[Paper Review] Intersection cohomology of Drinfeld's compactifications

Alexander BRAVERMAN, D. Gaitsgory+2 more|arXiv (Cornell University)|Dec 15, 2000

Advanced Algebra and Geometry4 citations

TL;DR

This paper provides an explicit description of the intersection cohomology sheaf on Drinfeld's compactification of the moduli stack of $P$-bundles on a curve, using combinatorics of the Langlands dual Lie algebra and Zastava spaces as local models. The key result identifies the IC sheaf as a direct sum of pushforwards of tensor products of local systems on symmetric products of the curve, indexed by weights in the coroot lattice, with shifts determined by the degree of the parabolic.

ABSTRACT

Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification $\widetilde{\on{Bun}}_P$ of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\on{Bun}}_P$. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.

Motivation & Objective

• To provide a geometric and representation-theoretic description of the intersection cohomology sheaf on Drinfeld's compactification $\widetilde{\operatorname{Bun}}_P$.
• To resolve the issue of non-properness in the naive Eisenstein series functor by constructing a corrected geometric Eisenstein series functor via the compactified stack.
• To establish a link between the singularities of $\widetilde{\operatorname{Bun}}_P$ and the semi-infinite flag variety through finite-dimensional models (Zastava spaces).
• To realize Lusztig's combinatorial patterns in terms of intersection cohomology, particularly via periodic Kazhdan-Lusztig polynomials.

Proposed method

• Use Zastava spaces $Z^\mu$ as finite-dimensional local models for the singularities of $\widetilde{\operatorname{Bun}}_P$, generalizing them to arbitrary parabolic subgroups.
• Define the semigroup $\mathbb{Z}_+\Theta$ of positive combinations of roots $\theta \in \Theta \subset \Lambda_{G,P}$, where $\Theta$ indexes non-zero weight spaces in the unipotent radical of the Langlands dual parabolic.
• Construct sheaf complexes $\mathcal{R}_{\mathfrak{P}(\theta)}(V)$ on symmetric products $X^{\mathfrak{P}(\theta)}$ via tensor products of localized sheaves associated to representations of the dual Levi $\check{M}$.
• Use the normalization map $j_{P,\mathfrak{P}(\theta)}$ to relate the IC sheaf on $\widetilde{\operatorname{Bun}}_P$ to pushforwards of these local systems along strata $^\mathfrak{P}(\theta)\overline{\operatorname{Bun}}_P$.
• Apply induction over strata, starting from the open stratum, to compute the total IC complex as a direct sum over all $\mathfrak{P}(\theta) \in \mathbb{Z}_+\Theta$, with shifts by $2|\mathfrak{P}(\theta)|$.
• Leverage self-duality of $\operatorname{Im}(e)[1]$ for the principal nilpotent $e$ on $\check{\mathfrak{u}}(P)$ to ensure duality properties of the resulting sheaf complex.

Experimental results

Research questions

• RQ1 How can the intersection cohomology sheaf on Drinfeld's compactification $\widetilde{\operatorname{Bun}}_P$ be explicitly described in terms of geometric and representation-theoretic data?
• RQ2 What is the precise relationship between the corrected geometric Eisenstein series functor and the naive one, and how does it relate to $L$-functions in the geometric setting?
• RQ3 To what extent do Zastava spaces serve as local models for the singularities of $\widetilde{\operatorname{Bun}}_P$, and how do they reflect the structure of the semi-infinite flag variety?
• RQ4 How do the stalks of the IC sheaf on $\widetilde{\operatorname{Bun}}_P$ encode Lusztig’s periodic Kazhdan-Lusztig polynomials?
• RQ5 Can the IC sheaf on $\widetilde{\operatorname{Bun}}_P$ be expressed as a direct sum of pushforwards of sheaves on symmetric products of the curve, with coefficients in representations of the dual Levi group?

Key findings

• The intersection cohomology sheaf on $\widetilde{\operatorname{Bun}}_P$ is isomorphic to a direct sum over all $\mathfrak{P}(\theta) \in \mathbb{Z}_+\Theta$ of the pushforwards $ (j_{P,\mathfrak{P}(\theta)})_! $ applied to the sheaf $ \mathcal{R}_{\mathfrak{P}(\theta)}(\operatorname{Im} e) \boxtimes \operatorname{IC}(\overline{\operatorname{Bun}}_P) $, shifted by $ 2|\mathfrak{P}(\theta)| $.
• The sheaf $ \mathcal{R}_{\mathfrak{P}(\theta)}(V) $ on $ X^{\mathfrak{P}(\theta)} $ is constructed as a box product of localized sheaves $ \overline{\operatorname{Loc}}(V_{\theta_s})^{(n_s)} $, where $ V $ is a $ \Theta $-adapted $ \mathbb{Z} $-graded $ \check{M} $-module.
• The complex $ \mathfrak{Q} := \mathfrak{r}_! \operatorname{IC} $ on $ \overline{\operatorname{Bun}}_P $ is isomorphic to the direct sum over all $ \theta \in \Lambda_{G,P}^{\text{pos}} $ and $ \mathfrak{P}(\theta) $, with the given pushforward structure.
• The sheaf $ \mathcal{R}_{\mathfrak{P}(\theta)}(\operatorname{Im} e)[2|\mathfrak{P}(\theta)|] $ on $ X^{\mathfrak{P}(\theta)} $ is self-dual, reflecting the self-duality of $ (\operatorname{Im} e)[1] $ in the category of finite-dimensional vector spaces.
• The construction is proven by induction over strata, starting from the dense open stratum, with the normalization map $ j_{P,\mathfrak{P}(\theta)} $ providing a resolution of singularities for the strata $ {}^{\mathfrak{P}(\theta)}\overline{\operatorname{Bun}}_P $.
• The result establishes a precise link between the geometry of $ \widetilde{\operatorname{Bun}}_P $ and the representation theory of the quantum group at a root of unity, via the IC sheaf's decomposition.