The convolution algebra of constructible sheaves

Abstract: Let (E) be a finite-dimensional real vector space. We study invertible objects in the monoidal category of constructible sheaves on (E), endowed with the convolution product (\star). We show that the inverse of an invertible constructible sheaf (F) is the dual of its antipodal transform. We also prove that a compactly supported constant sheaf is invertible if and only if its support is convex. We also introduce a microlocal transform (B(F)), obtained by projecting the characteristic cycle of $F$ to (E^*), and prove that it is compatible with convolution. This yields a necessary condition for invertibility.

• The paper demonstrates that a constructible sheaf with compact support is invertible under convolution if and only if its inverse is given by the Verdier dual of its antipodal transform.
• Employing microlocal analysis, it introduces a novel microlocal transform B(F) linking convolution to Lagrangian cycle invariants and establishing ring homomorphism properties.
• It establishes that the convexity of the support for constant sheaves is equivalent to invertibility, offering insights for applications in symplectic topology and representation theory.

The Convolution Algebra of Constructible Sheaves: Invertibility, Microlocal Analysis, and Lagrangian Cycles

Introduction and Principal Contributions

This work provides a comprehensive analysis of the convolution algebra structure on the derived category of constructible sheaves with compact support over a real finite-dimensional vector space $E$. Utilizing advanced tools from microlocal sheaf theory and the formalism of characteristic cycles developed by Kashiwara and Schapira, the paper characterizes invertible objects under the convolution operation $\star$, establishes explicit formulas for their inverses, and explores their microlocal invariants. The study further introduces and analyzes a microlocal transform $B(F)$ derived from the projection of the characteristic cycle of $F$ onto the dual vector space $E^*$, establishing compatibility of this transform with convolution and deducing necessary invertibility criteria in terms of $B(F)$.

Algebraic Framework and Notation

Let $D^b(k_E)$ denote the bounded derived category of sheaves of $k$-vector spaces (with $k=\mathbb{R}$ or $\mathbb{C}$) on $\star$0, and $\star$1 its subcategory of $\star$2-constructible sheaves. The convolution product is defined as

$\star$3

where $\star$4 is the addition map. The notion of invertibility concerns the existence of a two-sided inverse for $\star$5 under $\star$6, i.e., $\star$7, the Dirac sheaf at the origin, for some $\star$8.

Microlocal techniques are heavily employed, in particular, the use of the singular support $\star$9 of $B(F)$0, and the characteristic cycle $B(F)$1 as an element of the module of Lagrangian cycles. The antipodal transform $B(F)$2 is the pushforward of $B(F)$3 under the fiberwise antipodal map on $B(F)$4.

Characterization of Invertibility for Constructible Sheaves

The main algebraic result asserts that a constructible sheaf $B(F)$5 with compact support is invertible under convolution if and only if its inverse is given by $B(F)$6, i.e., the Verdier dual of the antipodal transform. This is established through a reduction to the one-dimensional case, leveraging decompositions of sheaves into direct sums of interval sheaves as classified by Guillermou. For $B(F)$7 of any dimension, invertibility of a compactly supported constant sheaf $B(F)$8 under $B(F)$9 is shown to be equivalent to the convexity of $F$0.

Explicit Construction of Inverses

For any $F$1 with compact support, if $F$2 is invertible, then its inverse satisfies

$F$3

where $F$4 denotes the Verdier dual. In the special case of constant sheaves on compact convex subsets $F$5, the inverse is up to shift the constant sheaf on the interior of $F$6.

Convexity versus Invertibility

A strong equivalence is proved: among compactly supported constant sheaves, invertibility is equivalent to convexity of the support. The proof employs both direct calculations for one-dimensional cases and induction on the dimension, combined with microlocal techniques and properties of direct images.

Microlocal Transform and Lagrangian Cycle Invariants

A significant part of the paper is devoted to the construction and analysis of a microlocal invariant $F$7 in terms of the characteristic cycle. Given a constructible sheaf $F$8, $F$9 is defined by projecting $E^*$0 onto $E^*$1:

$E^*$2

where $E^*$3 is the canonical projection.

A commutative and associative product $E^*$4 is defined on these microlocal invariants by an explicit morphism on the level of hypercohomology and supports.

Compatibility with Convolution

A central theorem shows that $E^*$5 is a ring homomorphism from the monoidal category $E^*$6 to the algebra $E^*$7 of microlocal invariants:

$E^*$8

for any $E^*$9 with compact support. This compatibility is established through a detailed analysis of characteristic cycles using direct image formulas and Künneth-type isomorphisms.

Invertibility Criterion for Microlocal Invariants

For an invertible sheaf $B(F)$0, one obtains the relation

$B(F)$1

where $B(F)$2 denotes the distinguished section in $B(F)$3 arising from the Dirac sheaf at $B(F)$4.

Theoretical and Practical Implications

The invertibility results provide a rigorous microlocal characterization of "atomic" operations in the convolution algebra of constructible sheaves, with potential applications in symplectic topology, representation theory (notably in the context of the geometric Satake equivalence), and micro-local Morse theory. The correspondence between invertibility and convexity establishes a direct link between algebraic properties of the monoidal category and geometrical-topological properties of supports.

The introduction of the microlocal transform $B(F)$5 and its functoriality with respect to convolution points to new invariants classifying convolution orbits, which may be influential in sheaf-theoretic approaches to integral geometry (e.g., Radon transforms) and symplectic geometry.

Directions for Future Work

Further investigations may include:

• Extension to real analytic or subanalytic sheaf settings where convexity conditions are subtler.
• Relationships to multiplicative convolution on (singular) toric varieties, and generalizations to symplectic groupoids.
• Explicit calculation of $B(F)$6 for more general (possibly perverse) sheaves, and connection with representation-theoretic categories.
• Analysis in the context of microlocal quantization and Fukaya category enhancements.

Conclusion

This work rigorously characterizes invertibility in the convolution algebra of constructible sheaves on real vector spaces, making precise the relationship between invertibility, Verdier duality, and the antipodal transformation. It introduces a microlocal invariant fundamentally compatible with the convolution product, tying together the algebraic and microlocal structures involved. The results promote deeper connections between sheaf-theoretic and symplectic techniques and open new avenues for the classification and manipulation of sheaf algebras with rich geometric content.