Abstract
In this twopart paper, we introduce a \(p\)adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when \(q\) is not a root of unity and \(\vert q1\vert<1\). We then define a category of \(\lambda\)twisted \(D\)modules on this analytic quantum flag variety. We show that when \(\lambda\) is regular and dominant and when the characteristic of the residue field does not divide the order of the Weyl group, the global section functor gives an equivalence of categories between the coherent \(\lambda\)twisted \(D\)modules and the category of finitely generated modules over \(\widehat{U_q^\lambda}\), where the latter is a completion of the adfinite part of the quantum group with central character corresponding to \(\lambda\). Along the way, we also show that Banach comodules over the Banach completion \( \widehat{ \mathcal{O}_q(B) } \) of the quantum coordinate algebra of the Borel can be naturally identified with certain topologically integrable modules.
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Notes
 1.
We note that there exists a different approach to quantum \(D\)modules and BeilinsonBernstein by Tanisaki [49].
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Acknowledgments
Most of the material in this twopart paper formed a significant part of the author’s PhD thesis, which was being produced under the supervision of Simon Wadsley. We are very grateful to him for his continued support and encouragement throughout this research, without which writing this paper would not have been possible. We would also like to thank him for communicating privately a proof to us which inspired our arguments in Section 4. We are also thankful to Andreas Bode for his continued interest in our work. Finally, we wish to thank Kobi Kremnizer for a useful conversation on quantum groups and proj categories. The author’s PhD was funded by EPSRC.
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A. Appendix A. Hopf duals of $$R$$ Hopf algebras and their comodules
We wish to establish some duality facts to do with Hopf algebras over \(R\) which are well known when working over fields but for which we couldn’t find references for Hopf algebras over more general commutative rings. Most of our proofs work using the usual arguments but one has to be a bit careful when dealing with torsion.
A.1. Hopf Duals Over \(R\)
For the entirety of this Section, \(H\) will denote a fixed Hopf \(R\)algebra. For our purposes, it will be enough to work in the case where \(H\) is torsionfree. First we wish to define a notion of Hopf dual. Since \(H\) has no torsion, it embeds as a subHopf algebra of \(H_L=H\otimes_R L\). We will define the Hopf dual to be a subHopf algebra of \((H_L)^\circ\). Let \(\mathcal{J}\) denote the set of ideal \(I\) in \(H\) such that \(H/I\) is a finitely generated \(R\)module. Moreover, denote by \(\mathcal{J}'\) the set of ideals \(I\) in \(H\) such that \(H/I\) is free of finite rank. Finally, write \(H^*\) for \( \text{Hom} _R(H, R)\). Note that \(H^*\) is always torsionfree since \(R\) is a domain: if \(\pi f=0\) then \(\pi f(u)=0\) for all \(u\in H\) and so \(f(u)=0\) for all \(u\).
Definition A.1.
We define the Hopf dual of \(H\) to be
If \(n\geq 0\) and \(x\in H\) we have for any \(f\in H^*\) that \(f(x)=0\) if and only if \(f(\pi^n x)=0\). Thus if \(0\neq f\in H^\circ\) then \(f_I=0\) for some \(I\in\mathcal{J}\) where \(H/I\) is not torsion. Moreover we then have \(f_{I_L\cap H}=0\) and so by replacing \(I\) with \(I_L\cap H\) we may in addition assume that \(H/I\) is torsionfree. Since \(R\) is a PID this shows that
Lemma A.2.
\(H^\circ\) is an subHopf \(R\) algebra of \(H_L^\circ\) . In particular the algebra maps on \(H^\circ\) are just the dual maps of the coalgebra maps on \(H\) and viceversa.
Remark A.3.
Some of the above arguments were implicit in Lusztig’s work, see [34, 7.1].
A.2. \(H^\circ\)Comodules as \(H\)Modules
We now wish to establish some correspondence between comodules over \(H^\circ\) and certain \(H\)modules. We call an \(H\)module \(M\) locally finite if for all \(m\in M\), \(Hm\) is finitely generated over \(R\).
Proposition A.4.
Every \(H^\circ\) comodule has a canonical structure of a locally finite \(H\) module with respect to which every comodule homomorphism is an \(H\) modules homomorphism. In other words there is a canonical faithful embedding of categories between the category of \(H^\circ\) comodules and the category of locally finite \(H\) modules.
Proof.
This is just the usual argument. If \(M\) is an \(H^\circ\)comodule with coaction \(\rho:M\to M\otimes_R H^\circ\), write \(\rho(m)=\sum m_1\otimes m_2\). Then we set
Next, we want to show that the functor we just defined is full, i.e that every \(H\)module map between two \(H^\circ\)comodules is a comodule homomorphism. We first need a technical result. Suppose \(M\) is a locally finite \(H\)module. Note that we have an \(R\)module injection \(\phi_M:M\to \text{Hom} _R(H,M)\) given by \(\phi_M(m)(u)=um\) for all \(u\in H\) and \(m\in M\). Moreover we have a map
Lemma A.5.
Let \(A\) and \(B\) be \(R\) modules, \(A^*= \text{Hom} _R(A,R)\) and suppose \(C\) is any \(R\) submodule of \(A^*\) such that \(A^*/C\) has no \(\pi\) torsion. Let \(M\) be any \(R\) module and set
Proof.
Suppose that \(0\neq u=\sum_{i=1}^s g_i\otimes f_i\in \text{Hom} _R(B,M)\otimes_R C\). The \(R\)submodule \(N\) of \( \text{Hom} _R(B,M)\) generated by the \(g_i\) is finitely generated, so since \(R\) is a PID we can pick a generating set \(n_1,\ldots, n_l, t_1,\ldots, t_m\) for \(N\) such that \(n_1,\ldots, n_l\) are torsionfree while \(t_1,\ldots, t_m\) are \(\pi\)torsion, and
Now, to show that \(\theta_{M,C}(u)\neq 0\) it suffices to show that the restriction of \(\theta_{M,C}\) to the span of the \(n_i\otimes f_j\) and \(t_k\otimes f_j\) is injective. So suppose we are given
Therefore we have
Corollary A.6.

1.
The map \(\theta_M: M\otimes_R H^\circ \to \text{Hom} _R(H,M)\) is injective.

2.
The map \(M\otimes_R H^\circ\otimes_R H^\circ\to \text{Hom} _R(H\otimes_R H,M)\) sending \(m\otimes f\otimes g\) to \(x\otimes y\mapsto f(x)g(y)m\) is injective.
Proof.
Let \(A=H\) and \(C=H^\circ\). From the definition of \(H^\circ\) it follows that \(A^*/C\) is torsionfree. Then (i) follows immediately from the Lemma by putting \(B=R\). For (ii) note that this map is simply the composite
We can now deduce the result we were aiming for.
Proposition A.7.
The functor associating any \(H^\circ\) comodule to the corresponding \(H\) module is a fully faithful embedding.
Proof.
From what we have done already we just need to show that any \(H\)module map \(f:M\to N\) between two \(H^\circ\)comodules is a comodule homomorphism. Write \(\rho_M\) and \(\rho_N\) for the coactions on \(M\) and \(N\) respectively, and pick \(m\in M\) and \(u\in H\). Then we know that \(um=\sum m_2(u)m_1\) and we have \(uf(m)=\sum m_2(u)f(m_1)\) since \(f\) is a module homomorphism. On the other hand by definition of the action on \(N\) we have \(uf(m)=\sum f(m)_2(u)f(m)_1\). To show that \(f\) is a comodule map we need to show that
Write \(\tilde{\rho}_1=\rho_N\circ f\) and \(\tilde{\rho}_2=(f\otimes 1)\circ\rho_M\). Moreover recall the map \(\phi: M\to \text{Hom} _R(H,M)\) given by \(\phi(m)(u)=um\). Then let
From now on, if \(M\) is a locally finite \(H\)module we will say that it is an \(H^\circ\)comodule to mean that its \(H\)module structure arises from an \(H^\circ\)comodule structure.
In order for the above functor to be an isomorphism of categories we therefore just need to show that it is surjective. This may not be true in general, however we can write a very simple necessary and sufficient condition for an isomorphism of categories to hold. Suppose \(M\) is a locally finite \(H\)module and let \(\phi_M: M\to \text{Hom} _R(H,M)\) be given by \(\phi_M(m)(x)=x\cdot m\). We have the map \(\theta_M:M\otimes_R H^\circ\) as before.
Proposition A.8.
A locally finite \(H\) module \(M\) is an \(H^\circ\) comodule if and only if \(\phi_M(m)\) belongs to the image of \(\theta_M\) for all \(m\in M\) .
Proof.
If \(M\) is a comodule with coaction \(\rho\), then by our observation preceding Lemma A.5 we have \(\phi_M=\theta_M\circ\rho\) where \(\phi_M\) comes from the induced \(H\)module structure, and the result is clear. Conversely assume \(\phi_M(m)\) belongs to the image of \(\theta_M\) for all \(m\in M\). Fix \(m\in M\). Then there exists \(m_1, \ldots, m_n\in M\) and \(f_1, \ldots, f_n\in H^\circ\) such that for all \(x\in H\), \(x\cdot m=\sum_{i=1}^n f_i(x)m_i\) and we define
Since Lemma A.5 was quite general, the same argument as in the above proof shows the following
Lemma A.9.
Suppose \(M\) is a locally finite \(H\) module and let \(C\) be a subcoalgebra of \(H^\circ\) such that \(H^*/C\) is torsionfree. If \(\phi_M(m)\) belongs to the image of \(\theta_{M,C}\) for all \(m\in M\) then \(M\) is a \(C\) comodule.
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Dupré, N. A BeilinsonBernstein Theorem for Analytic Quantum Groups. I. PAdic Num Ultrametr Anal Appl 13, 44–82 (2021). https://doi.org/10.1134/S2070046621010027
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Keywords
 quantum groups
 noncommutative geometry
 \(p\)adic representation theory