Semi-Infinite Parabolic IC-Sheaf

• Semi-Infinite Parabolic IC-Sheaf is a central object in geometric representation theory that extends perverse sheaf techniques to infinite-dimensional settings.
• It employs a colimit construction over dominant M-coweights and semi-infinite t-structures to capture precise intersection cohomology features.
• The sheaf's compatibility with Hecke eigenproperties and factorization formalism underpins its critical role in the geometric Langlands program.

The semi-infinite parabolic intersection cohomology sheaf (semi-infinite parabolic IC-sheaf) is a central object in geometric representation theory, synthesizing infinite-dimensional geometry, factorization, and t-structure methods. Originally arising in the paper of the affine Grassmannian and the geometric Langlands program, its parabolic incarnations relate deeply to the geometry of moduli stacks, Zastava spaces, and quantum group representation theory.

1. Formal Definition and Construction

Let $G$ be a connected reductive group, $P \subset G$ a parabolic subgroup with Levi factor $M$ and unipotent radical $U$. The foundational space is the affine Grassmannian $\mathrm{Gr}_G = G((t))/G[[t]]$, stratified by orbits of various subgroups. The semi-infinite category $\operatorname{SI}(\mathrm{Gr}_G)$ consists of $G((t))$-equivariant sheaves that are, in addition, equivariant with respect to a loop group $N((t))$, with $N$ the unipotent radical of a Borel.

The semi-infinite parabolic IC-sheaf is constructed as a colimit over the lattice of dominant $M$-coweights, generalizing the Borel case: $$\operatorname{IC}^{\infty/2}_P \simeq \operatorname*{colim}_{\lambda \in \Lambda_{M, ab}^+} t^\lambda \cdot \mathrm{Sat}(V^\lambda)[\mathrm{shift}]$$ where $t^\lambda$ is the translation by $\lambda$, $\mathrm{Sat}(V^\lambda)$ is the image under the geometric Satake equivalence of the $G$-representation $V^\lambda$, and the shift encodes the semi-infinite dimension (typically involving $(1, 2\rho)$ or similar, with $\rho$ the half sum of positive roots) (Dhillon et al., 2023, Dhillon et al., 3 Aug 2025). The transition maps are canonical morphisms arising from the Drinfeld–Plücker formalism.

2. Semi-Infinite Category and t-Structures

In the semi-infinite setup, the relevant categories of sheaves (or D-modules, or étale sheaves) are defined equivariantly for subgroups such as $H = M(\mathcal{O}) \cdot U(F)$. The heart of the semi-infinite t-structure is designed to adapt the concept of perverse sheaves and IC-extensions to infinite-dimensional settings—typically using concepts like pseudodimension (psdim) for appropriate cohomological normalization (Achar et al., 24 Mar 2025).

A key innovation is the semi-infinite t-structure, particularly in the effective context (see Zastava spaces and Ran space versions below), which determines intermediate extension properties for the IC-sheaf and controls the degrees of stalks and costalks along strata indexed by coweights or defects.

3. Factorization, Ran Space, and Zastava Space Realizations

A Ran space version encodes the essential factorization property: the construction is described as the intermediate extension of a dualizing (or constant) sheaf from the basic (generic) stratum in the Ran-graded affine Grassmannian $\mathrm{Gr}_{G,\mathrm{Ran}}$. The object is expressed as a colimit over finite-dimensional approximations indexed by finite sets or coweights (Gaitsgory, 2017, Dhillon et al., 3 Aug 2025). This approach brings factorization algebras and coalgebras into the description of the IC-sheaf, with !- and *-restrictions to closed strata identified, up to cohomological shifts, with the structure sheaf or enveloping algebras over the dual unipotent group.

Moreover, the semi-infinite parabolic IC-sheaf admits a geometric model via compactified Zastava spaces. These finite-dimensional moduli spaces come equipped with factorization structures, and the semi-infinite IC-sheaf is realized as the intermediate extension of the dualizing sheaf along the open Zastava locus in the semi-infinite t-structure (Hayash, 2023). The equivalence of categories of sheaves on Zastava and certain factorization modules on the Ran space underlines the robustness of these constructions.

4. Relationship to Drinfeld Compactification, Global-Local Compatibility

One of the principal features is the relationship between the locally defined semi-infinite parabolic IC-sheaf (on $\mathrm{Gr}_G$ or its Ran version) and the global intersection cohomology sheaf on Drinfeld's compactification of $\mathrm{Bun}_P$, the moduli stack of $P$-torsors on a curve $X$. There is an explicit isomorphism (possibly up to cohomological shift) between the pullback of the global IC-sheaf and the local semi-infinite IC-sheaf under the “evaluation at the point” (or uniformization) map (Dhillon et al., 2023, Dhillon et al., 3 Aug 2025): $$T^*_\mathrm{glob}\mathrm{IC}_{\mathrm{glob}}\left[(g-1)\dim P + (\mu,2\rho-2\rho_M)\right] \simeq \mathrm{IC}^{\infty/2}_P$$ with $T^*_\mathrm{glob}$ a globalization/pullback functor and $g$ the genus of $X$. This canonical identification demonstrates that the semi-infinite parabolic IC-sheaf is the local geometric counterpart of the global parabolic IC-sheaf.

5. Hecke Structures and Drinfeld–Plücker Formalism

The semi-infinite parabolic IC-sheaf is naturally equipped with a Hecke eigenobject structure, formalized and encoded using the Drinfeld–Plücker formalism (Gaitsgory, 2017, Gaitsgory, 2017, Dhillon et al., 2023). That is, for every representation $V$ of the Langlands dual group $\hat{G}$, there are canonical isomorphisms: $$\operatorname{IC}^{\infty/2}_P * \mathrm{Sat}(V) \simeq (\operatorname{Res} V) \otimes \operatorname{IC}^{\infty/2}_P$$ reflecting compatibility with convolution and restriction functors. These structures are essential for the role of the IC-sheaf in geometric Langlands duality and for its connection with dual baby Verma modules on the spectral side.

6. Extensions: Flag Varieties, Modular Settings, and Kac–Moody Generalizations

The theory extends naturally to semi-infinite sheaves on (partial) affine flag varieties, where parabolic semi-infinite IC-sheaves are constructed as the unique common subquotients of standard and costandard objects attached to strata indexed by elements of the extended affine Weyl group (Achar et al., 24 Mar 2025). The resulting theory generalizes the classical intersection cohomology sheaf in a way compatible with modular coefficients, perverse t-structures, and recollement.

Moreover, parabolic and Kac–Moody generalizations are established using factorizable Zastava spaces and semi-infinite t-structures associated to Białynicki–Birula decompositions (Hayash, 2023). In these settings, the construction of the IC-sheaf hinges on compatibility with factorization and t-structure methods, facilitating applications to geometric Eisenstein series and higher categorical representation theory.

7. Applications, Representation Theory, and Geometric Langlands

The semi-infinite parabolic IC-sheaf plays a central role in the geometric Langlands program. It provides a bridge between local geometric data (on semi-infinite orbits in the affine Grassmannian or flag variety) and global objects (on moduli stacks of bundles), reflecting the structure of modular representation categories and acting as a geometric avatar for dual baby Verma modules (Dhillon et al., 2023). Its factorization and Hecke properties ensure compatibility with Eisenstein series constructions, and through explicit formulas relating stalks/costalks to (generalized) Kazhdan–Lusztig polynomials and Kostant’s partition function, it provides a vehicle for transferring representation-theoretic results into the field of sheaf theory (Achar et al., 24 Mar 2025).

The precise control over perversity, support, and intermediate extension ensures that “error terms” from singularities or non-open loci contribute only in positive perverse degrees and do not interfere with the isolation of intersection cohomology structures (Dhillon et al., 3 Aug 2025).

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The synthesis of colimit construction, semi-infinite t-structures, Hecke and factorization formalisms, and compatibility with both local-to-global correspondences and quantum group modules situates the semi-infinite parabolic IC-sheaf as a canonical invariant underpinning the interface of infinite-dimensional geometry, moduli space theory, and categorical representation theory.