p-adic Cherednik algebras on rigid analytic spaces

Let $X$ be a smooth rigid space with an action of a finite group $G$ satisfying that $X/G$ is represented by a rigid space. We construct sheaves of $p$-adic Cherednik algebras on the small étale site of the quotient $X/G$, and study some of their properties. The sheaves of $p$-adic Cherednik algebras are sheaves of Fréchet-Stein $K$-algebras on $X/G$, which can be regarded as $p$-adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof.

💡 Research Summary

The paper develops a theory of p‑adic Cherednik algebras attached to the action of a finite group (G) on a smooth rigid analytic space (X) over a complete discretely valued field (K) of characteristic 0. The author first constructs sheaves of twisted differential operators (\mathcal D_{\omega}(X)) on (X) and extends the classical construction of rational Cherednik algebras to the rigid‑analytic setting. For a unit (t\in K^{\times}), a (G)-invariant de Rham class (\omega\in H^2_{\mathrm{dR}}(X)^G), and a reflection function (c\in\operatorname{Ref}(X,G)) (assigning a scalar to each reflection hypersurface), a sheaf (\mathcal H_{t,c,\omega}(X,G)) of filtered associative algebras is defined on the small étale site of the quotient (X/G). The construction mimics Etingof’s algebraic Cherednik algebras but requires careful analytic justification: the Dunkl–Opdam operators are expressed using rigid‑analytic functions defining the reflection hypersurfaces, and the resulting algebra sits inside the skew‑group algebra (G\ltimes\mathcal D_{\omega}(X_{\mathrm{reg}})).

A key structural result is a PBW theorem for (\mathcal H_{t,c,\omega}(X,G)): the associated graded algebra with respect to the natural Dunkl–Opdam filtration is canonically isomorphic to the skew‑group algebra (G\ltimes\operatorname{Sym}_K(\mathcal T_X\oplus\mathcal T_X^{\vee})). Consequently, the algebra admits a triangular decomposition analogous to the classical rational Cherednik algebra, which underlies its representation‑theoretic richness.

The second major part of the work introduces the p‑adic Cherednik algebra (\widehat{\mathcal H}{t,c,\omega}(X,G)) as a Fréchet completion of (\mathcal H{t,c,\omega}(X,G)). By endowing (\mathcal H_{t,c,\omega}(X,G)) with a countable family of Banach norms (|\cdot|n) (derived from the natural norms on the sheaf of infinite‑order differential operators (\widehat{\mathcal D}X) of Ardakov–Wadsley), one obtains a projective system of Banach algebras (\mathcal H{t,c,\omega}^{(n)}). The author proves that this system satisfies the axioms of a Fréchet‑Stein algebra: each transition map (\mathcal H{t,c,\omega}^{(n+1)}\to\mathcal H_{t,c,\omega}^{(n)}) is flat and the algebras are Noetherian. This structure allows the import of the co‑admissible module theory developed for (\widehat{\mathcal D}X); in particular, co‑admissible (\widehat{\mathcal H}{t,c,\omega}(X,G))-modules form an abelian category with good homological properties.

The paper also establishes the (c)-flatness of transition maps on the étale site. For an étale cover ({U_i\to X/G}), the restriction maps (\widehat{\mathcal H}{t,c,\omega}(U_i)\to\widehat{\mathcal H}{t,c,\omega}(U_i\cap U_j)) are shown to be flat and strictly continuous, ensuring that the sheaf of Fréchet‑Stein algebras glues correctly. This result is essential for defining a sheaf of p‑adic Cherednik algebras on the whole quotient space and for developing a localization theory for modules.

Finally, the author develops a localization functor for co‑admissible modules: given a co‑admissible (\widehat{\mathcal H}_{t,c,\omega}(X/G))-module (M), its sheafified version on the étale site is defined by \