Pull-back and push-forward functors for holonomic modules over Cherednik algebras

Abstract

In this article we continue the study of holonomic modules over sheaves of Cherednik algebras, initiated by the third author in [Tho18]. Working with arbitrary parameters, we first develop a theory of $b$-functions to prove that push-forward along open embeddings preserves holonomicity. This implies that pull-back along closed embeddings also preserves holonomicity. We use these facts to show that both push-forward and pull-back under any melys morphism preserves holonomicity. Since duality preserves holonomicity, we deduce that extraordinary push-forward and extraordinary pull-back also exist for holonomic modules. As a consequence, we give a general classification of irreducible holonomic modules similar to the classification of irreducible holonomic $\mathscr{D}$-modules as minimal extensions of integrable connections on locally closed subsets. Finally, we prove that ext-groups between holonomic modules are finite-dimensional and explore applications of our work to the classification of aspherical parameters and existence of finite-dimensional modules for sheaves of Cherednik algebras.

Theorems & Definitions (194)

• Proposition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Proposition 1.10