Category O for p-adic rational Cherednik algebras

We introduce the concept of a triangular decomposition for Banach and Fréchet-Stein algebras over $p$-adic fields, which allows us to define a category $\mathcal{O}$ for a wide array of topological algebras. In particular, we apply this concept to $p$-adic rational Cherednik algebras, which allows us to obtain an analytic version of the category $\mathcal{O}$ developed by Ginzburg, Guay, Opdam and Rouquier. Along the way, we study the global sections of $p$-adic Cherednik algebras on smooth Stein spaces, and determine their behavior with respect to the rigid analytic GAGA functor.

💡 Research Summary

The paper develops an analytic analogue of the highest‑weight category O for p‑adic rational Cherednik algebras by first establishing a general theory of triangular decompositions for Banach and Fréchet‑Stein algebras over a non‑archimedean field K.

In Section 2 the author recalls the classical definition of a triangular decomposition (A ⊗ H ⊗ B) for graded algebras and introduces the necessary functional‑analytic tools to handle weight‑space decompositions of bounded operators on Banach spaces. Building on the work of C.T. Féaux de Lacroix, a notion of “weight‑space decomposition” is defined for Banach modules, allowing one to speak of generalized eigenspaces even when the operator is not diagonalizable.

Section 3 defines a Banach algebra R to have a triangular decomposition if it contains a dense graded subalgebra R₀ that admits such a decomposition with subalgebras A, B, H satisfying the usual compatibility conditions (commutation with H, finite‑dimensional graded pieces, H semisimple, inner grading via an element ∂∈R₀⁰). The category “O” consists of R‑modules that are finitely generated over A and admit a ∂‑weight decomposition. The main result (Theorem A) shows that “O” is an abelian, highest‑weight subcategory of Mod(R); every simple object is the unique simple quotient of an analytic Verma module Δ(W)=A⊗ₖW for an irreducible H‑module W, and the simple objects are in bijection with Irr(H).

The theory is then lifted to Fréchet‑Stein algebras R=lim← Rₙ, where each Rₙ is a Noetherian Banach algebra with a compatible triangular decomposition. For each level one obtains a category Oₙ, and the transition functors Rₙ₊₁⊗_{Rₙ}– give equivalences of highest‑weight categories. The inverse limit Ō=lim← Oₙ is defined (Theorem B) and shown to be a full abelian subcategory of the co‑admissible module category C(R), closed under closed submodules and finite direct sums, and again a highest‑weight category with simples indexed by Irr(H).

Having built the abstract framework, the paper applies it to p‑adic rational Cherednik algebras. Let h be a finite‑dimensional K‑vector space and G⊂GL(h) a finite group generated by pseudo‑reflections. The classical rational Cherednik algebra H_c(h,G) is known to possess a triangular decomposition. The author considers the rigid‑analytic space h^{an} (the analytification of h) and defines the sheaf of p‑adic Cherednik algebras 𝓗_c on the quotient h^{an}/G. Its global sections H_c(h^{an},G)=Γ(h^{an}/G,𝓗_c) are the p‑adic rational Cherednik algebra of interest.

Section 4 proves that H_c(h^{an},G) is a Fréchet‑Stein algebra and even a nuclear Fréchet space (Theorem C). Moreover, every co‑admissible module over it is nuclear. The author then studies the natural map from the algebraic Cherednik algebra H_c(h,G) to its analytic counterpart. Using a rigid‑analytic GAGA functor, it is shown (Theorem D) that this map is faithfully flat, its image is dense, and H_c(h^{an},G) is precisely the Arens‑Michael envelope of H_c(h,G). Consequently, any Banach‑space representation of the algebraic Cherednik algebra lifts uniquely to a continuous representation of the analytic one.

Section 5 constructs an explicit Fréchet‑Stein presentation H_c(h^{an},G)=lim← H_c^{(m)} and verifies that the triangular decomposition persists at the Fréchet‑Stein level. Hence a category Ō_c⊂C(H_c(h^{an},G)) is defined (Theorem E). This category is abelian, closed under closed submodules and finite direct sums, and is a highest‑weight category whose simple objects correspond bijectively to the irreducible K‑linear representations of G. The canonical map H_c(h,G)→H_c(h^{an},G) induces an equivalence of highest‑weight categories H_c(h,G)⊗_{H_c(h,G)}– : O_c ↔ Ō_c.

The paper concludes by outlining potential applications: the analytic nature of H_c(h^{an},G) opens the door to p‑adic analytic techniques (e.g., locally analytic representation theory, p‑adic Langlands program) in the study of rational Cherednik algebras. The author also sketches how the integral model H_c(h_R,G) (with R the Witt vectors of a finite field of characteristic p) could be used to lift modular representations to characteristic 0 via the p‑adic analytic algebra, suggesting connections with deformation theory, D‑module theory in positive characteristic, and Weyl modules. Finally, a broader vision is presented: extending the construction to more general p‑adic Lie groups, possibly linking Cherednik‑type algebras with Ardakov‑Wadsley’s sheaves of completed p‑adic differential operators and the Beilinson‑Bernstein localization in the non‑archimedean setting.

Overall, the work provides a robust analytic framework for highest‑weight categories attached to p‑adic rational Cherednik algebras, bridging algebraic representation theory with p‑adic functional analysis and opening several avenues for future research.