Grothendieck Group of Poisson Sheaves

• Grothendieck group of Poisson sheaves is an algebraic structure capturing classes of coherent Poisson modules on singular symplectic varieties.
• Perverse coherent sheaves form a canonical perverse basis, linking intersection cohomology with geometric and representation-theoretic invariants.
• Equivariant and t-structure techniques facilitate explicit computations in settings like the nilpotent cone and affine Grassmannian.

The Grothendieck group of Poisson sheaves encapsulates the algebraic and categorical structure of coherent sheaves with a compatible Poisson module structure on singular symplectic varieties. For symplectic singularities and related settings, perverse coherent sheaves provide a canonical basis for this Grothendieck group, reflecting and generalizing both geometric and representation-theoretic invariants. The interplay between Poisson geometry, derived categories, and categorical t-structures leads to a robust structure theory, recoverable in explicit cases such as the nilpotent cone and the affine Grassmannian.

1. Definitions and Foundational Structures

Let $X$ be a complex Poisson variety, commonly a conical symplectic singularity. The Poisson bracket on $X$ extends to a structure of Lie algebroid on the sheaf of Kähler differentials $\Omega_X$. A Poisson sheaf (also called a Poisson module) on $X$ is a coherent $\mathcal{O}_X$-module equipped with a compatible $\Omega_X$-module structure, with the compatibility abstractly analogous to Lie algebra modules covariant with respect to the Poisson bracket.

The abelian category $\mathrm{Coh}^{\Omega_X}(X)$ of such Poisson sheaves admits a Grothendieck group,

$$K^{\Omega_X}(X) := K_0(\mathrm{Coh}^{\Omega_X}(X)),$$

which records formal differences of isomorphism classes of Poisson sheaves modulo short exact sequences. If a group $G$ of Poisson automorphisms or a conical grading torus $\mathbb{C}^\times$ acts on $X$, the equivariant category yields

$K^{\mathbb{C}^\times \times G,\, \Omega_X}(X).$

Perverse coherent sheaves (as formulated by Bezrukavnikov et al. and extended in the Poisson setting) are complexes in the derived category of Poisson sheaves, constructed with respect to a t-structure adapted to the stratification by symplectic leaves: $$\mathcal{P}_\mathrm{coh}^{\Omega_X}(X) \subset D^b_\mathrm{coh}(\mathrm{Qcoh}^{\Omega_X}(X)).$$ A standard perversity is $p(S) = -\frac12 \dim S$, where $S$ is a symplectic leaf.

2. Perverse t-Structure and Canonical Basis

A main theoretical result is the existence of a perverse t-structure on the bounded derived category of coherent Poisson sheaves for $X$ with finitely many symplectic leaves (all even-dimensional) and the existence of a duality functor. The heart $\mathcal{P}_\mathrm{coh}^{\Omega_X}(X)$ is both Artinian and Noetherian.

Simple objects in $\mathcal{P}_\mathrm{coh}^{\Omega_X}(X)$ correspond bijectively to data $(S, V)$:

• $S$: a symplectic leaf,
• $V$: a simple coherent Poisson sheaf on $S$.

These are constructed via intersection cohomology (IC) extension functors from sheaves on the leaves: $\operatorname{IC}(S, V).$ The Grothendieck group thus admits a decomposition

$$K^{\Omega_X}(X) = \bigoplus_{(S, V)} \mathbb{Z} \cdot [\operatorname{IC}(S, V)],$$

with the perverse basis formed by the classes of simple perverse coherent sheaves.

Equivariance under $G$, $\mathbb{C}^\times$, or both, is incorporated by forming the corresponding equivariant categories and Grothendieck groups, where analogous statements hold.

The equivalence of K-theory for Poisson coherent sheaves and quasi-coherent Poisson sheaves is established: $$K_0(\mathrm{Coh}^{(G, \mathcal{L})}(X)) \cong K_0(D^b_{\mathrm{coh}} \mathrm{Qcoh}^{(G, \mathcal{L})}(X)).$$

3. Explicit Examples: Nilpotent Cone and Affine Grassmannian

Nilpotent cone ($\mathcal{N} \subset \mathfrak{g}$):

Stratification by $G$-orbits coincides with symplectic leaves. The perverse coherent $G \times \mathbb{C}^\times$-equivariant sheaves category gives

$$K^{G \times \mathbb{C}^\times}(\mathcal{N}) = \bigoplus_{(O,V)} \mathbb{Z} \cdot [\operatorname{IC}(O, V)],$$

where $O$ is a nilpotent orbit and $V$ runs over irreducible equivariant bundles. The perverse basis coincides with the canonical/Kazhdan–Lusztig basis in Hecke algebras and K-theory.

Affine Grassmannian ($\mathrm{Gr}_G$):

Symplectic leaves are orbit-type subvarieties (e.g., $\mathcal{W}_0^\lambda$), and the perverse basis in

$$K^{\mathbb{C}^\times,\Omega}(\mathcal{W}_0^\lambda)$$

matches, via restriction, the basis in the coherent Satake category: $$K^{G(\mathcal{O}) \rtimes \mathbb{C}^\times}(\overline{\mathrm{Gr}^\lambda}).$$ In the colimit $\lambda\to\infty$, this is an isomorphism. In type A, the perverse basis corresponds to Lusztig's dual canonical basis; for other types, it generalizes to expected triangular bases in cluster algebras.

4. Categorical Formalism and Functoriality

The perverse t-structure is constructed via a stratification by symplectic leaves, consistent with the sheaf-theoretic local structure along each stratum. The heart is an abelian Artinian/Noetherian category. Extensions of simple Poisson sheaves from leaves are realized via intersection cohomology.

Equivariance is handled using Harish-Chandra Lie algebroids. For non-nilpotent or singular cases (e.g., symplectic slices or degenerations), the construction persists after passing to completions or normalizations. Functorial properties, including forgetful and restriction functors, preserve the structure of the perverse basis under appropriate morphisms (e.g., between open leaves and closures, or under restriction from larger equivariant groups).

5. Canonical Bases, Algebraic Structures, and Central Formulas

The distinguished perverse basis in the Grothendieck group is given by

$$K^{(\mathbb{C}^\times \times G, \Omega_X)}(X) \cong \bigoplus_{(S,V)} \mathbb{Z} \cdot [\operatorname{IC}(S, V)],$$

parametrized by symplectic leaves and their irreducible (equivariant) Poisson sheaves.

For the nilpotent cone,

$$K^{G \times \mathbb{C}^\times}(\mathcal{N}) \cong K^{\mathbb{C}^\times, \Omega}(\mathcal{N});$$

simple perverse coherent sheaf classes yield the canonical basis matched with Hecke algebra theory.

For the affine Grassmannian,

$$K^{\mathbb{C}^\times,\, \Omega}(\mathcal{W}_0^\lambda) \hookrightarrow K^{G(\mathcal{O}) \rtimes \mathbb{C}^\times}(\overline{\mathrm{Gr}^\lambda}),$$

with the perverse basis preserved under the embedding.

6. Broader Context, Implications, and Future Directions

Perverse coherent sheaves via Poisson Lie algebroid methods establish a canonical and geometric basis for the Grothendieck group of Poisson sheaves on symplectic singularities, unifying and extending constructions from representation theory (e.g., Hecke algebras and canonical bases) and algebraic geometry (intersection cohomology, Satake equivalence).

In prominent cases (nilpotent cone, affine Grassmannian), this approach recovers and generalizes well-known bases (canonical/Kazhdan–Lusztig, dual canonical, and cluster-theoretic bases). The formalism is highly robust, admitting equivariance, functoriality, and compatibility with derived categorical frameworks.

A plausible implication is that this structure provides a unifying categorical framework for various representation-theoretic and geometric correspondences, and suggests paths towards categorified Hall algebras, quantum group actions, and potential connections to shifted Poisson and symplectic geometry. The explicit formulas and basis theorems furnish concrete computational tools for describing K-theory and intersection phenomena in singular symplectic contexts.