in a neighborhood of that point, modulo compact operators.
• This result characterizes the Fredholm property via the invertibility of the operator's symbol in a local sense over the orbit space, rather than requiring global ellipticity.

2. Characterization for Finite Groups

For the case where the group $\Gamma$ is finite, the authors provide a more explicit characterization in terms of the principal symbol.

Theorem:
Let $D$ be a $\Gamma$-pseudodifferential operator. The following are equivalent:

1. $D$ is Fredholm.
2. The associated invariant operator $\hat{D}$ on $L^2(M, E \otimes \mathbb{C}[\Gamma])$ is Fredholm when restricted to invariant sections.
3. The restriction of the principal symbol $\sigma_0(\hat{D})(\xi)$ to the subspace of vectors invariant under a minimal isotropy subgroup $\Gamma_0$ is invertible for all $\xi$ in the unit cosphere bundle of the principal orbit type.

Specifically, if $M/\Gamma$ is connected and $\Gamma_0$ is a minimal isotropy subgroup, $D$ is Fredholm if and only if:
$$\sigma_0(\hat{D})(\xi): (E_\xi \otimes \mathbb{C}[\Gamma])^{\Gamma_0} \to (E_\xi \otimes \mathbb{C}[\Gamma])^{\Gamma_0}$$
is invertible for all $\xi \in S^*M_{\Gamma_0}$.

3. Handling Disconnected Quotients

The authors address the case where the quotient space $M/\Gamma$ is not connected. They demonstrate that the manifold can be decomposed into $\Gamma$-invariant submanifolds $M_i$ corresponding to the connected components of the quotient. The operator $D$ is Fredholm if and only if its restriction to each $M_i$ is Fredholm. This reduces the general case to the connected case analyzed in the main theorem.

4. Index Formula

For Fredholm operators, the paper provides a formula for the Fredholm index using $K$-theory. The index is identified as the image of the symbol under the boundary map in the exact sequence associated with the $C^*$-algebra of symbols and compact operators.

Significance and Claims

The authors position this work as a refinement and extension of existing results in the theory of singular operators and group-invariant pseudodifferential operators.

• Generalization of Antonevich's Result: The paper notes that for "efficient" actions (where the minimal isotropy group is trivial), their condition recovers a known result by Antonevich [4], where the invertibility of the principal symbol on the full bundle is sufficient. However, the authors claim a new necessary and sufficient condition for cases where the minimal isotropy group is non-trivial. In these cases, the invertibility must be checked only on the subspace of vectors invariant under the minimal isotropy group, a stricter and more precise requirement.
• Unified Framework: By utilizing the Simonenko local principle, the paper unifies the treatment of $G$-operators, avoiding the need for separate ad-hoc arguments for different types of group actions.
• Explicit Symbolic Criteria: For finite groups, the paper moves beyond abstract local invertibility to provide a concrete algebraic condition on the principal symbol, making the Fredholm criterion computationally accessible.

The authors explicitly state that they do not introduce new experimental proposals or future applications but rather focus on the rigorous derivation of these criteria using established $C^*$-algebraic and microlocal analysis techniques. The work is dedicated to the memory of M. Benameur, highlighting the authors' commitment to the theoretical foundations of operator algebras and index theory.