Type A algebraic coherence conjecture of Pappas and Rapoport

1. Problem Statement

The paper addresses the Pappas-Rapoport Coherence Conjecture, a theorem proved by Xinwen Zhu. Geometrically, the conjecture states that the dimensions of spaces of sections of certain line bundles coincide in two distinct settings:

1. General Fiber: Sections of line bundles on spherical Schubert varieties ($X_\lambda$) within the affine Grassmannian ($Gr$).
2. Special Fiber: Sections of line bundles on unions of Schubert varieties ($Y_w$) within the affine flag variety ($Fl$).

Algebraically, this translates to an equality between:

• The dimension of a specific Demazure module (cyclic representation of the current algebra $\mathfrak{sl}_n[z]$).
• The dimension of a sum of Demazure modules (representations of the Iwahori algebra $\mathfrak{I}$).

While Zhu's proof is geometric (using global affine Grassmannians and degeneration), this paper aims to provide an explicit algebraic construction that links these two sides directly, specifically in Type A ($\mathfrak{sl}_n$). Furthermore, the authors seek to generalize this construction beyond the specific cases appearing in the geometric conjecture to a wider class of representations, including Kostant-Kumar modules.

2. Methodology

The authors introduce a deformation procedure based on quiver representations and Inonu-Wigner contractions.

A. The Deformation Setup

• Input: A collection of $n$ cyclic representations $D_b$ of the current algebra $\mathfrak{sl}_n[z]$ (or $\mathfrak{sl}_n(O)$), indexed by vertices of a cyclic quiver $\Delta$ (where vertices are $b \in \mathbb{Z}/n\mathbb{Z}$).
• Tensor Product: Consider the tensor product $\bigotimes_b D_b$. Inside this, identify the Cartan component (the submodule generated by the tensor product of cyclic vectors).
• Deformation Parameter ($\varepsilon$): The authors define a family of Lie algebras $\mathfrak{a}(\varepsilon)$ acting on the tensor product.
  • For $\varepsilon \neq 0$, $\mathfrak{a}(\varepsilon) \cong \mathfrak{gl}_n(O)$ (the current algebra).
  • For $\varepsilon = 0$, $\mathfrak{a}(0)$ degenerates to a specific quotient of the Iwahori algebra $\mathfrak{I}$.
• The Construction:
  1. Define a subspace $D(\varepsilon)$ generated by the action of $\mathfrak{a}(\varepsilon)$ on the tensor product of cyclic vectors.
  2. Take the limit as $\varepsilon \to 0$ to obtain $D(0)$. This is defined via a filtration on the space of polynomials in $\varepsilon$, effectively taking the "leading term" of the action.
  3. $D(0)$ is a representation of the Iwahori algebra $\mathfrak{I}$.

B. Admissibility and Extremal Vectors

• The construction requires the input modules $D_b$ to be $b$-admissible, meaning they can be twisted by specific endomorphisms of the Lie algebra without changing their isomorphism class.
• The authors prove that affine Demazure modules satisfy this admissibility condition.
• Key Conjecture (Conjecture B/6.14): The resulting Iwahori module $D(0)$ is generated not just by the tensor product of cyclic vectors, but by the action of $\mathfrak{I}$ on the tensor products of extremal weight vectors (vectors of weight $\sigma(\lambda)$ for all $\sigma$ in the Weyl group).

3. Key Contributions and Results

A. Theorem A (The Algebraic Link)

If the input weights are multiples of a single fundamental weight (the geometric coherence case), the constructed module $D(0)$ is isomorphic to a direct sum of affine Demazure modules inside an integrable highest weight representation of the affine Kac-Moody algebra.

• This provides an explicit algebraic realization of the geometric degeneration from the affine Grassmannian to the affine flag variety.

B. Generalization to Kostant-Kumar Modules

The paper extends the construction to arbitrary collections of weights (not just multiples of one fundamental weight).

• Result: In the general case, the module $D(0)$ is shown to be isomorphic to a sum of affine Kostant-Kumar modules.
• Significance: This connects the coherence conjecture to a broader class of representations that appear in the study of Schubert varieties in products of flag varieties.

C. Verification and Counter-Examples

• Conjecture Verification: Using a computer program (Appendix A), the authors verified Conjecture 6.14 (that $D(0)$ is generated by extremal vectors) for numerous low-rank examples.
• Non-Examples: In Section 8.4, the authors provide a counter-example where the input is a Demazure module, but the resulting Iwahori module generated only by extremal vectors is strictly smaller than the full degeneration $D(0)$. This highlights that the "algebraic coherence" (generation by extremal vectors) holds for specific classes (like multiples of fundamental weights) but fails in the most general setting without additional constraints.

D. Geometric Interpretation

The authors propose a geometric counterpart to their algebraic construction:

• The degeneration of the tensor product of finite-dimensional modules corresponds to the degeneration of the flag variety into a union of Kostant-Kumar Schubert varieties.
• They expect $D(0)$ to be realized as the space of sections of line bundles on these specific subvarieties of quiver flag varieties.

4. Significance

1. Algebraic Proof of Coherence: It offers a purely representation-theoretic proof of the Pappas-Rapoport coherence conjecture in Type A, bypassing the need for global geometric arguments in the final algebraic formulation.
2. New Class of Modules: It introduces and characterizes a new family of Iwahori modules constructed via quiver degenerations, linking them to Kostant-Kumar modules.
3. Quiver Connections: The work bridges the gap between affine Lie algebra representations and the representation theory of cyclic quivers (specifically the endomorphism algebras of equioriented quivers).
4. Computational Framework: The inclusion of a computer program allows for the systematic testing of these conjectures in higher ranks, providing a tool for future research in affine representation theory.

Summary of Structure

• Sections 2-4: Preliminaries on Lie algebras, affine Grassmannians, and the lattice formalism for Type A.
• Section 5: The core construction of the degeneration $D(\varepsilon) \to D(0)$ and the definition of the algebra $\mathfrak{a}(\varepsilon)$.
• Section 6: Application to finite-dimensional representations; proof of the coherence conjecture for multiples of fundamental weights; formulation of the generation conjecture.
• Section 7: Generalization to affine Demazure modules and the identification with Kostant-Kumar modules.
• Section 8: Explicit low-rank examples (Rank 1 and 2) illustrating the construction, including cases where the conjecture holds and where it fails.
• Appendix A: Description of the Python algorithm used to verify dimensions and linear independence of generated subspaces.