 Thank you for this introduction. And I also thank the organizers for the invitation to this great place. Yeah, let's start with this. So in this talk, I will mostly stick to the group GL2QP. And it's about a relation between distribution algebras of this group, GL2QP, and arithmetic D modules on certain formal models of P1 to get into the mood and to fix some notation, I will start with a small reminder of where the problem comes from. So this is the Balans and Banstein theorem. So this is so to speak the solution of the problem on the Lie algebra level. So I will also do this just for GL2. So G is the Lie algebra of GL2 over QP. UG is the universal enveloping algebra. So this comes with an augmentation. The augmentation character I call theta. And then I can form a central reduction. I form UG theta. This is simply UG tensile over the center relative to the restriction of theta to the center of UG numbers QP. So this is an algebra on the one hand. On the other hand, I can look at the scheme P1, the projective line over QP. And then the globally defined differential operators on this scheme can be easily written down. I mean, this algebra I generated by three generators. So Z is an affine coordinate on the P1. So I can view this inside the white algebra. So Z delta, which is the algebra of differential operators of the affine line. And then the group GL2 QP acts on P1. I mean, even the algebraic group GL2 acts on P1. I can differentiate the action. I get a map from the Lie algebra G into the differential operators. And the point is on the central reduction, this is an isomorphism. And it can be made explicit quite easily. So if I fix basis vectors of the Lie algebra E. Oh, thank you. Right. So this is E. Then I have toral elements H1 and H2. And then this map actually is given quite explicitly. E is mapped to delta. F is mapped to minus z squared delta. H1 is mapped to minus z delta. And then, of course, H2 has to be mapped to z delta. So this gives a connection between D modules on P1 and representations of the D algebra, which is very tight actually. One has an equivalence of categories between all the modules over this algebra. And on the other hand, D modules on P1. This is not specific to GL2. No, no. This can be done for any split reductive group over QP. So this is for you. And yeah, so going in the other direction, going in one direction is taking global sections. And then viewing the global sections of the D module via this first result as a Lie algebra representation. And the other way is so-called localization function, which I don't want to go into detail at this point. But what is more important is that if one brings in the G action, then one can write down an equivalent version of this equivalence. So this sheet of differential operators is an equivalent sheet with respect to the G action on P1. So I have a subcategory of equivalent D modules on P1. And this is equivalent to certain representations of G, which come equipped with a compatible action of the D algebra. So now let D LAG be the locally analytic distribution algebra of G. And I can, again, form a central reduction. And since this distribution algebra contains, on the one hand, the group, and on the other hand, the universal enveloping algebra, a module over this reduced algebra is a special case of module on this left-hand side. And so the question is, natural question is, can one modify the geometric side on this equivalence in a periodic analytic way such that going from right to left really is adapted to this subcategory of modules, that it really gives locally analytic representations? And maybe there's another function in the other way that this is an equivalence. So can one have a periodic analytic version of this for locally analytic G representations? So by periodic analytic version, I mean on the, for example, on the rigid analytic projective line. So but one soon runs into the following problem. So in the locally analytic representation theory of G, there occurs some power series envelopes of this universal enveloping algebra. So if one just takes the rigid analytic P1 with the naive differential operators, one will always end up with polynomial expressions and the Liegema will never get these power series envelopes. So one needs the chief of completed differential operators. But on the other hand, there is a theory that provides completed differential operators. So the idea is to approximate the rigid analytic P1 by a sequence of formal models, semi-stable formal models, and use then battalos arithmetic differential operators on each of the model. I mean the link to the analytic P1 then, and this is meant by approximators, then given by Reynolds theorem, which says that the rigid analytic P1 is more or less the same as the collection of all formal models up to admissible blow-up. Now, yeah. The choice of the size model will correspond to something. On the other side? You mean on the Liegema side? Yes. Yes, you will get different things on the Liegema side. I will come to this. And I will only deal with a certain sequence of semi-stable models, not with all the semi-stable models. And the ones which we're working with actually, I should have said this right away. This is joint work with D-Pompatel. And it's actually, at the same time, it's still work in progress. So the models will be related to certain parts of the Breutitz tree of GL2. And the parts actually will be combinatorial balls. And the stabilizers of these balls are congruent subgroups. So this gives a link to congruent subgroups of GL2. And I will discuss this in the next paragraph. So congruent subgroups. So this works in greater generality. So let G be a split-reductive affine group scheme over set P with the algebra G. And let M be the natural number greater or equal to 0. Let I be the augmentation ideal. So it's the kernel of the augmentation map. And suppose we fix a set of positive roots with respect to a maximal toral sub-scheme acting on the Lie algebra. So fix a triangular decomposition into a maximal toral sub-algebra, the Neil-Potent radical, and then the opposite Neil-Potent radical. And then fix a Lie algebra basis that consists of individual basis vectors on each of the summits. So x1 to xd in basis of G. And one can form the following algebra in Gm. So as a module, it's a union over N grades are equal to 0 on the morphisms set P N plus 1. So that's the MPD envelope of the coordinate ring relative to the ideal I, modded out by the N plus 1 filtration step of the MPD filtration instead of giving more details down. Is it better? So instead of making it this precise, I mean, if you don't know what this means, then never mind. Because there is a explicit description just given. So this is a multi-index 1.80. So according to this choice of basis, there's a Pancré-Bier-Coffit representation of this algebra. So it's this one. The symbols mean the following. So coefficients are in set P. Then this qk up M is the integral part of dividing k by P to the M. And then if I underline it, I take the products of it corresponding to the components of k. And then similarly on the last factor, xk, this means x to the k modulo k factorial in case x is nil potent. So in case it comes from the first summit or the third summit, boy, it is binomial expression x over k in case x is total. It's easy to see that this is not just a submodule of, so I can view this inside the universal enveloping algebra of G10.q. And it's easy to see it's not just a free zp submodule, but it's in fact a free zp algebra. This is because this MPD envelope has a co-multiplication by Hohm turns into a multiplication. Some examples. So if M is 0, then this is just k. So this factorial then cancels with the k factorial here. So we get x to the k in the nil potent case and a polynomial in x in the total case. So it's easy to see from this that UG0 is just the universal enveloping algebra. And secondly, if M prime is greater or equal to l, then since this MPD ideal is then also an M prime PD ideal, the universal property of this enveloping algebra gives a homomorphism from UGM to UGM prime. And since all of these algebras are enveloping algebra of the coordinate ring relative to i, the coordinate ring maps into all enveloping algebras. And then dually, they all map into the coordinate ring. So the inductive limit maps to the construction where we replace the enveloping algebra by the coordinate ring and the MPD ideal by the augmentation ideal. So this is just algebra of distributions on the group scheme G. And in fact, it's easy to see that this is an isomorphism. And then, of course, one has a natural version of central reduction. So theta is just UGM tensile over the center of this algebra set P. And then one also has completed versions. So this is the periodic completion in GM. And similarly, with the theta here, and then finally, U hat GM Q, this is the periodic completion tensile. So what has this to do with concurrent subgroups? So here's the application. So I let GN inside my smooth group scheme G be the end concurrent subgroup scheme. So points in ZP algebras are just the kernel of the reduction map. And it's very easy to see that the l-algebra of this smooth sub-scheme is just P to the n times the l-algebra of G. So it's P to the n times G. And what I can do now is the following. These are the rigid analytic groups that will show up when it comes to the differential operators. So I let GN 0 be the generic fiber of the formal completion of GN along the unit element in the special fiber. So this is a rigid analytic group over QP. So for example, in the case of GL 2, then the concurrent subgroup scheme is, I mean, is just the concurrent subgroup, the usual. And the CP value points of GN 0 is just 1 plus P to the n, n2 times CP, where mCP inside OCP is the maximal ideal. So it's not a quasi-compact rigid space. It's actually in the GL 2 case. It's isomorphic to four copies of the open unit disk around 0, or whatever. We're interested in the rigid analytic functions on this group. These are the holomorphic functions. So for example, in this case, if I take an increasing affinoid covering of these open disks by affinoid disks, I can realize this as a projective limit over Banach algebras. And of course, the Lie algebra of the rigid analytic group, but then also of the group scheme acts on this space by differentiating the translation action. So XF is d mod of dt e x t acting on F. In the case of GL 2, this is just the exponential series for 2 cross 2 matrices, which will converge if t is sufficiently small enough. So this gives me a formal power series with coefficients in this space. And I take the formal derivative and I take the evaluation at t equals 0. And this implies if I define XF to be the function and then evaluate it by the unit element, then I get a map from UG into the linear forms on this space, even into the continuous linear forms. So this is the continuous tool. Similarly to the distribution algebra in the local litic case, this analytic distribution algebra has a product and is an algebra. It's very simple. I mean, what's the product of two distributions evaluated on F? That's integrating the function g goes to, yeah, it's this way around. So integrating the function mu goes to g times F, where this is a translation action g and 0 on c and g and 0. So this is then an algebra homomorphism. And then the first result, which actually was proved in partial cases already by Emerton, is that this map gives an isomorphism of the following algebra, UG dagger q. This is the inductive limit over U hat pn gm q. We have an algebra isomorphism into dn gn 0. So this construction can be used to describe the analytic distribution algebra of a wide open concurrent subgroup. Now I come to the formal models of p1. So let t be the Brouillard-Hitz tree of pjl2qp. So this is a homogeneous tree. The edges that come to a fixed vertex are in bijection with p1 fp. So in the case p equals 2, it looks like this. Here's an edge, here's an edge, and there go two edges. So here's a vertex, here's a vertex, here's two edges. Similarly here. The vertices are homothetic classes of lattices. So m is a lattice in qp square. So if I have a neighboring vertex, v, then this has a representative m prime, which satisfies inequality. Inequality is like this. And this means that I get a map from m to m, model m prime. I identify this with fp. To fp, therefore I get an fp rational point in the special fiber of pm. So if I have pm here, then here's a rational point in the special fiber that corresponds to v. And so if I blow up pm in all its rational points in the special fiber for p equals 2, that's three points, I get a scheme whose special fiber looks like this. And I can look at the adjacent vertices, they correspond to the smooth points on the n components. If I blow up again in the smooth points, I get a scheme which looks like this, and so on. I can go on like this. Let's call this x0v, this is x1v, this is x2v, and so on. And I have blow up morphisms going from the higher one to the smaller one, and so on. At this point, I have to clean the board. So in this way, one obtains, so generally, xnv is the blow-up of xn minus 1v in xn minus 1v, special fiber smooth points at p. So this is a semi-stable scheme over zp. The dual graph, so I mean the graph that prescribes the intersection behavior in the special fiber is then obviously isomorphic to the ball around x of radius less or equal to n inside t, v, thank you. And of course, nothing has happened on the generic fiber, so xnv tends to qp is just p1qp. And finally, I mean the group acts on the tree, g acts on the tree, and for any group element, I get an isomorphism from xnv to xng to the v, g acting on the tree. So in a sense, g acts on the collection of all these semi-stable schemes. Now, the next proposition says that it actually suffices to do this for a fixed vertex, so v0 is the vertex corresponding to the standard lattice. And say xn is xn relative to v0, then first of all, g already acts on the projective limit of these schemes xn. Because any ball at any vertex is eventually contained in a ball around v0, so this is a co-final family in all the xnv's. Then more specifically, the group g0, I mean I should have said it also from the beginning on, I mean from now on, this is true for the whole paragraph, from now on, g is gl2qp, g0 is gl2zp, and what else do I need? G, that's the group scheme, gl2 over zp and g0 is the Lie algebra. So then this group gl2zp already acts on each finite level because this can be proved by induction, I mean g0 acts certainly on x0, that's just p1zp. And then suppose it acts on x to the n minus 1, and it certainly respects the special fiber, it respects the smooth points, and then because it's formed out of rational points, it also preserves the rational points of the smooth fiber. So by the universal property of blow-up, it will then act on xn to the v, so for all n, this is one thing. Secondly, the schemes, the congruent schemes, the nth congruent scheme acts also on xn as a scheme, so one has morphism, so one has a morphism gn, satisfying the usual properties. Again, this can be proved by induction. So the scheme g0, of course, acts again on the p1 of zp, and then suppose we have an action of, so gn minus 1 acts on xn minus 1, and then look at gn, gn is contained, gn is contained in gn minus 1, and it has the property that this inclusion sends gn special fiber to 1. So if I restrict the gn minus 1 action to gn actually here, then on the special fiber, it will act trivial, so it will preserve this blow-up locus, and then by the universal property of blow-up, it will lift on to xn. So we have this level-wise action of the gn's, and finally, of course, if I take the periodic completion, I mean the formal completion of xn p equals 0, then this is an admissible formal scheme and the formal model of p1 weak qp, and then, of course, the maps are then so-called admissible blow-ups. OK, now I come to finally to the differential operators. So I treat the smooth case, I mean the case of x0 and the case of xn, where n is greater than 0 separately. So the smooth case, 0, and the aim is to compute the global sections of the arithmetic differential operators on the x0. So in the algebraic x0, S and f are covering ux, where ux is back zpx, and I have the level m differential operators on x0. I will give a description in local coordinates. So I will not define it formally at this point, or I'll give a description in local coordinates. And dx0 hat m is the periodic completion. And finally, dx0 dagger is the inductive limit of all m. So in local coordinates, I mean so the formal completion of x0, this has a covering then by the formal completion of x, ux, and uy. And so I can write the sections over this affine formal scheme, and these are just series akqmk hat with partial x to the k, where ak is a section of x0 over ux hat. And so this goes to 0, for k goes to infinity. Now, this is not good writing. So akx, which is, of course, just qpx, zpx. And it goes to 0, for k goes to infinity. qmk, I have already defined. And delta x, delta x is just differentiation with respect to x, and delta xk, in this case, is delta x to the k. Well, look, k faculty. This is how this sheaf looks like locally. And one can do the following now. So g0x on x0, and this gives a map from g into the vector fields on x0. And this induces the following. So this is our first result, 0 and 1. So let's call this psi0. So psi0 induces a map from g0. So this actually should be g0. This is the zp in the algebra, m hat q into dx0 hat on x0 for all m. So the global sections, indeed, all come from the d algebra. As soon as 1 tends us with q, this is not true on the integral level. I mean, at least not for m greater than 0. And if one takes the inductive limit over all m, then one gets that this algebra is isomorphic to dx0 dagger x0. And I should remark that for m equals 0, so in case where this is just the universal enveloping algebra, then this first isomorphism was already established by Adakoff-Watzle. And this generalizes this result for arbitrary m. Now in the semi-stable case, n is greater than 0. I can do the following. So the special fiber of xn inside xn, this contains all the singularities. So this is a normal crossing divisor. That means all the singularities are ordinary double points. And then one has the action of the scheme gn on xn. And if one differentiates, one gets a map which is the analog of psi 0 for n. So from gn, cn to the tangent sheaf on xn. And in fact, one can look at the vector fields, the logarithmic vector fields. In this case, this is just a sub-sheaf corresponding of all the derivations that preserve the ideal sheaves corresponding to the irreducible components in this divisor. And the point is that this map already lands in the sub-sheaf because if one takes an irreducible component in xn, so first of all it acts on the special fiber, and if one takes an irreducible component in there, and this lies over some smooth rational point in xn minus 1 as smooth fp. And so an element in this takes this irreducible component to an irreducible component that sits over gx. But I mean if I restrict the action of gn minus 1 to gn, then this x trivial. So one has gx is x. And this implies that g operates already on the component. So someone gets this map. And then there's the second theorem. So psi n induces inclusions. So in this case, one does not get a neat statement like in the smooth case. But one has inclusions of dn, gn, 0 inside xn dagger q xn inside dn, gn prime 0 for n prime, the greatest integer less or equal to n times p minus 1 over p plus 1. So you see in the case n equals 0, this is n prime is then equal to, I mean n prime is then also 0. And then these become inclusions. These become in fact bijections. So this is the smooth case. In the semi-stable case, one has just the sandwich behavior. And I mean this is meant, of course, in the sense that if p is the blow-up map from xn to xn prime, then one has a map from p lower star d xn m to d lower star xn prime m. And this, in fact, is injective if I tens up with q. So this means I can view the global sections of xn inside the global sections of dxn prime. And then it makes sense to ask that they lie, in fact, in the subring dn, gn prime. And that one has this algebra, homomorphism lies, of course, on the fact that o xn is just o xn prime, because it's a blow-up map. OK. And then back to the initial problem. It will be a bit brief here. So having computed the global sections, one can do the following. So if one takes the projective limit over all xn as a locally-ringed space and pulls back all the dxn to this limit, so pn minus 1 dxn dagger q, where pn is the projection to a factor. Then on the limit, the group gx, and this is an equivariant sheaf on x, and what one can see is that if m is an equivalent d hat module, then it's global sections. Then global sections over x, this will be a module over the individual global sections of the dxn's. And they will build up a module structure over locally analytic distribution algebra of g0. And at the same time, one still has a g action on the global sections, so this is a module over this pair. But a module over this distribution algebra plus a compatible g action is then a module over the full distribution algebra. So in a sense, here's a space which is somehow related to the rigid p1 and a natural sheaf, so to speak, such that any equivariant module has global sections or module over this distribution algebra. Of course, if one wants to talk about locally analytic representations, then one has to take into account topologies. I mean, a locally analytic representation is a module over this ring plus certain topological prerequisites. But I have not talked about this here, but if one carefully goes through all the arguments and keeps track of the topology, then one might eventually put natural conditions on this equivariant module here, such that one actually gets on global sections a true locally analytic representation. So this is where we are at the moment. And this is my talk. Thank you. In contrast with the very interesting result in Terrain-Muat, you do not have any central reduction for the unblocking algebra. Can you comment on this? Yeah. That's a good point. I just forgot. You can have this result without what kind of control on the center of this unblocking algebra. In Baguinson-Belschein case, one knows that this is generated by the Casimir elements, but here? Yeah, for the SL2 case, it's generated by the Casimir element in the algebraic setting. And here it should be a power series ring in the Casimir element. Yeah. Thank you. On the right, it's the Casimir element. Yeah. Yeah, thank you. This is the standard queue. It's not true on the integral level. It's always injective, but it's not true. On the integral level, it has a proper co-kernel. Is it 0 and 1 for SL2 case? Yeah, it descends to the SL2 case, GL2 case, SL2 case. Yeah. But the result of the code for what state is more general? It's more general, yeah. So they prove it only for m equals 0, but in fact, they will prove it for any reductive split group. So under the assumption that the prime number is good for the root system, or very good for the root system. But for GL2, any prime number is good. But do you expect to have an equivalence of categories at this point? We're not sure yet. It is not as neat as in the d-algebra case. So it's not the global sections over all the differential operators here. They are actually, I mean, so from this it follows that the limit of the dxnq dagger xn. And this is just a projective limit over the dngn. But this is only connected to the d-algebra list. Now you lose the group action, so to speak. This is the so-called Arendt-Michael envelope of Li. So this is the closure of Udqp inside the distribution algebra. So the global sections of this sheaf are just the closure of the universal envelope in algebra. So if there's a function in the other direction, it comes not only from tensoring over the global sections with the sheaf. This will give something which is too small, actually. But can you actually, by this construction, obtain some representations which say morally not principle series locally? We have so far not computed examples. So it's a good time to start with it, but not done so far.