Parabolic Semi-Infinite Categories

• Parabolic semi-infinite categories are advanced structures that extend finite highest weight categories to infinite-dimensional, stratified settings via parabolic induction and graded methods.
• They employ semi-infinite t-structures and IC-sheaves to capture deep representation-theoretic properties and facilitate local-to-global correspondences in geometric Langlands applications.
• Diagrammatic, combinatorial, and geometric models of these categories offer tangible frameworks for studying Kazhdan–Lusztig theory, categorification, and homological algebra.

A parabolic semi-infinite category is an advanced categorical structure at the intersection of infinite-dimensional representation theory, geometry of flag varieties, and modern higher algebra. It arises in contexts where classical finite highest weight categories are extended to infinite or stratified settings, often in relation to parabolic subgroups, semi-infinite stratifications, factorization spaces, and equivariance conditions. Parabolic semi-infinite categories play a pivotal role in the paper of (parabolic) category $\mathcal{O}$, geometric representation theory, semi-infinite flag varieties, categorification phenomena, and the geometric Langlands program.

1. Foundational Constructs: Parabolic Induction and Graded Category $\mathcal{O}$

Parabolic semi-infinite categories are rooted in the representation theory of complex reductive Lie algebras and, crucially, in the structure of category $\mathcal{O}$ for both finite and infinite rank types. These categories are constructed by employing the parabolic induction functor, which embeds modules from a Levi subalgebra $\mathfrak{l}$ (inside a parabolic subalgebra $\mathfrak{p} \subset \mathfrak{g}$) into the representations of the full Lie algebra $\mathfrak{g}$ via: $$\mathrm{Ind}_{\mathfrak{p}}^{\mathfrak{g}}(M) \cong U(\mathfrak{g}) \otimes_{U(\mathfrak{p})} M,$$ where $M$ is a module in category $\mathcal{O}(\mathfrak{l})$ (Eberhardt, 2016). This functor is exact and compatible with the weight structure, providing a powerful mechanism to bridge finite and infinite rank theories via truncation functors and inductive limits (Chen et al., 2018).

Graded versions of category $\mathcal{O}$, specifically those carrying a Koszul grading as developed by Beilinson–Ginzburg–Soergel (BGS), form the natural habitat for the semi-infinite versions, as these gradings facilitate homological and stratification techniques (Eberhardt, 2016, Heidersdorf et al., 10 Jun 2025). The graded aspect is central for formulating equivalences with geometric categories, such as those of stratified mixed Tate motives on partial flag varieties.

2. Semi-Infinite and Parabolic Stratification: t-Structures and Zastava Spaces

The semi-infinite structure emerges in the geometry of the affine Grassmannian $\mathrm{Gr}_G$ and its parabolic variants. Parabolic semi-infinite categories are defined as categories of sheaves (or $D$-modules) on spaces such as the Ran version of the affine Grassmannian $\mathrm{Gr}_{G, \mathrm{Ran}}$, equipped with a stratification by orbits of the loop unipotent group $N_P((t))$ (where $N_P$ is the unipotent radical of a parabolic $P$).

A salient feature is the introduction of semi-infinite $t$-structures, defined using G$_m$-actions and Białynicki-Birula decompositions, which enable one to formulate perverse-like conditions even in infinite-dimensional colimit settings (Hayash, 2023). The open, "non-degenerate" locus of these stratifications is typically compactified via Zastava spaces—generalized to the parabolic context by considering generically transverse pairs of reductions to the unipotent radical $N_P$ and to the opposite parabolic $P^-$ (Hayash, 2023). The semi-infinite $t$-structure is then used to define intermediate extensions of dualizing sheaves across singularities in these spaces.

The parabolic compactified Zastava space, $Z_{P, P^-}$, is constructed as a fiber product of the moduli of $N_P$ and $P^-$-reductions, and carries a factorization structure over configuration spaces indexed by negative coweights of the Levi subgroup (Hayash, 2023, Dhillon et al., 3 Aug 2025). This factorization, crucial for formulating the local-to-global correspondences, manifests via colimit and descent constructions on the Ran space.

3. The Parabolic Semi-Infinite IC-Sheaf and Its Properties

A key object is the parabolic semi-infinite intersection cohomology sheaf (parabolic semi-infinite IC-sheaf), denoted $\operatorname{IC}^{\infty/2}_P$ or variants thereof. This object is constructed in several equivalent ways:

• As a colimit over dominant coweights, mimicking the construction

$$\operatorname{IC}^{\infty/2} = \operatorname{colim}_{\lambda \in \Lambda^+} t^\lambda \cdot \mathrm{Sat}(V^\lambda)[-\langle 2\rho, \lambda\rangle],$$

where $t^\lambda$ is translation and $\mathrm{Sat}$ is the geometric Satake functor (Gaitsgory, 2017);

• As an intermediate extension (in the semi-infinite $t$-structure) of the dualizing sheaf on the open stratum of the parabolic Zastava space (Hayash, 2023, Dhillon et al., 3 Aug 2025).

This IC-sheaf is factorizable and encodes rich representation-theoretic data: its stalks and costalks—calculated via !-fibers along parabolic strata—record stable (or periodic) affine Kazhdan–Lusztig polynomials and symmetric algebra invariants of the Langlands dual Lie algebra (Gaitsgory, 2017). Its construction is independent of auxiliary choices such as finite-type approximations (after renormalization), and it exhibits canonical Verdier self-duality as detailed in the framework of Ind-completed constructible categories (Dhillon et al., 2023).

The factorization property imposes that the tensor product of local IC-sheaves over disjoint divisors is isomorphic to the IC-sheaf over their union: $$\operatorname{fact}_y : \bigotimes_{i=1}^n \mathrm{IC}_{P, \mathrm{Ran}}(D_i) \xrightarrow{\sim} \mathrm{IC}_{P, \mathrm{Ran}}\left(\bigcup_{i=1}^n D_i\right)$$ (Hayash, 2023). This is foundational for the local-to-global compatibility of the category and for applications to factorization homology and geometric Eisenstein series.

4. Connections to Representation Theory and Homological Algebra

Homological methods underlie parabolic semi-infinite categories. The equivalence between graded category $\mathcal{O}$ and stratified mixed Tate motives, as well as the Koszulity of semi-infinite highest weight categories, ensure that Ext-groups and projective/injective resolutions retain a linear and tractable form even in the infinite or semi-infinite regime (Eberhardt, 2016, Heidersdorf et al., 10 Jun 2025).

In the infinite rank Lie algebra context, projective covers of simple modules and the existence of finite-length analogues of the BGG reciprocity formula are established by truncation functors and direct limit techniques (Chen et al., 2018). The combinatorics of extension groups in parabolic singular blocks are governed by (anti-)spherical Kazhdan–Lusztig polynomials, made explicit via graded endomorphism algebras with standard Koszul gradings (Gruber, 2022).

Super duality phenomena link parabolic semi-infinite categories for Lie algebras and Lie superalgebras, preserving properties such as Koszulity and homological dimensions (Chen et al., 2018). These features support categorification, as demonstrated by the realization of endomorphism algebras such as cyclotomic Nazarov–Wenzl algebras within diagrammatic categories associated to parabolic $\mathcal{O}$ in types B, C, D (Rui et al., 2023).

5. Combinatorial, Diagrammatic, and Geometric Realizations

Parabolic semi-infinite categories admit diverse explicit models:

• Diagrammatic and algebraic: Realized via Soergel modules, cellular and diagrammatic algebras (e.g., Khovanov algebras, Khovanov–Lauda–Rouquier categorifications), affine or cyclotomic Brauer-type categories, and their Schur–Weyl dual relations to parabolic $\mathcal{O}$ (Eberhardt, 2016, Rui et al., 2023, Heidersdorf et al., 10 Jun 2025). These provide combinatorial control over homological invariants and decomposition matrices via Kazhdan–Lusztig theory.
• Combinatorial/weighted categories: Equivalences to categories of FI(n)-modules or representations of weighted set categories are established for equivariant modules over infinite polynomial rings under parabolic group actions (Yu, 2 Jul 2024). These combinatorial interpretations yield stratifications by support conditions and allow control over local cohomology, injective dimensions, and Hilbert series.
• Tableau and crystal models: For quantum affine algebras and their parabolic settings, tableau models using Kashiwara–Nakashima columns and Lakshmibai–Seshadri paths afford concrete combinatorial realizations of the semi-infinite Bruhat order and its parabolic analogues (Ishii, 2021).

Parabolic semi-infinite categories also connect with geometric entities such as the semi-infinite flag variety, Drinfeld's compactifications, and Zastava spaces. Through categorical and motivic techniques (such as perverse sheaves, mixed Tate motives, and factorization properites), these connections bridge the combinatorics of Hecke modules and the geometry of infinite-dimensional moduli spaces (Morton-Ferguson, 2022, Hayash, 2023).

6. Applications in Geometric Representation Theory and Langlands Correspondence

The foundational place of parabolic semi-infinite categories in the geometric Langlands program is established through their role as categorified kernels mediating local-to-global functoriality, Eisenstein series correspondences, and spectral-side constructions. The semi-infinite IC-sheaf provides the geometric incarnation of representation-theoretic invariants, encoding Hecke eigenvalues and equivalences between sheaf and module categories (Gaitsgory, 2017, Dhillon et al., 2023, Hayash, 2023, Dhillon et al., 3 Aug 2025).

Factored versions on the Ran space facilitate the passage from local to global, yielding compatibility with the intersection cohomology sheaf of the Drinfeld compactification of $\mathrm{Bun}_P$ and the spectral realization of dual baby Verma objects (Dhillon et al., 2023, Dhillon et al., 3 Aug 2025). These structures underlie categorification frameworks, topological field theory approaches, and higher representation-theoretic constructions in both finite and Kac–Moody settings.

7. Structural Finiteness, Koszulity, and Diagrammatic Methods

Despite the infinite or semi-infinite nature, many parabolic semi-infinite categories exhibit strong finiteness and regularity properties:

• Krull–Gabriel dimension coinciding with the number of parabolic strata (Yu, 2 Jul 2024);
• Existence of filtrations by prime supports, yielding Serre quotients corresponding to representation categories of explicit combinatorial subgroups;
• Rationality of Hilbert series, finite generation of local cohomology, and finite injective dimension under P-equivariance (Yu, 2 Jul 2024);
• Koszulity in semi-infinite highest weight categories, entailing that standard modules admit linear projective resolutions—this holds for Khovanov algebras, Deligne categories, and their semi-infinite generalizations (Heidersdorf et al., 10 Jun 2025);
• The equivalence between stratified highest weight categories and categories of modules over based quasi-hereditary algebras or cellular diagrammatic algebras, a perspective that unifies algebraic, geometric, and combinatorial incarnations (Brundan et al., 2018, Heidersdorf et al., 10 Jun 2025).

These finiteness phenomena and equivalences are foundational for explicit computations, categorification, and the transfer of classical representation-theoretic techniques to the infinite or stratified setting.

---

In conclusion, parabolic semi-infinite categories constitute a rich and unifying framework at the interface of infinite-dimensional Lie theory, geometric representation theory, and categorification. They intertwine homological algebra, combinatorics, and geometry via foundational constructs such as parabolic induction, semi-infinite stratifications, IC-sheaves, and diagrammatic or tableau models. These categories serve as the backbone for deep applications in the geometric Langlands program, categorification of quantum invariants, and the paper of modern infinite-dimensional symmetry.