Symplectic duality for the constant term of the geometric Eisenstein series

Technical Summary: Symplectic Duality for the Constant Term of the Geometric Eisenstein Series

Problem Statement
This paper investigates the cohomology of a specific moduli stack of quasimaps, denoted $QM_{\vec{N}}^{\vec{M}}$, which categorifies the constant term of the geometric Eisenstein series for the mirabolic parabolic subgroup of $GL_{n+1}$ over the function field $\mathbb{F}_q(C)$ of a smooth projective curve $C$. The central problem is to understand the structure of this cohomology, which carries an action of a Hecke-type modification algebra, and to identify it with a more tractable geometric object. Specifically, the author seeks to establish a "symplectic duality" (or 3d-mirror symmetry) between the Higgs branch description (the quasimap stack) and a Coulomb branch description (a singular surface and its resolution).

Methodology
The author employs a combination of geometric representation theory, derived algebraic geometry, and equivariant cohomology techniques. The methodology proceeds through the following steps:

1. Derived Stack Construction: The moduli stack of quasimaps $QM_{\vec{N}}^{\vec{M}}$ is defined as a derived stack. Its tangent complex is computed explicitly, establishing that the stack is quasi-smooth. This allows for the definition of a virtual tangent space and a virtual dimension.
2. Algebra of Correspondences: A modification correspondence $\iota$ is introduced, which replaces a line bundle $L$ with $L(-x)$ at a point $x \in C$. Pullback and pushforward operations along this correspondence generate an algebra $A_\chi$ of operators acting on the cohomology (with coefficients in a local system $\chi$). The relations in this algebra (commutators and Casimir identities) are computed in terms of tautological Chern classes.
3. Symplectic Dual Variety: The author constructs the "Coulomb branch" $X^\vee_0 = \text{Spec}(\mathbb{Q}_l[x, y, \eta]/(xy - \eta^{n+1}))$, which is an $A_n$-surface singularity. This surface is identified as the symplectic dual to the Higgs branch $X = T^*\mathbb{P}^n$.
4. Resolution and Fixed Loci: A minimal resolution $\rho: X^\vee \to X^\vee_0$ is constructed by gluing affine charts $U_k \cong \mathbb{A}^2$. The author utilizes a Białynicki-Birula decomposition of the quasimap stack induced by a torus action. This decomposition stratifies the stack into attracting orbits $Attr(k)$, whose fixed loci are symmetric products of the curve.
5. Comparison via Filtration: The core of the proof involves comparing two filtrations:
   • On the quasimap side: A filtration induced by the Białynicki-Birula decomposition.
   • On the dual side: A filtration of the local cohomology of a specific vector bundle on $X^\vee$ supported on the attracting Lagrangian subvariety $L \subset X^\vee$.
     The author proves that the associated graded pieces of these filtrations are isomorphic as modules over the respective algebras. The extension problem (whether the filtrations split) is then analyzed to determine the full isomorphism.

Key Contributions and Results

• Identification of the Algebra: The subalgebra of the correspondence algebra generated by degree-two operators is identified with the ring of regular functions on the $A_n$-surface singularity. The degree-one operators form a spin module (a Clifford algebra) over this surface, and degree-zero operators act as first-order differential operators.
• The Trivial Character Case (Theorem 1): For the trivial local system, the paper establishes a torus-equivariant isomorphism between the shifted cohomology of the quasimap stack and the local cohomology of a twisted sheaf on the resolution $X^\vee$:
  $$\hat{H}^*_c(QM_{\vec{N}}^{\vec{M}}) \cong H^*_L(X^\vee, \hat{\mathcal{O}}_{vir} \otimes \mathcal{O}(\vec{M}))$$
  Here, $\hat{\mathcal{O}}_{vir}$ is a "symmetrised virtual structure sheaf" (a spin bundle twisted by the square root of the canonical bundle), and $\mathcal{O}(\vec{M})$ is a line bundle determined by the degrees of the input bundles.
• The Nontrivial Character Case (Theorem 2): For a nontrivial rank-one local system $\chi$, the isomorphism holds if and only if certain "obstruction" conditions are not met. Specifically, if $\chi^{n+1}_0 \neq 1$, the isomorphism fails precisely when there exist indices $k < s$ such that the bundles $M_k$ and $M_s$ are isomorphic, and a specific Euler characteristic condition involving the canonical bundle $K_C$ is satisfied. In these exceptional cases, the filtration on the cohomology does not split, reflecting the non-reduced structure of the fixed locus on the singular side versus the reduced fixed locus on the resolved side.
• Explicit Computation of Local Cohomology: The paper explicitly computes the local cohomology of the resolution $X^\vee$ with support in the attracting set $L$, showing it is generated by specific cyclic vectors with well-defined Frobenius and torus weights.

Significance and Claims
The paper claims to provide a geometric realization of the constant term of the geometric Eisenstein series in terms of the local cohomology of a resolution of a singular surface. This result generalizes previous work on $GL_2$ (specifically the analysis in [KO23]) to $GL_{n+1}$.

The significance lies in:

1. Categorification: It categorifies the constant term of the Eisenstein series, lifting it from a trace of Frobenius to a full cohomological object with an algebra action.
2. Symplectic Duality: It explicitly constructs the symplectic dual (Coulomb branch) for this specific geometric Eisenstein series setup, identifying the $A_n$-singularity as the dual to the Higgs branch $T^*\mathbb{P}^n$.
3. Obstruction Analysis: The paper provides a precise characterization of when the duality is an isomorphism and when it fails due to non-splitting extensions, linking these failures to specific geometric conditions on the curve and the bundles (involving the genus $g$ and degrees of the bundles).

The author notes that the proof relies on the purity of the cohomology of the attracting orbits, which follows from their smoothness, and utilizes equivariant localization techniques to match the weights and structures on both sides of the duality. The work is presented as a step towards understanding the geometric Langlands program and the structure of automorphic forms via geometric methods.