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Type A algebraic coherence conjecture of Pappas and Rapoport

Evgeny Feigin, an appendix in collaboration with Andrey Karenskih

TL;DR

The paper constructs an explicit algebraic degeneration from a collection of cyclic $ rak{sl}_n[z]$-modules to a non-cyclic Iwahori-module ${f D}(0)$, whose Cartan component mirrors the tensor product Cartan. In type A, when weights are multiples of a fixed level-one weight, ${f D}(0)$ decomposes as a sum of affine Demazure modules, thereby providing an algebraic realization of the Pappas-Rapoport coherence conjecture as established by Zhu. The construction relies on a one-parameter family of Lie algebras ${ rak a}(oldsymbol{ u})$, an Inönü–Wigner contraction, and Kostant–Kumar modules to describe the degeneration, connecting to global affine Grassmannians and Kostant–Kumar Schubert varieties. The work yields explicit low-rank examples that both corroborate the approach and expose its boundaries, offering a deeper understanding of how degenerations encode Demazure–Kostant–Kumar structures and their geometric counterparts.

Abstract

The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles on spherical Schubert varieties in the affine Grassmannians and to the line bundles on unions of Schubert varieties in affine flag varieties. Algebraically the claim can be reformulated as an equality between dimensions of certain Demazure modules and certain sums of Demazure modules. The goal of this paper is to formulate an algebraic construction providing an explicit link between the above mentioned Demazure modules. Our construction works only in type A, but it is applicable to a much wider class of representations than whose popping up in the geometric coherence conjecture. In the general case one side of conjectural equality involves the affine Kostant-Kumar modules.

Type A algebraic coherence conjecture of Pappas and Rapoport

TL;DR

The paper constructs an explicit algebraic degeneration from a collection of cyclic -modules to a non-cyclic Iwahori-module , whose Cartan component mirrors the tensor product Cartan. In type A, when weights are multiples of a fixed level-one weight, decomposes as a sum of affine Demazure modules, thereby providing an algebraic realization of the Pappas-Rapoport coherence conjecture as established by Zhu. The construction relies on a one-parameter family of Lie algebras , an Inönü–Wigner contraction, and Kostant–Kumar modules to describe the degeneration, connecting to global affine Grassmannians and Kostant–Kumar Schubert varieties. The work yields explicit low-rank examples that both corroborate the approach and expose its boundaries, offering a deeper understanding of how degenerations encode Demazure–Kostant–Kumar structures and their geometric counterparts.

Abstract

The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles on spherical Schubert varieties in the affine Grassmannians and to the line bundles on unions of Schubert varieties in affine flag varieties. Algebraically the claim can be reformulated as an equality between dimensions of certain Demazure modules and certain sums of Demazure modules. The goal of this paper is to formulate an algebraic construction providing an explicit link between the above mentioned Demazure modules. Our construction works only in type A, but it is applicable to a much wider class of representations than whose popping up in the geometric coherence conjecture. In the general case one side of conjectural equality involves the affine Kostant-Kumar modules.
Paper Structure (27 sections, 18 theorems, 48 equations, 1 algorithm)

This paper contains 27 sections, 18 theorems, 48 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite-dimensional algebras
  4. Infinite-dimensional algebras
  5. The coherence conjecture
  6. Global affine Grassmannians in type A
  7. Lattices
  8. Global affine Grassmannian
  9. The construction
  10. Finite-dimensional representations
  11. Lie algebras
  12. Fundamental representations
  13. Family of modules
  14. Algebraic coherence conjecture
  15. Embeddings
  16. ...and 12 more sections

Key Result

Theorem 1

If all $\lambda^{(b)}$ are multiples of one fundamental weight $\omega_j$, $\lambda^{(b)}=k_b\omega _j$, then $V_{\bar{\lambda}}$ is isomorphic to a sum of affine Demazure modules inside $L(\sum_{b} k_b \Lambda_b)$.

Theorems & Definitions (56)

  • Theorem 1
  • Conjecture 2
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 46 more