Koszul Perverse Heart

• Koszul perverse heart is an abelian category defined as the heart of a t-structure with a graded highest weight structure and quadratic Ext-algebra properties.
• It arises from precise stratifications and parity conditions on spaces like flag varieties and Grassmannians, integrating geometric and modular representation theory.
• The framework enables explicit computations using weight separation, degrading functors, and Koszul duality to bridge graded and ungraded perverse categories.

A Koszul perverse heart is an abelian category arising as the heart of a t-structure in a derived category of sheaf-theoretic (or motivic, or Hodge-theoretic) origin, equipped with both a graded highest weight structure and the property that the Ext-algebra of its simple objects is Koszul—i.e., generated in degree one with relations in degree two. The Koszul perverse heart combines geometric, representation-theoretic, and homological features central to the modern structure theory of categories attached to flag varieties, Grassmannians, and their modular or motivic counterparts. Its existence and properties are established by precise structural theorems across several settings, including modular representation theory, parity sheaves, and mixed (Hodge or Tate) motives.

1. Geometric and Algebraic Origins

The Koszul perverse heart emerges in categories constructed from a variety $X$ (typically a flag variety, partial flag, or isotropic Grassmannian) stratified by orbits of a Borel or parabolic subgroup. The crucial examples are categories of perverse sheaves (complex or modular) with respect to such stratifications, categories of mixed (Tate or Hodge) motives, or their modular and diagrammatic models (Riche, 2024, Weidner, 2013, Soergel et al., 2014, Ehrig et al., 2015, Ehrig et al., 2013, Achar et al., 2011, Achar et al., 2014). In all cases, the stratification fulfills strong parity and purity conditions (e.g., "affine even" stratifications), ensuring that Ext-groups between intersection cohomology objects and parity sheaves vanish outside single prescribed degrees.

The construction of the perverse t-structure yields an abelian heart $\mathcal{P}$, which under technical hypotheses (weight separation, parity of IC complexes, and BGS-purity) manifests a graded highest weight structure. In the modular and mixed settings, this grading is physically realized via Tate twists (or Hodge weights), and is encoded algebraically in the structure of the endomorphism algebra (or Ext-algebra) of projective generators.

2. Koszulity Criterion and Structure Theorems

The defining feature of a Koszul perverse heart is that its Ext-algebra $$E = \bigoplus_{i\geq 0} \mathrm{Ext}^i_{\mathcal{P}}(L, L)$$—where $L$ runs over simples in $\mathcal{P}$—admits a grading such that $E_0$ is semisimple, $E$ is generated in degree 1, and all relations are quadratic (generated in degree 2) (Riche, 2024, Weidner, 2013, Achar et al., 2014). The canonical model is the category of graded modules over a Koszul algebra, and the heart $\mathcal{P}$ itself is explicitly equivalent to such a module category.

Precise criteria for this structure include:

• (Acyclic stratification) Each stratum is isomorphic to affine space, and the closure relations are acyclic.
• (Parity and BGS-purity) IC complexes are parity sheaves, and the eigenvalues of Frobenius (or Hodge weights) are separated (Weidner, 2013, Achar et al., 2011).
• (Weight separation) For modular coefficients, the characteristic or $\ell$ must exceed the "weight range" $\operatorname{wr}(X)$ defined via Frobenius eigenvalues (Weidner, 2013).

When these criteria are satisfied, there exists an equivalence: $\mathcal{P} \simeq \mathrm{grmod}\,A$ where $A$ is a (possibly non-positively graded) Koszul algebra, such as the endomorphism ring of a projective generator in the perverse category or an explicit diagrammatic model (see Table 1).

Setting	$\mathcal{P}$	$A$
Modular perverse sheaves	$P_{(B)}(X,F)$	$\mathrm{End}(P^F)$
Mixed Tate motives	$\Perv_S(X)$	$\mathrm{Ext}^\bullet(\Delta, \Delta)$
Diagrammatic—Grassmannians	$\mathbb{D}_k$-mod (isomorphic to perverse sheaves)	$\mathbb{D}_k$

3. Canonical Examples and Explicit Models

Affine Flag Varieties and Mixed Modular Sheaves

For an affine flag variety $\mathrm{Fl} = LG/I$ and characteristic $p$, the category $P^{\mathrm{mix}}_I(\mathrm{Fl})$ of mixed modular Iwahori-equivariant perverse sheaves forms a genuine Koszul highest-weight category (Riche, 2024). Mixed standards, costandards, and tilting objects are constructed as parity sheaf extensions; their Ext-algebra is quadratic and Koszul via comparison with the affine Soergel bimodule category and the Elias–Williamson theorem.

Grassmannians and Parity Sheaf settings

On Grassmannians $\mathrm{Gr}(k,n)$ or partial flags $G/P$, Weidner and others establish that, under separation of weight eigenvalues (e.g., $\ell > \min(k,n-k)+1$ for characteristic $\ell$), the perverse heart identifies with graded modules over an explicit Koszul algebra: vertices correspond to Young diagrams, and arrows to box additions. The algebraic structure is governed by quiver relations arising from the geometry of Schubert cells and their closures (Weidner, 2013).

Isotropic Grassmannians and Diagrammatics

For isotropic Grassmannians of type $D_k$ (or $B_{k-1}$), the perverse category is modeled by the diagrammatic algebra $\mathbb{D}_k$, constructed by a folding procedure from type A arc algebras (Ehrig et al., 2015, Ehrig et al., 2013). The basis elements correspond to symmetric cup-diagrams with orientation data, and the algebra is presented as a path algebra on a quiver with relations determined by diagrammatic moves (diamonds, loops, extendibility), all homogeneous of degree 2. The resulting algebra is positively graded, quasi-hereditary, and Koszul; its module category matches the perverse heart under derived equivalence.

Mixed Tate Motives and Hodge Modules

In the setting of Whitney–Tate stratified varieties, the category of perverse mixed Tate motives, or after a "winnowing" process, mixed Hodge modules, is Koszul if pointwise purity and affine evenness hold (Soergel et al., 2014, Achar et al., 2011). The perverse heart then identifies with graded modules over a quadratic Ext-algebra, with tilting equivalence relating the derived and homotopy categories directly.

4. Functorialities: The Degrading Functor and Koszul Duality

The Koszul perverse heart often admits a canonical "degrading" (forgetful) functor $\nu$ to the ungraded perverse category, compatible with the structure of standard and costandard objects, and satisfying

$\nu \circ \langle 1 \rangle \cong \nu,$

with isomorphisms on Hom and Ext spaces via direct sum over shifts (Riche, 2024). This functor is constructed by identifying the Koszul heart with Soergel bimodules, applying the forgetful functor from graded to ordinary bimodules, and translating back via geometric Satake or tilting-Soergel equivalences.

Koszul duality itself is realized as an exact equivalence of triangulated categories,

$$\kappa: D^{\mathrm{mix}}_I(\mathrm{Fl}) \xrightarrow{\sim} D^b(P^{\mathrm{mix}}_I(\mathrm{Fl}))$$

sending mixed standards to tiltings, and vice versa, with the Ext-algebra being quadratic self-dual or Ringel-dual. On the level of graded highest-weight categories, this identifies the perverse heart with the Koszul dual module category, and recovers geometric Langlands–type dualities in examples with Langlands dual groups (Achar et al., 2014, Riche, 2024).

5. Multiplicities, Kazhdan–Lusztig Theory, and Cellularity

A crucial invariant of the Koszul perverse heart is the explicit computability of graded multiplicities for standard objects within tiltings and projectives. In the modular and mixed modular contexts, these are governed by $p$-Kazhdan–Lusztig polynomials: $$(T^{\mathrm{mix}}_w : A^{\mathrm{mix}}_y \langle n \rangle) = [v^n] \underline{H}_{y,w}^p(v)$$ with $\underline{H}_{y,w}^p(v)$ the $p$-canonical basis coefficient in the Hecke algebra (Riche, 2024). In the Grassmannian and diagrammatic algebra settings, similar closed formulas express graded decomposition numbers in terms of combinatorial data—Dyck paths or oriented circle diagrams—matched to Kazhdan–Lusztig polynomials (Ehrig et al., 2015, Ehrig et al., 2013).

Additional structure includes graded cellularity: there exist explicit graded cellular bases and standard ("cell") modules with known decompositions, reflecting the highest weight and Koszul properties simultaneously.

6. Generalizations and Contexts

The Koszul perverse heart framework extends to:

• Parabolic and Whittaker variants, where local systems replace the constant sheaf as standards, without altering the Koszul structure (Riche, 2024).
• Motive categories with weight structures, where tilting equivalences underlie equivalence with modules over Koszul algebras (Soergel et al., 2014).
• Categories $\mathcal{O}$ (modular and classical), whose blocks are governed by the same Ext-algebra and Koszulity criteria, with the geometric and representation-theoretic sides unified via diagrammatic algebras and Soergel modules (Weidner, 2013, Ehrig et al., 2015, Ehrig et al., 2013).

7. Significance and Further Developments

The theory of Koszul perverse hearts unifies geometric, algebraic, and categorical approaches to highest weight categories arising from stratified spaces and modular representation theory. It provides a conceptual and computationally explicit framework for phenomena such as the gradability of category $\mathcal{O}$, Kazhdan–Lusztig theory for flag varieties and Grassmannians, and structural properties (formality, duality, cellularity) in modular, mixed, and motivic settings. Ongoing work, such as extending to larger classes of parahoric or Whittaker sheaf categories or broader classes of motivic sheaves, continues to exploit the robust algebraic invariants (quadratic Ext algebra, graded cellularity, duality) exhibited by the Koszul perverse heart (Riche, 2024, Weidner, 2013, Ehrig et al., 2015, Ehrig et al., 2013, Soergel et al., 2014, Achar et al., 2014, Achar et al., 2011).