A Beilinson-Bernstein Theorem for analytic quantum groups
We introduce a -adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when is not a root of unity. We then define a category of -twisted -modules on this analytic quantum flag variety. This category has a distinguished object which plays the role of the sheaf of -twisted differential operators. We show that when is regular and dominant, the global section functor gives an equivalence of categories between the coherent -twisted -modules and the finitely presented modules over the global sections of . Along the way, we also show that Banach comodules over the Banach completion of the quantum coordinate algebra of the Borel can be naturally identified with certain topologically integrable modules.
1.1. Background and motivation
Let be a complete discrete valuation field of mixed characteristic , with discrete valuation ring , uniformizer and residue field . We fix an element and assume that and that is not a root of unity. Ardakov and Wadsley have recently started an ongoing program aiming to develop -adic analytic analogues of -modules in order to understand -adic representation theory, see [4, 5, 6, 3]. Their aim is to use -adic analytic localisation results analogous to the classical theorem of Beilinson-Bernstein  in order to better understand locally analytic representations of -adic groups, which were introduced by Schneider and Teitelbaum in a series of papers including [45, 43, 44]. There have also been other approaches at using localisation techniques to understand locally analytic representations, notably by Schmidt  and Patel, Schmidt and Strauch [37, 38, 39].
Let us briefly recall one of Ardakov and Wadsley’s main results. Let be a simply connected split semisimple algebraic group over with -Lie algebra and let be its flag scheme . In , they defined a family of Banach completions of the envloping algebra of the -Lie algebra . Moreover, for a weight , they introduced a family of sheaves of completed deformed twisted crystalline differential operators on . Their theorem then states:
For any and for regular and dominant, the global section functor gives an equivalence of categories between coherent sheaves of -modules and finitely generated -modules with central character corresponding to .
Our aim is to prove an analogue of the above Theorem when working with quantum groups, where for simplicity we only treat the case in this paper. The study of quantum groups in a -adic analytic setting was first proposed by Soibelman in , where he introduced quantum deformations of the algebras of locally analytic functions on -adic Lie groups and of the corresponding distribution algebra. His ideas were also heavily influenced by the aforementioned work of Schneider and Teitelbaum. This paper of Soibelman then inspired a short note of Lyubinin  and also a different approach for GL in . Recently, there has also been a new approach at constructing -adic analytic quantum groups using Nichols algebra in . However, besides these, not much work has been done in this area. In , we constructed quantum analogues of the Arens-Michael envelope of and of the algebra of rigid analytic functions on the analytification of , and proved that these were Fréchet-Stein algebras. We also constructed several Banach completions of those algebras, and some of these objects feature in this paper. Our hope is that more work will be done to pursue these efforts. The theory of quantum groups has strong links with the representation theory of algebraic groups in positive characteristic. We expect that a successful theory of -adic analytic quantum groups would have similar links with the representation theory of -adic groups, and we view our work as a first effort towards developing such a theory.
Recently, there has also been some work hinting at noncommutative analogues of rigid analytic geometry in . In this light, we think that defining noncommutative analogues of analytic flag varieties as we do in this paper is interesting in its own right. It would be interesting to compare our constructions with their general framework.
1.2. Quantum flag varieties and quantum -modules
The proof of Theorem 1.1 relied on the classical Beilinson-Bernstein theorem, and similarly we will use a quantum group analogue of it due to Backelin and Kremnizer 111We note that there exists a different approach to quantum -modules and Beilinson-Bernstein by Tanisaki .. We briefly recall their constructions. Let be the quantized enveloping algebra of . Let be the quantized coordinate algebra of , and let be the quotient Hopf algebra of corresponding to a Borel subgroup of . Backelin and Kremnizer then define the quantum flag variety to be the category of -equivariant -modules. Specifically, an object of this category is an -module equipped with a right -comodule structure such that -action map is a comodule homomorphism. In this language, the global section functor is the functor of taking -coinvariants. They then define the ring of quantum differential operators on to be the smash product algebra , and a -twisted -module becomes an object of the quantum flag variety equipped with an additional -action such that the -coaction and the action of the quantum Borel subalgebra ‘differ by ’ (here is an element of the character group of the weight lattice). There is also a distinguished object which represents global sections in the category of -twisted -modules. The precise definitions are made in Section 3. Their main theorem is that, when is regular and dominant, the global section functor gives an equivalence of categories between -twisted -modules and modules over 222We were informed late in the writing process that there may be gaps in the proof of [10, Proposition 4.8], i.e. in the computation of global sections, see [50, Remark 5.4]. This does not stop the equivalence of categories as we describe it from holding. Indeed, the proof of that only relies on an analogue the Beilinson-Bernstein ‘key lemma’, which itself only requires for there to be a map in order to hold..
Nothing stops us from making completely analogous definitions using certain Banach completions , and of these algebras (see section 1.3 below). That allows us to define what we call the analytic quantum flag variety as the category of -equivariant Banach -modules, meaning that the objects of this category are Banach -modules which are also Banach -comodules such that the -action map is a comodule homomorphism (see section 6.7). We note that this category is not abelian. Instead it fits into Schneiders’ framework of quasi-abelian categories . In particular it has a derived category and, under suitable conditions, we can right derive left exact functors. The global section functor here is also the functor of taking -coinvariants, and we use this framework of quasi-abelian categories to make sense of the cohomology of . We can then define -twisted -modules to be objects in which are equipped with an additional -action such that the -coaction and the action of differ by . There is also a distinguished object which represents global sections. Again, the precise definitions are made in section 7.3.
1.3. General strategy
They first work with integral versions of classical algebraic -modules and show that large enough twists of coherent -modules are acyclic and generated by their global sections. Using this, they then show that the category of coherent -modules has a family of generators obtained from taking certain twists of . In particular those are -adic completions of algebraic -modules.
The first step essentially reduces the problem to working with those coherent -modules which can be ‘uncompleted’. They then show that these are generated by their global sections. This uses the classical Beilinson-Bernstein theorem.
Finally, they show that completions of acyclic coherent -modules are also acyclic. This uses technical facts about the cohomology of a projective limit of sheaves.
Once you know that coherent -modules are acyclic and generated by their global sections, the result follows from standard general facts.
In order to adapt this, we are first required to work with integral forms of quantum groups and the corresponding integral quantum flag variety, see sections 2.2, 2.4 & 4.1. Specifically, there is an integral form of which was first defined by Andersen, Polo and Wen . By taking to be its image in the quotient Hopf algebra , we are then able to define the category of -equivariant -modules. We can also define an integral form of the ring , and use it to define -twisted -modules in (here is an element of , the character group over of the weight lattice). These integral forms allow us to define the Banach completions we mentioned above by simply setting , and respectively.
Unlike in the first step above, we are not able to show that large enough twists of coherent -modules are acyclic and generated by global sections, but we manage to show it for those which are annihilated by . This turns out to be enough for the first two steps to work. Most of this paper is then spent developing the correct tools from noncommutative algebraic geometry in the category in order for the ideas used in the third step to even make sense.
1.4. Čech complexes
To have a version of step (iii) above, we need to work with the right sort of complexes, computing the cohomology of global sections, in order to apply the argument on the cohomology of a projective limit. To do so, it is convenient to work with proj categories. Indeed, the classical flag variety is isomorphic to Proj, and Backelin-Kremnizer showed that is equivalent to Proj in the sense of Artin-Zhang . We show that the integral quantum flag variety enjoys the same property. To obtain this result, one problem we ran into is that, while it is well-known that the algebra is Noetherian, it isn’t known in general whether its integral form is also Noetherian (in type , it is known to be true from Polo’s appendix in ). That makes it non-trivial to define the objects which should play the role of coherent modules. Thankfully, we were able to prove that the integral form of is Noetherian, and using this we showed that the Noetherian objects in are precisely those which are finitely generated over , see Theorem 4.6. Once this obstacle is cleared, the proof that we have a noncommutative projective scheme is essentially identical to the one in .
This result is essential because it allows us to define our promised complex which computes the cohomology of global sections for these integral forms. We think of this as a Čech-like complex. Using the Proj description of , one can in a suitable sense cover the category with analogues of the Weyl group translates of the big cell, see sections 4.7 & 4.8. The complexes are then obtained using general constructions from Rosenberg . After taking -adic completions, the objects of are then naturally sent to another intermediate category, which we unoriginally call and which is in some sense an integral form of . We use the Weyl group localisations mentionned above to write down an analogue of our Čech-like complexes in this new integral category. After extending scalars, this gives us a Čech-like complex in the category . This is the right object in order to apply the arguments from step (iii).
1.5. Main results
At several stages of this paper, we work with Banach comodules over . We first give a more explicit description of these objects. We begin by defining what we call topologically integrable modules over a certain completion of , see section 5.3. Roughly, these are modules where the torus acts topologically semisimply and the positive part acts locally topologically nilpotently. The definition is partly inspired from work of Féaux de Lacroix , who developed a notion of semisimplicity for topological Fréchet modules (note that we already used the notion of topological semisimplicity in our previous work [21, Section 5]). Our first main result is then:
The category of Banach right -comodules is canonically equivalent to the category of topologically integrable -modules.
This result allows for a more intuitive understanding of what these comodules are, and also draws further parallels between our constructions and standard notions that appear in -adic representation theory. We note that Banach comodules over a Banach coalgebra have also been studied in a more general, categorical setting in .
Our next result is that the cohomology of in can be computed using the Čech-like complexes described above:
For any , the standard complex computes .
The precise definition of this complex is made in section 6.3. We note that as a consequence of this, we obtain in Corollary 6.12 that has finite cohomological dimension (something which wasn’t obvious beforehand!). Both of these are essential in order to obtain a Beilinson-Bernstein theorem, but we also think of them as interesting results in their own right. We view our analytic quantum flag variety as being in some sense a noncommutative analytic space, and these results make it feasible to work with it.
Finally, with all the above at hand, we are able to run the strategy fom section 1.3 to obtain our version of Beilinson-Bernstein localisation. We call a -module in coherent if it is finitely generated over .
Suppose is regular and dominant. Then the functor of global sections and the localisation functor Loc are quasi-inverse equivalences of categories between the category of -twisted coherent -modules on the analytic quantum flag variety and the category of finitely presented modules over .
1.6. Future work
We are at the moment unable to compute the global sections . Similarly to the situation with and , while it is known that is Noetherian (see ), it doesn’t appear to be known whether its integral form is as well. Under the hypothesis that it is Noetherian, we are able to prove that where the latter ring is a Banach completion of the ad-finite part of modulo the central character corresponding to , see Theorem 7.8333This on the other hand does assume that the computation of global sections in [10, Proposition 4.8] holds.. We hope to resolve these issues in future work. Also note that we only deal with the analogue of the case from Theorem 1.1 in this paper, and we plan to extend our results to an analogue of the case.
1.7. Structure of the paper
In Section 2, we recall all the necessary facts about quantum groups and their integral forms. In particular we give an explicit description of the categories of -comodules and -comodules as the categories of integrable modules over Lusztig’s integral forms of and respectively. We believe this to be well-known, but we could not find any suitable reference for this, so we included a proof. This needed some general facts about Hopf -algebras which we included in an appendix. In Section 3, we recall all the main definitions and constructions from . We also include a proof that is Noetherian.
The main body of our work starts in Section 4, where we replicate the constructions from Section 3 working with integral forms. We then prove that this integral category is a noncommutative projective scheme. In doing so, we make heavy use of results about the cohomology of the induction functor for quantum groups from Andersen, Polo and Wen . Then we use this to construct a Čech-like complex which computes the cohomology of global sections. Finally, we define -modules and, using a result of Andersen and Jantzen , we prove that if a coherent -modules is annihilated by , then large enough twists of it are acyclic and generated by their global sections.
In Section 5, we recall facts about completed tensor products and introduce Banach comodules over a Banach coalgebra. We then introduce topologically integrable modules over the Banach completion of Lusztig’s integral form for , and show that these are equivalent to Banach -comodules. Using results on topological semisimplicity from our previous work , it follows from the fact that any Banach -comodule embeds topologically into , equipped with the comodule structure .
In Section 6, we then introduce the categories and , and recall all the necessary facts on quasi-abelian categories. We then construct a Čech-like complex and prove that it computes the cohomology of global sections. The main technical tool we need here is some flatness results for completed tensor products from . The theorem then follows essentially by using the fact that it holds for lattices modulo for every . Finally, in Section 7 we put everything together to prove our Beilinson-Bernstein theorem. The arguments here are essentially those from , with some small adjustments. We reproduce them nevertheless.
This paper is going to be a significant part of the author’s PhD thesis, which is 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 5. We are also thankful to Andreas Bode for his continued interest in our work, and for communicating Proposition 6.9 to us before his work was written up. Finally, we wish to thank Kobi Kremnizer for a useful conversation on quantum groups and proj categories.
1.9. Conventions and notation
Unless explicitly stated otherwise, the term “module” will be used to mean left module, and Noetherian rings are both left and right Noetherian. Also, all of our filtrations on modules or algebras will be positive and exhaustive unless specified otherwise. Following [4, Def 2.7], an -submodule of an -vector space will be called a lattice if and is -adically separated, i.e . Given an -module , we denote by its -adic completion and write .
Given an -normed vector space , we denote by its unit ball. Given a Banach algebra , a Banach -module will always be assumed to have action map of norm at most 1, i.e will always be assumed to be an -module.
In a Hopf algebra , we use Sweedler’s notation for the comultiplication, i.e we write . All our comodules will be right comodules unless stated otherwise.
Finally, while we talked about -group schemes and their corresponding Lie algebras in this introduction, quantum groups are defined purely in terms of the root system and are traditionally defined starting from complex Lie algebras and algebraic groups, regardless of what the base field is. This is the convention we follow as well. Hence we let be a complex semisimple Lie algebra. We fix a Cartan subalgebra contained in a Borel subalgebra. We choose a positive root system and we denote the simple roots by . Let denote the Cartan matrix. We let be the simply connected semisimple algebraic group corresponding to , and we let be the Borel subgroup corresponding to the positive root system, and let be its unipotent radical. Let and . Let be the Weyl group of , and let denote the standard normalised -invariant bilinear form on . Let be the weight lattice and be the root lattice. Let denote the abelian group . We will use the additive notation for this group. Let be the smallest natural number such that for all . Let and write .
We make the following two assumptions. First, we assume that exists in and that . Then for each , we have an associated element in sending a given to , which we will also denote by . Secondly, we assume that and, if has a component of type , we furthermore restrict to .
All the above algebraic groups and Lie algebras have -forms, and we write etc to denote them.
2. Preliminaries on quantum groups and their integral forms
2.1. Quantized enveloping algebra
We begin by recalling basic facts about quantized enveloping algebras (see eg [17, Chapter I.6] for more details). For and , we write . We then set the quantum factorial numbers to be given by and for . Then we set
The simply connected quantized enveloping algebra is defined to be the -algebra with generators , , , , satisfying the following relations:
We will also abbreviate to when no confusion can arise as to the choice of Lie algebra . We can define Borel and nilpotent subalgebras, namely is the subalgebra generated by all the and the , and is the subalgebra generated by all the . Similarly, is defined to be the subalgebra generated by all the . There is also a Cartan subalgebra given by , which is isomorphic to the group algebra . There is an algebra automorphism of defined by , and .
Recall that is a Hopf algebra with operations given by
for and all . Then is a sub-Hopf algebra of .
Also recall that there is a triangular decomposition
and that have bases consisting of PBW type monomials. More specifically, if are the positive roots, ordered in a particular way, then there are elements of such that the set of all ordered monomials forms a basis for . We now let and the corresponding monomials in the ’s will form a basis of . The triangular decomposition immediately gives a PBW type basis for , namely it consists of monomials of the form
where . We recall that the height of such a monomial is defined to be
where for a positive root . This gives rise to a positive filtration on defined by
This filtration can actually be extended to a multifiltration as follows. The associated graded algebra can be seen to have the same presentation as , with the exception that now all the ’s commute with all the ’s. Moreover it is isomorphic to as a vector space. We can then make into a -filtered algebra, by assigning to each monomial the degree where we impose the reverse lexicographic orderin ordering on . Denote the corresponding associated graded algebra of by . This algebra is known to be -commutative over (see [19, Proposition 10.1]). Here we say that an -algebra is -commutative over a subalgebra if it is finitely generated over , say by , such that the normalise and for all there are such that . We regord here a noncommutative analogue of Hilbert’s basis theorem, which follows directly from [36, Theorem 1.2.10] and induction.
If is -commutative over and is Noetherian, then so is .
Hence we see that is a Noetherian -algebra.
2.2. Integral forms of
We now recall details about two integral forms that we will work with. First recall the notation:
for any integer . Then Lusztig’s integral form is defined to be the -subalgebra of generated by () and all and for and . Recall that for , with we define
There is an -subalgebra generated by all and all . We let denote the -subalgebra of generated by and all for and . By [2, Lemma 1.1], for each there is a unique character defined by
We will say these characters are of type .
Given a -module and a character as above of , we write for the elements such that for all . We now recall the notion of integrable module from [2, 1.6]:
A -module is said to be integrable of type if it is a sum of weight spaces which all correspond to a character of type as described above and if in addition, for every , there is such that is killed by and . Similarly we define a -module to be integrable of type if it is the sum of its weight spaces corresponding to type characters and for every , for .
Since all our characters will always be of type we will often just say ‘integrable’ to mean ‘integrable of type ’.
The second integral form we will need is the De Concini-Kac integral form . This is defined to be the -subalgebra of generated by . This algebra has a similar presentation to . If we write for and , then is generated as an -algebra by with the same relations as except that the commutator relation between and is replaced by the two relations
Note that is a Hopf -algebra. For example we have the identity
Note that we also have the equality
for all , and so contains all .
We showed in [21, Section 4] that has a triangular decomposition where is the -subalgebra generated by the ’s, respectively ’s, and is the -subalgebra generated by . Moreover for all by the above. We also showed that has a PBW basis, more specifically that the PBW monomials which form an -basis of are also an -basis of .
Note that both of these integral forms are -adically separated since and is free over .
2.3. The ad-finite part
Recall that since it is a Hopf algebra, has a left adjoint action on itself given by . This action is not in general locally finite, so we define the finite part of to be
is then the largest integrable submodule of with respect to the adjoint action. It is a subalgebra of (see [30, Corollary 2.3]). It is in fact quite large:
For every , , , .
A quick computation shows that and are scalar multiples of and respectively, so it’s enough to show that . Now another quick computation shows that
For each , let be the -subalgebra of generated by . Then the above shows that is finite dimensional for every by [30, Lemma 6.2]. But then it follows from [30, Proposition 6.5] that is finite dimensional. ∎
A completely analogous computation was made in [11, Lemma 2.3] working with the right adjoint action rather than the left adjoint action.
For each , there is a Verma module which is the cyclic -module with a single generator and relations
for all . The above lemma implies that the natural surjection restricts to a surjection .
Now working with integral forms, the -Hopf algebra acts on itself via the adjoint action and, moreover, this action preserves by [50, Lemma 1.2]. Hence we may define
Note that . Indeed, clearly the right hand side contains . Conversely, if then is a lattice inside which by definition is finite dimensional over . Hence this lattice is finitely generated over by [4, Proposition 2.7].
2.4. Quantized coordinate rings and their integral forms
We now recall the construction of the quantized coordinate algebra . For any module over an -Hopf algebra , and for any and , the matrix coefficient is defined by
Also recall from [26, Theorem 5.10] that for each there is a unique irreducible representation of type , , of and that these form a complete list of such representations. The quantized coordinate ring is then defined to be the -subalgebra of the Hopf dual generated by the matrix coefficients of the modules for . In fact, from [17, I.7-I.8], it is a finitely generated, Noetherian -algebra, and it is a sub-Hopf algebra of . There is also a quantized coordinate algebra of the Borel . Since is a Hopf-subalgebra of , the restriction maps yields a Hopf algebra homomorphism and we let denote its image.
We now recall how the integral forms of and are defined. Let be Lusztig’s integral form defined in above. Let denote the set of ideals in such that is a finite free -module. We now consider the set consisting of ideals such that contains a finite intersection of ideals . Note that for any -module , we may view as a -module via for all . In [2, Definition 1.10], a so-called induction functor from the trivial subalgebra was defined. It takes any -module to the subrepresentation of given by all elements in the sum the weight spaces in which are killed by all and for . In other words is the largest integrable subrepresentation of . We then define the integral form of the quantized coordinate algebra to be . By [2, Corollary 1.30], we have if and only if kills an ideal . In particular,
So is a sub-Hopf algebra of (see Definition A.1) and it may be viewed as the algebra of matrix coefficients of finite free -modules of type . In particular the comultiplication on it makes it into a -comodule and hence we may view it as a -module by Proposition A.2 (and that agrees with the definition of the -action on ). Moreover by [2, Theorem 1.33], is free over .
Next, we look at the Borel subalgebra of . Let be the set of such that . The Hopf algebra homomorphism given by restriction has kernel precisely and so we see that is a Hopf ideal and that is a Hopf algebra. Similarly to the above,  defined an induction functor from the trivial subalgebra to in a completely analogous way: if is an -module, we define to be the largest integrable submodule of . By [2, Proposition 2.7(ii) and (iii)] we have that and so it is integrable, and it is free as an -module.
2.5. The categories of comodules
We now recall how the category of -comodules (respectively -comodules) can be identified with integrable -modules (respectively -modules). We expect this to be well-known but we did not find a suitable reference for it, so we provide proofs. To that end, we use general results about -Hopf algebras which we’ve written in the appendix.
Since , it is integrable with the -module structure described above. Note that for any -module there is a natural map which is the composite of the map , coming from the inclusion , and the map from Corollary A.2. By abuse of notation we also denote this map by . For the Borel, we have again a map for any -module . Moreover we have again that if and only if kills an ideal of such that is finitely generated and contains a finite intersection of ideals .
The next result immediately follows from the above:
If is torsion-free as an -module then and are torsion-free. In particular and are torsion free.
If kills an ideal in , then so does as is torsion-free. An analogous argument applies to . The last part follows by putting . ∎
Since and are sub Hopf algebras of and respectively, it follows that any comodule over (respectively ) is a comodule over (respectively ). Thus we may view comodules over and as locally finite modules over and respectively. This defines functors from the categories of -comodules and -comodules to the categories of locally finite -modules and -modules respectively.
The following observations will be useful in the next proof and also at several points later on. Suppose that is a -comodule, with coaction . Note that by the axioms of comodules, the composite
so that the map splits and is a direct summand of as an -module. Moreover, the diagram
commutes. But note that the map makes into a -comodule, so that the above diagram and the splitting says that identifies via with a subcomodule of where the latter is given the comodule structure . Of course all of the above applies more generally to a comodule over an arbitrary coalgebra.
The category of -comodules, respectively -comodules, is isomorphic to the category of integrable -modules, respectively -modules.
We first show that the above functors are fully faithful. This is the exact same argument as in Proposition A.3, using Lemma A.2 with , and for -comodules and with , and for -comodules. For these to apply we need to show that and are torsion-free, but this is just the previous Lemma.
Next, the key fact we use is [2, Theorem 1.31(iii)]: for any -module the natural map is an isomorphism onto . Now suppose that is an integrable -module. Then for all , the action map belongs to . So by the above facts the maps all belong to the image of . By Lemma A.4 with we conclude that must be an -comodule. An analogous argument shows that integrable -modules are -comodules using [2, Proposition 2.7(iv)], which states that the natural map is an isomorphism onto .
Thus since the functors are fully faithful we are now reduced to showing that any -comodule (respectively -comodule) is integrable when viewed as a -module (respectively -module). We prove it for , the proof for being entirely analogous. Suppose is a -comodule. Then by the above remark the map is an injective comodule homomorphism where the right hand side is given the coaction map . In other words, in the langu