Thank you for the introduction and thank you for the invitation to speak here in what looks

like it will be a really really great conference. So I have no special stories about Godman because I never met him but I just wanted to say that I heard his name before hearing about automorphic form because when I was 18, so I was a really young student and one of my math teacher was just telling me that the best lecture she ever had in algebra was Godman's

lecture and she just said that I should go and read this Cour d'Algebr and so I opened it and when I was 18 and I didn't understand a lot of what was going on in there so that was kind of my first real contact with algebra but it didn't prompt me to go further. So that's all for the

Godman part. So today I will report on a joint work with Auguste Hebert which is just there about Iwori Hecker algebras and Mazure for split-cast-moody groups. So it has very little to do with automorphic form, I'm sorry, but the motivation comes from reductive groups. So maybe I should start with an introduction. So everything started when Auguste

advisor Stéphane Gosson asked me one day whether I've ever read about Schneider and Stuhler's papers and if I think that it may apply to cast-moody groups. So I could answer the first

question that I read Schneider's and Stuhler's paper but that I had no idea about what was a cast-moody group so I wouldn't be very helpful. So maybe I should recall what is in this paper. So I will not recall everything just what was interested Gosson at that time.

So here the, let me say, the motivation comes from Schneider and Stuhler in the 97 papers in the publication of Mathematica delle Aschues. So in this paper, starting from a smooth

representation, let me say irreducible even if we don't need this, a smooth representation of G which is the F point of a connected reductive group. So here G will be connected

reductive defined over F and F is non-archimedean local field of residue field. So positive

characteristic P. So starting from such representation, so it's complex representation which is not usual in my work but here I am. So starting from such a representation,

they attach to this coefficient system, so a G-equivariant coefficient system on the ratchet building. And from this G-equivariant coefficient system,

they get a functorial way to get projective resolution of pi. And so basically what

Gosson was interested in is trying to see whether if now we don't take a connected reductive group but a Cas-Moody group, maybe a split Cas-Moody group, can we to such

representation attach such resolution? And here just the two first question I had on the top of my head without knowing anything about Cas-Moody groups were, OK, do we know what is a smooth representation? And do we have something that would behave like ratchet building

for Cas-Moody groups? Because if you know a little bit about Cas-Moody groups, you know that under some in some settings, you can attach to them a pair of twin buildings but not one building. And while it's not very convenient in this setting. But here, why Gosson has this

is that with Rousseau, they built something that could replace this ratchet building, which is called a maison. So I will come on this a bit later just because it's a bit fun.

Why did they call this maison? Because as you will see, it's well, it's like a building, but it's really, really crappy. And it's very not convenient. We don't want to live in there. So that's why you go like this. OK, so the two brown underlined words are really what we will start from to explain how we arrived

to Iwo Rijeck algebra. So of course, they didn't know what smooth representation were. And it was even worse. So I will just phrase it like this. So maybe, OK, let me just say it. To say that pi is smooth means that any vector has open stabilizer in G by representation of G.

So here, you have some topology of G that is appearing, meaning that you have to know what is an open subgroup of G. And if you go a little bit further and are kind of more optimistic, you can more generally do we have, let me say, a nice topology on G. So in the

kasmodi setting, that gives back. So when I say the usual, it's really the topology of G of

F coming from F, like when you used to work with when you work with reductive group, with

periodic groups, should I say. So the usual topology comes from in the reductive case.

So I will be more specific on this later. But maybe I can already do a little spoiler here. The answer is no. So far, we don't have such a topology. But smoothness, it's a bit tricky. So we were kind of, OK, let's not talk about smoothness for now and maybe focus on the other

parts. And having the other part, so when you have this generalization of gratis buildings, which is this measure, you can also define Hecke algebras. So not as many Hecke algebras

as we have in the reductive case, but at least two main Hecke algebras. So first,

the spherical Hecke algebra. So basically, just think of this as the convolution of a bin variant function under the integral points of G, and the classical Euler-Hecke algebra.

So I will explain a bit more about this later. But just to kind of point out what we did with August. So Gausson, Rousseau, and Bardipence, and other people I will mention later,

they managed to define such algebras and to check that when you work with reductive groups, you really got back the algebra you used to. But the thing is that, so in the spherical case here, Gausson and Rousseau managed to prove that we have

an isomorphism between the spherical Hecke algebra and the same analog of what you got in the reductive case using Sateke isomorphism. And here, well, if you believe in the fact that you want to generalize what happened in the reductive case, then you should have the

center of this algebra being isomorphic to this spherical Hecke algebra. And one thing we checked is that, well, if you take this construction, it doesn't work. It's not true. So starting from this, we'll build another Euler-Hecke algebra, and this will be the third part of my talk that really, again, kind of generalize what happened in the reductive group

case and which moreover satisfies this analog of Bernstein theorem. So that's kind of the motivation in production. So maybe kind of a summary of what

we did so far. And then I'll go into more precise notation, definition, and statements. So first thing is that we proved that we cannot,

let me say, mimic what exists in the reductive case to get a topological group

when chorology is gas-mooting and reductive.

We showed that the center of this Euler-Hecke algebra is too small.

And sometimes it's really, really small, like it's just colors. And last thing, we built a completed Euler-Hecke algebra, which satisfies

an analog of Bernstein theorem, meaning that the center is really isomorphic to the spherical Hecke algebra. So this is what I want to discuss. If I have enough time, I will discuss another

construction we did that brings more Hecke algebras in the game, more in what already exists in the reductive case, but only if I have time. So this is all in our joint paper.

So if you want to take this. OK. So now, maybe if you want the introduction is the first part,

and this is the second part. So maybe I want to recall very, very briefly what are gas-mooting and how you attach to this following Thieb's construction gas-mooting groups.

So here, I've used this paper of Bertrand Remy in Asterix. So it's Asterix number 8082 in French. But everything is really, really, really well explained.

So if you have any kind of requirement on that part, I really recommend that you read that paper. OK. So first definition. So here, I'm sorry, I will go with a bunch of notation and definitions. It will not be the funniest part of the talk, but it's really, really necessary. So what we call a gas-mooty data. So it's really like in the reductive case

that you will have a tuple of data, and you will attach to them some kind of combinatorial and geometric objects. So it's just a tuple of five elements and where the notation are really

the usual ones. So you have two models, roots, co-roots. So indexed by a finite set.

So A here will be just a generalized Cartesian matrix. X and Y free the model of finite rank

that are dual from each other. So the duality map is part of the data. Alpha i are elements of X. These are the roots. And the alpha chach are in Y. It's a co-root.

And OK, let me really loosely say they are compatible with A, meaning that the coefficients of A are given by the duality between alpha A and alpha chach G

for A-G. So if you have such a data, as in the reductive case, you can define an apartment, you can define by groups, you can define other things. So in particular, to such a tuple,

so you can define the model apartment as usual is just the scalar extension

of Y to R. So I will denote it by curly A. So there will be many As. OK, so all the alpha i now can be seen as linear form on A, just by duality.

So using this remark, I can define automorphism of A. So it's a reflection formula as usual.

So you take an element v of Y and you map it on v minus alpha i. So here, you're still in A. And you then can define the vectorial vale group.

So it's a vale group, but we like to refer it as a vectorial one because of this construction. It's just a group generated by all these ri's in GLA.

And should I make the remark? Yes, OK. Note, and it's really important,

saying that dual uv is finite is the same. So when, OK, maybe I'm coming, no, I will say this later, when you have defined G and then I will make this remark.

So we also have an assigned vale group and an order.

So I mentioned here the existence of this fine vale group, though we will not explicitly see it appear in what we will see today. But it's really related to this Iowari-Hecker algebra. It allows you to get a presentation of this Iowari-Hecker algebra, for instance.

It's just the usual way to define it. You just submit direct products. I always got it the wrong way. Qth is just the co-root lattice.

OK, and the thing is that when you get such data, you can follow what it did and you get a functor. So there is currently G. So it defines the category of rings, has value in the

groups. I will be really slow here. So attach to S satisfying, so there are nine

axioms, the cosmological group's axioms, and determined by its value on the field.

And when I say that we consider split cosmological groups, it means that the group we are considering are f values of G for f being a field. OK, so split cosmological group.

So everything I wrote up there is just about split cosmological groups.

You can work in more general setting. What we did really work for split cosmological groups, and I don't want to go further now. If you have more questions on this, I think it should wait a bit later. So it's then G of f with f field and G like this.

OK, and here, well, the first motivation was when f was a non-occupant local field of finite residue field, as usual. So maybe I can give an example so that you see that it's not that bad. So for instance, we have the so-called fine SLN. If you consider just the

functor that takes a ring and map it to SLN plus 1 of R of T minus 1, then this cosmoly.

OK, well, I don't need the code. And it's an instance of a fine cosmoly group if we want to go into detail. But anyway, just to say that, well, it's not kind of, if you look at this,

it's not so different from the group I used to work with, like periodic groups or everything, but a little bit. OK, maybe some remarks here. So first thing, I started to work earlier.

So you have this cosmoly group. So if you take a split cosmoly group, so here you put a field here, for instance. And you look at the corresponding, for instance, value group here saying that wv, so the vector of value group is finite, is the same as saying

that g is reductive. So that's why in the second, the results we got with Eber,

most of the time you will see when wv is infinite, because when it's finite, well, it's reductive case, and people already worked on this. Another remark which really shows the difference with the usual reductive case

is that in general, so you have all these linear forms here, alpha i, and you can look at the intersection of the kernels. And it's what is called the inessential part

of the problem. And this is non-zero. And this really brings trouble when you want to look at this building construction, as you will see in a moment. Are there any questions? So more definitions. So the second part is about,

I call it filters and measure. So I will say it now. So if you really want a precise,

so the English term for measure is Hovel, but the mathematical notion corresponding is a fine ordered Hovel. So there are extra conditions, like you can define marginally what is a Hovel, and then if you add this condition, you have this measure here.

And this measure really should be seen as the corresponding object for Cast-Mudy groups, as Brett's building, or for connected reductive groups. OK. So here, the point is not to do a lecture on filters and measure everything, just to show you that it's really, really more crappy than in the reductive case.

So for instance, I will start with kind of nice definitions. So we have, for instance, we have, as in the reductive case, we have vectorial chambers. So here,

they will just be of the form, so you can have a sign, but even if you don't care about the sign, it will be complicated. You just start from the fundamental chamber. You take an element of the vectorial value group, and you make the second one act on the first one. So this is

just the fundamental chamber, like all the elements on which all the alpha i's are positive. So far, it's not very complicated.

We also have vectorial faces. So basically, the same kind of construction. You take an element of the failed group, and you will take some part of E and look at elements

that vanishes when indexed by G and are positive when not indexed by G. So here, we have W, WV, and for G included in E, you just set this. It's vanishing on D, and positive otherwise.

So if we could just use this, well, we would just have buildings, and everything will be fine. But the thing is that we can't just use those. Maybe I will recall that the Titz cone,

so it's this quality. So it really plays an important role in what we do here. It's just the union of basically all the image of this fundamental, the closure of the fundamental chamber by the action of the failed group. And the thing is that

this Titz cone here can contain, well, infinitely many copies of this closure, which is not the case in general. Because here, WV can be infinite. And it's one of the reasons, not the only one, but one of the reasons why

when you really want to have the corresponding object, you can't just work with this vectorial object, but you have to go to filters. So more precisely, what will be a face, a facet? Of, let me see the measure to be, because I haven't defined it yet. So it will be

indexed by some element of the apartment, and some vectorial face like this. And it will be the germ at x of such a sector. So this is a sector. So it's, again,

filters. So it's a filter. It's a set of sets that satisfy some included condition. I mean, not aiming to make the thing. Maybe I can just say that if you take any of

any x in A and any omega containing it, this will just be the set of all the subsets in A

that contains a neighborhood of x in W. In omega, I said W in omega. So this is definitely not as nice as all the things here. It's really a filter. It can be

really, really, really big. And well, there's no nice topology on that kind of thing. And so you really have to try to figure things out. Maybe what I can say is that, for instance, in the Doretic theory, you're used to the fact that,

well, locally, you have finitely many chambers and everything's going well. Here, even for really, really nice group as SL2, a fine SL2, you can have infinitely many chambers, even locally. So it's really a mess. But anyway, even if it's a mess,

what goes on also, and so I will put three names here, I'm going to say, goes on. So we saw an event. So goes on. So they define the measure. They say, OK, we can

build something that will generalize what it's building. Russo made the whole axiomatic saying, OK, it's really behaving the same way, like you have five axioms and it's really the counterpart. And Hebert proved that, well, among these five axioms, there are a few of them you can exchange for one axiom, which is really nicer to work with. So we have an axiomatic setting

that attach to A and WV, a set currently in, called the measure attached to A.

And I phrase it like this because really, this construction only depends on the fact that you take an apartment and a while group, something that would behave like a while group. So in particular, you can start from G, in particular. We can consider measure attached

to a k-th-mod group. I mean, attached to its root datum.

The bold face A is the same as the curly A? No, wait, wait, wait. I changed the notation. Yes, this is this A here. Sorry. Thank you. Can I do this? The curly A is bold A. It's more used to this, sorry. It's because here, usually, the curly A is covering by apartment of the building.

And it's really the same here. You have this set, and part of the data is also covering by apartment, which are sets that have a structure isomorphic to this A here. So it's really the same thing as in the reductive case. OK.

And OK, I said it several times, but I never said it. When J is reductive, we recover. And really one, not a pair or anything.

This measure here is really at the heart of the construction of these two algebra here. So it's a lot of work done by all these people. So I will not resume everything in two minutes.

But maybe I can, before going further, I can define some groups that are of specific interest when one are interested in smooth representations. So starting from a face f, having a vectorial direction fv, we can consider

its pointwise stabilizer, which will basically only depend on fv. But I will just note it

f like this. So stabilizer under the action of g on i. So maybe let me

make it, if not yet, really clear. So from now on, I really fixed a split-cast multi-group. So I just look at g of f or f, even a local non-archimedean field.

I consider the measure attached to this group. And so the group acts on its measure. And under this action, I can look at the facet, at the face, and look at its pointwise stabilizer, its kf. And this kf here, well, if you think that what's happening in the Brett's building theory, it's kind of parahoric subgroups. And well, for the nicest one of them,

they are open compact. And they are really used to define this spherical Hecke algebra and Iwari Hecke algebra. So the question here, maybe example, just so that you see that it's really the nice objects we expect to. So if you, for instance, look at the

subgroup coming from just the origin of the measure, you just have the integral points. So this, well, you want it to be a maximal open compact, if you can make

this happen. Another case, when you look at the fundamental chamber, where you have what Bardiport, Goussaint, and Rousseau called an Iwari subgroup, because it's really,

I would love to write it e, but I already have an e, so I will just not do it. But it's really the Iwari subgroup you get when you're in the reductive setting, really just the pointwise stabilizer of this chamber here, this fundamental chamber.

And if you start from this k0 here, you will get the spherical Hecke algebra. So I will be a bit, a little bit correct, but a bit vague. You just look at the invariant function under k0 here, this is k. Well, not really on j, but on something a bit smaller. And

if you look at this, then you can see that you have a convolution algebra, and this really gives you back the spherical Hecke algebra in the reductive case. So that's what Goussaint Rousseau did. To be really, really complete, I should say that there are also a work of

Braverman, Kashdan, and Patnaik, but it's under a little bit more restrictive assumptions, like they assume the group to be split, untwisted, and fine, I think. So in this setting, so Braverman, Kashdan, and Patnaik, they also have this spherical Hecke

algebra. But Goussaint and Rousseau went a bit further, because they do this in full generality for any cosmological groups, and they got this Satake isomorphism also. And if you start from this Iwohori subgroup, then so again under the same assumption,

Braverman, Kashdan, and Patnaik, and on the other hand, so in full generality, Bardipense, Goussaint, and Rousseau, they define this Iwohiri Hecke algebra that we will really be interested in from now on. And again, you can see this as a convolution algebra of

iB invariant function. But you have to be a bit careful, because to define this convolution

product, you have to go in a bigger algebra. You have to go in a Bernstein realistic algebra. What you don't need to do in the reductive case, but in the cosmological groups, you have to go into a bigger algebra, so that this convolution product, a priori is well-defined, and then you show that you really stay in this set, what's make, turns it into real algebra.

So here, kind of the naive hope was, well, maybe we can put a topology on j such that this would be, well, at least this one would be open compact, maybe maximal, and maybe this one would be at least open. And the first result we have with Hébert is that this cannot happen. So this is a consequence of something a little bit more general,

that I will state now. So we assume that the vector of a group is infinite. Otherwise,

we're in the reductive setting, and well, we already know what's happening in this case. We take f type 0 facet. So what does it mean to be type 0? Just mean that all the

vertex are in the wv orbit of O, complicated. Then there is no topology on j such that kf

is open compact. So in particular, there is no topology on j such that this k0, which is ks,

or this kcvf, maybe open compact. So it was a bit not disappointed, because we didn't know what to expect. But it was kind of a bit frustrating, because we were already like, OK, if we have this, then we can make compact induction and everything. So now we have to be a bit more careful.

But then we say, OK, if this smoothness thing doesn't work now, maybe we can see whether this Hecke algebra are the kind of reasonable Hecke algebra. And here we had another surprise. So maybe I should start another part here. So we say, OK, well, we have this

Hecke algebra. Let's try to compute its center. So the goal was to see whether it is isomorphic

to the spherical Hecke algebra. So maybe from now on, I will just put these two notations here.

I will try to stick to them until the end. And so by Gosan and Rousseau, we know that there is a Sateke isomorphism here that says that this spherical Hecke algebra

is the fixed point in a so-called Ljunga algebra. So I will define this. I will tell you precisely what are these objects here. But what you have to notice here is that this element can have infinite support. I mean, they can have infinitely many non-zero coefficients, if you think

about them as a formal series. It can have infinitely many non-zero coefficients. And the thing is that, unfortunately, in this Euler-Hecke algebra, that cannot happen. So more precisely, what we proved here, I will put this as a proposition. So the center is isomorphic

to this. So these two things are a spot of y. But it's very explicit. I mean, you can compute this. But what you have to remember from this is that here, you have finite support. I mean, you only can have

finitely many non-zero coefficients. Well, hey, you don't. And in particular, it is not isomorphic to H. And I mean, it's even worse than that, in the sense that,

well, there are some instances of G such that here, you just have scholars. You just have C. OK, maybe what I will do in this part here, I will write everything over C, because the main motivation was complex representation. But it works in a more general

setting. Like if you take a ring here that contains some z brackets, sigma, sigma prime, with sigma and sigma prime being parameters for Hecke algebras, then it's really working well too. So you don't have to stick to complex theory. So here, the question was, OK, is this

results? Just say that, in some sense, this Euler-Hecke algebra is too small. But if you think a bit about it, it's not that surprising, because as I said earlier, in this measure, well, you have infinitely many chambers. And this group being infinite,

you can well say that maybe finiteness is too strong. And maybe you want almost finiteness. And this is what I will discuss now. So I said that I will tell you what is this. So I will

tell you the first definition. So if you start from a subset of y, you will say it's

almost finite if the following condition holds. So if there is, so I don't know,

finite subset, let me call it j, such that. I have to write the condition correctly.

Can I ask a very stupid question? What is C of y in double square brackets? That's what I will define, that the point of this part is to define this object here. It is Loijenga algebra. So it's in a paper of Loijenga. Yes. Here, you mean? Yes. So that one, I will define it here. The point of all this section

is to tell you what this is, and then to explain why we made a bigger algebra to see this happening. So this is this condition here, which is that.

If you take an element of this subset, then you can always dominate it by an amount of a finite set. So here, this domination thing, it's a pre-order on the building, and it just means that nu minus lambda is with positive coefficient. You can always write this

as a linear combination of co-routes, and what you want is that all the coefficients are positive. Okay. And having this, so there is a lemma. It's a technical lemma, but it helps to make sense of the next definition.

If you take e, a normal finite subset of y, then for any e prime,

so here, no condition on e prime, there is a finite part g of y, such that the intersection

of e and e prime is included in finitely many translates of this Kachetz plus here.

So this tells us in some sense that, okay, it's not finite, but if you look in finitely many slices that may be infinite, then you can manage to make things work. And now, what is this c double bracket here?

So a priori is just the set. So it's formal series, so indexed by y, as you can guess.

So what do you assume here? So you want that this family here, so it's a part of y, and you want it to be almost finite. So when I say this, it's support,

like the set of lambda such that a lambda is non-zero is almost finite. And this e lambda is just a family of symbols that are multiplicative symbols,

like what you want for a basis. You want that a lambda times a mu is a lambda plus mu

for any lambda in y. And the L'Huyenge theorem says, so that's what it's called. So you can really define a structure of algebra on this.

Okay, so it's an inventionist 61, if you want the full reference. This is an algebra, a complex algebra here. But again, you don't really need to be oversee, you can be over any kind of big enough ring. Okay. And similarly, one can define,

so following the same model, L'Huyenge algebra for y plus, and for y plus plus,

and I should tell you who these people are. So here y plus, it's just the trace of y on the titz cone, and y plus plus, it's even smaller, it's a trace of the closure

of the fundamental chamber. So again, just want to here, that would be finite if you were in the reductive case here. You have an essential part, so you have to be careful. But anyway, you can define the same thing. And again, you can have two algebras like this, that will be sub-algebras of this big one. And what we proved, so maybe I

give you a statement that interests me. So here, now you just look at the action of w, v, and y,

and you look at elements that are invariant under this action. What means just that here, if you take a lambda or aw dot lambda, it's the same thing, just usual action.

So if you look at this object there, what you can prove, what we prove, is that actually it's embedded in this thing here. So it's not that big. I mean, everything is in the positive part, everything in the titz cone, which was not obvious at all at first. So if you take the w, v invariant in this algebra, it's contained. Actually, we proved

something a bit stronger, but I just did this here. Because actually, what we did is that we

said that, OK, we know how to describe this in terms of support, in terms of these coefficients, and in particular, this proves that this is true. But having this, we can define this complicated algebra. You mean contained, or there is a monomorphism?

So there is a monomorphism, actually. So actually, we have a bijection, which is compatible to the algebra structure, and in particular, this is. But here, I phrased it like this, because what we are really interested is this support condition. I mean, we really want to have control on when this coefficient does or doesn't vanish.

And that's why I just need this here. So I've completed. So I could even say the completed algebra, because so far, we just have one algebra.

So inspired with this almost finite subset things, we define a notion of almost finiteness in y times wv. So a subset e wv times y plus is almost finite. So here,

we have a very strong condition on the w part. So this condition is a finite-test condition.

So what do you want is that the set of w's that may show up in this subset. So you want this one to be finite. And for any such w, you want this corresponding set of

element in y plus to be almost finite. And I can write it like this, because sometimes it's just empty. So almost finite, as here, up there. You can do the same for y plus.

And now we can, so we define a set. And we prove that this set is not just a set,

it's an algebra. And it's kind of the right algebra. So finally, so we let H naught

be as a set. OK, maybe I should write it like this, of elements. So it's formal series, really, really formal here. So here, just a notation for now. I will

just say a word on this later. So we set here being almost finite, as which is defined

here. OK, so here, I pre-read the symbol. But in reality, what we do here is that we see this in, again, the bigger algebra, Bernstein-Lussich, completed Bernstein-Lussich

algebra, where we have bases defined from such symbols. And we say, OK, let's look at just this formal thing and what we prove. And it's, well, one of the hard part of the thing. OK, maybe, yes, it's just, well, this one is just, what do you want?

First thing, this H-hot is actually a convolution algebra, in which

this Euler-Hickel algebra, defined by Bardipont, Gausson, and Rousseau, naturally embeds.

So I think it wasn't easy, because you really have to. I mean, we have really an explicit formula for this convolution product, but you have to check that it's well-defined, and then check that everything's fine. Well, it's a bit tricky. And then maybe I can say something further here.

So it naturally embeds as the subalgebra of elements with finite support.

So this shows up again that it's really a bit too small. And the main point in some sense, the center of this algebra is isomorphic

to the spherical algebra. So we really have this analog of what is happening in the reductive case. And well, just as a byproduct, the center of this Iwo Rijeka algebra that was built earlier,

it's really just a trace of the center of this completed algebra in the Iwo Rijeka algebra. So just two consequences, and then I think I will just stop here. Maybe a first remark. So in the reductive case, again, these centers will be the same, but it

doesn't mean at all that the algebras are the same. In the reductive case, we don't need that algebra here. And you can define it, but it's really, really, really big. I mean, you can have elements with infinite support even in the reductive case. The second thing, and maybe I'll stop here then, is that in the reductive case, I think you know that the Iwo Rijeka algebra is a finite type

as a model of its center. Here, if you look at this completed Iwo Rijeka algebra, then it's kind of obvious that it will not be of finite type over its center. So it's something really much more ugly in some sense, but you really need all these elements in there

if you want to really generalize what's happening on the reductive case. OK. Thank you for your attention. Are there more questions? Do you have some application of the existence

of this completed algebra? Some applications? Not yet. Not yet. I mean, we just worked this out. But I mean, now the idea is trying to understand what are the modules on this and maybe try to see whether it's the way you get what's happening

from the reductive case or not. Because this one here, again, looks a bit too small to have enough modules, for instance. But it's really in progress. The simple module, is it being functional? Oh. So if you allow your first one, is this no Jenga algebra

in the integral domain? I don't think it was. Project. Can you have vanishing from this? Not sure. I mean, when you compute this convolution project, you may have a theory.

But no, maybe because of this. Maybe because of this convolution. So how do you prove that this is well defined? You have kind of an argument, like the argument Zorgo used in a different context, saying that at some point,

you will have one coefficient only that can vanish if you take the support which is empty. And maybe using the same kind of argument, we could have some integrality here. But have them think about it.

So it's algebra as it has an interpretation as an endomorphism. That's the hope. That's really the hope. OK, my motivation behind all this is that trying to have some compact induced thing, like saying, OK, it's the endomorphism algebra of something that I would like to call the compact induced representation of the trivial character.

This one, well, it's not fully written yet, but basically does. But that one, we don't know yet. Thank you. Thank you.