 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 cours d'algebra 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 prevent me to go further. So that's all for the Godman part. So today I will report on a joint work with Auguste Hébert which is just there about Iwari Hecker Algebra and Mazur for Split Casmoudi Groups so it has very little to do with Automorphic Forms 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 Gausson asked me one day whether I've ever read about Schneider and Stuller's papers and if I think that it may apply to Casmoudi Groups so I could answer the first question that I read Schneider and Stuller's paper but that I had no idea but what was the Casmoudi Groups 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 Gausson at that time so here the let me say the motivation comes from Schneider and Stuller in the 97 papers in the Publication Mathématiques de l'IHS 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 a non-archimedeon local field resistive field okay 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 Brattitz building to this from this G-equivariant coefficient system they get a factorial way to get projective resolution of pi and so basically what Gausson was interested in is trying to see whether if now we don't take a connected reductive group but a Casmoudi group so maybe a split Casmoudi 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 Casmoudi groups where okay do we know what is a smooth representation and do we have some she something that would behave like Brattitz building for Casmoudi groups because if you know a little bit about Casmoudi groups you know that under some in some in some settings you can attach to them a pair of twin buildings but not one building and well it's not very convenient in this setting but here why Gausson has this is that with Rousseau they build something that could replace this Brattitz building which is called a mazure so I will come on this a bit later just because it's a bit fun why did they call this mazure 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 you really don't want to live in there so that's why you call like this okay so the two brown underlined words are really what we will start from to explain how how we arrived to Iwori Heckelgebra so of course they didn't know what smooth representation were and it was even worse so just phrase it like this so maybe okay 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 an open subgroup of G and if you go a little bit further and are kind of more optimistic you can modernary do we have let me say a nice topology on G so in the Kasmo G 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 when G comes from so in the in the reductive case so I will be more specific on this later but I maybe I can already do a little spoiler here the answer is no so far we don't have such a topology so about smoothness it's okay a bit tricky so we were kind of okay let's not talk about smoothness for now and maybe focus on the the other part and the having the other part so when you have this generalization of what is building which is this mazer this mazer you can also define HECA algebras so not as many HECA algebras as we have in the reductive case but at least two main HECA algebras so first the spherical HECA algebra so basically just think of this as the convolution of algebra of a bin variant function under the integral integral points of G and the classical HECA 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 the 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 that so in the in the in the spherical case here Gausson and Rousseau managed to prove that we have an isomorphism between this spherical HECA algebra and the same analog of what you got in the reductive case using satake 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 HECA algebra and one thing we check is that well if you take this construction it doesn't work it's not true so starting from this we'll build another Iwori HECA algebra and this will be the third part of my talk that really again kind of generalize what happened in the reductive groups case and which moreover satisfies this analog of Brunstein theorem okay so that's kind of the motivation introduction 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 prove that we cannot let me say mimic what exists in the reductive case to get a topological group when when ecology is cosmodine on reductive we showed that the center of this Iwori HECA algebra is too small and sometimes it's really really small like it's just colors and I think we built a completed Iwori HECA algebra which satisfies an analog of Brunstein theorem meaning that the center is really isomorphic to the spherical HECA algebra okay so this is what I want to discuss if I have enough time I will discuss another construction we did that gives more brings more HECA algebras in the game more in what already exists in the reductive case but only if I have time so this is all in a in our joint paper so if you want to check this okay so now so 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 a cosmodine algebra and how you attach to this following tits construction cosmodine 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 that paper okay so first definition so here I'm sorry I will go with a bunch of notation and definitions which will not be the funniest part of the talk but it's really really necessary um so what we call a casmudi data so it's really like in the in the reductive case that you would have a tuple of data and you will attach to them some kind of combinatorial and geometric objects and so it's just a tuple of five elements and where the notation are really the usual ones so you have two models roots core roots so indexed by a finite group by a finite set so a here will be just generalized carton matrix x and y free z model of finite rank data 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 tch i and y it's the core root and okay let me really loosely say they are compatible with a meaning that the coefficient of a are given by the duality between alpha a and alpha tch g for a g okay 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 usually is just the scalar extension of y to r so i will denote it by curly a so there will be many a's okay um 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 i'm always mistaking the v alpha i tch so here you're still in in a and you then can define the vectorial veil group so it's the veil group but we like to refer it as a vectorial one because of this construction it's just a group generated by all these r i's in gla and um okay should i make the remark now yes okay um note and it's really important uh saying that wv is finite is the same so when okay maybe uh i'm coming no i will say this later when you have defined g and then i will make this remark um so we also have an assigned veil group and an order so i mentioned here the existence of this defined veil group so we will not explicitly see it appear in what we we will see today but it's really related to this euoraheca algebra it allows you to get a presentation of this euoraheca algebra for instance um it's just a usual way to define it you just uh semi-direct pro let's go ahead and qch qch is just the core root that is okay um okay and the thing is that when you got such data you can follow what it's did and you get a function so there is currently g so it defines above on the category of rings has value in the groups okay um i will be really slowly here um so attach to s satisfying so there are nine axioms the cosmology groups axioms and uh determine by its value and when i say that we consider split cosmology groups it means that the group we are considering are f values of g for f being a field okay so split cosmology group so everything i wrote up there is just about split cosmology groups uh you can work in more general setting what we did really work for split cosmology groups and i don't want to go further now uh if you have more question on this i think uh it should wait a bit later um so it's then g of f with f field and g like this okay um and here well the first motivation was when f was a non non-argumentary local field of finite crusade you feel 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 a fine sln uh if you consider just the function that takes a ring and map it to sl and plus one of r of t t minus one okay um then this cosmology okay well i don't need the code and it's an an instance of a fine cosmology group if we want to go into details 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 was used to work with like pietic groups or everything but a little bit okay um maybe some remarks here so first thing i was i started to work earlier so um if you start so you have this cosmology group so if you take a split cosmology group so here you put a field here for instance and you look at the corresponding uh for instance veil group here um saying that w v so the vectorial by group is finite is the same as saying that g is reductive so that's why in the second the results we got with ever most of the time you will see when w v is infinite infinite because when it's finite well it's reductive case and people already worked on this okay um another remark which really shows the difference with usual reductive case that in in general so you have obviously no forms here alpha i then you can look at the intersection of the kernels and it's what is called the in and social in and send inessential part of the apartment and this is non-zero and this really brings trouble when you want to look at this uh building construction as you will see in a moment okay um um are there any questions so more definitions so the second part is about i will i called it filters and and measure so uh i will say it now so if you really want a precise um so the the English term for measure is hovel but the mathematical uh that notion corresponding is a fine ordered hovel order to find so there are extra conditions like you can define more generally what is a hovel and then if you had this condition you have this measure here and this measure really should be seen as the corresponding the corresponding object for casmody groups as brought to the building or for connected reductive groups okay so here the 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 uh for instance i will start with kind of nice definitions so we have uh for instance we have as in the reductive case we have victorial chambers so here there 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 victorial 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 so far it's not very complicated um we also have victorial faces so um basically the same kind of construction you take an element of the fail group and you will take some part of e and look at elements that vanishes uh when indexes by g and are positive when not indexed by g so here we have w w v and for g included in e you just set this it's vanishing only 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 um maybe i will recall that the tits 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 veil group and the thing is that um this tits cone here can contain well infinitely many copies of this closure which is not the case in general okay because here w v can be infinite um and it's one of the reason not the only one but one of the reason 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 uh a face a facet um of let me see the measure measure to be because i haven't defined it yet so it will be uh indexed by some element of the apartment and some vectorial face like this and it will be the germ such a sector okay so this is a sector so it's again filters so it's a filter it's a set of it's a set of set that satisfies some inclusion condition i mean not aiming to make the the thing maybe here i can okay maybe i can just say that uh germ if you take any of any x in a and any omega containing it this will just be the set of all the subsets a that contains a neighborhood of x in w in omega i said w in omega okay um so this is definitely not as nice as all the things here it's really a filter it can be really really big um and well there's no nice topology on that kind of thing and so you really have to try to figure things out um maybe what i can say is that for instance uh in the build brettitz theory you're used to the fact that well locally you have finally many chambers and everything's going well here even even for a really really nice group as a sl2 a fine sl2 you can have infinitely many chambers uh even locally so it's it's a really a mess but anyway even if it's a mess um um what gosson rousseau so i will put three names here i'm saying gosson rousseau rousseau and ebert so gosson rousseau they define the the measure they say okay we can build something that will generalize what is building rousseau made the whole axiomatic saying okay it's really behaving the same way like you have five axioms and it's really the counterpart and ebert 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 um attach to a and wv a set currently called the measure attached to a and okay and i i phrase it like this because really this construction only depends on the fact that you take an apartment and a vial group for something that would behave like a vial group so in particular you can start from g in particular one can consider to a cosmology group i mean here it attached to its good datum the boldface is the same as the curly a and uh wait wait wait i've changed your notation yes this is this a here sorry thank you um can i do this the curly a is a bold a it's more used to this sorry um it's because here usually the curly is covering by apartment of the building and it's it's really the same here you have this set uh and part of the data is also a covering by apartment which are sets that have a structure isomorphic to this a here so it's really the the same thing as in the relative case okay um and okay i said it several times but i never said it when j is reductive we recover and really one not a pair or anything okay and this measure here is really at the heart of the construction uh 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 um so starting from a face having a vectorial direction fv you can consider its point wise stabilizer which will basically only depend on fv but i will just note it uh f like this okay so stabilizer under the action of g on i so maybe let me uh make it if not yet really clear so from now on i really fix a split-cast moody group so i'll just look at g of f for f even a local non-archimedean field uh i consider the measure attached to this group and so the group acts on this on its measure and under this action i can look at the facet at the face and look at its point-wise stabilizer it's kf and this kf here well if you think that what's happening in the brettitz 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 hickory algebra and eowory hickory algebra so the question here maybe example just so that you see that it's really the nice objects we we expect to so if you for instance look at the subgroup coming from the just the the origin of the of the measure you just have the the integral point okay so this well you want it to be a maximal open compact if you can make this happen okay um another case if when you look at the fundamental chamber where you have what uh bar di porso gousson and rousseau called an eowory subgroup because it's really okay i would love to write it e but i already have a e so i will just not do it but it's really the eowory subgroup you get when you're in the redix setting really just the point-wise stabilizer of this chamber here this fundamental chamber and if you start from this k not here you will get the spherical hickory algebra so i will be a bit um so it would be correct but a bit vague like you just look at the invariant function under k k not here that is k um well not really on j but it's on something a bit smaller and if you look at this then you can see that you have a convolutionalgebra and this really gives you back the spherical hickory algebra in the reductive case so that's what gousson also did to be really really complete i should say that uh there are also a work of bravaman cashdan and pat naik but it's under a little bit more restrictive assumptions like they assume the group to be split and twisted and fine i think so in this setting so bravaman cashdan and pat naik they also have this spherical hickory algebra but gousson and rousseau went a bit further because they do this in full generality for nica smoothie groups and they got this satakai isomorphism also and if you start from this iwo horrisome group then so again under the same assumption bravaman cashdan and pat naik and on the other hand so in full generality bar dipens gousson and rousseau they define this iwo hickory algebra that we will really be interested in from now on um and again you can see this as a convolutional algebra of um i invariant i b invariant function but you have to be a bit careful because to define this convolution product you have to go in a bigger algebra okay you have to go in a Bernstein-Loustich algebra what you don't need to do in the reductive case but in the kasmody groups we have to go into a bigger algebra so that this convolution predict a priori is well defined and then you show that you really stay in this set what's make turns it into an algebra okay um 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 hiver is that this cannot happen so this is a consequence of something a little bit more general that I would state now so um so we assume the vectorial by room is infinite otherwise we're in the reductive setting and well we already know what's happening in this case um we take f type zero facet so what does it mean to be type zero just mean that all the vertex are in the w v orbit of o complicated then there is no topology on j such that k f is open compact okay so in particular there is no topology on j such that this k not which is k s or this k c v f maybe open compact so it was well it was a bit um not disappointed because we didn't know what to expect but it was kind of a bit frustrating because we were already like okay if we have this then we can make compact induction on everything so now we have to be a bit more careful um but then we say okay if this smoothness thing doesn't work now maybe we can see whether this hacker algebra are the kind of reasonable hacker algebra and here we had another surprise so maybe I should start another part here so we say okay well we have this e-worry hacker algebra let's try to compute its center so the the goal was to see whether is isomorphic to the spherical hacker algebra that I don't like this so maybe from now on I will just put these two notation here I will try to stick to them until the end and um so by Gosar and Rousseau we know that uh there is a sataki isomorphism here that says that this spherical hacker algebra is um the fixed point in a so-called logenga algebra so I will define this okay I will tell you precisely what are these objects here um but what you have to notice here is that this element can have um infinite support I mean they can have infinitely many non-zero coefficient okay if you think about them as a formal series uh it can have infinitely many non-zero coefficient and the thing is that unfortunately in this e-worry hacker algebra that cannot happen so more precisely what we proved here I will put this as a proposition so the center is isomorphic so it's so these two things are sub as as part of y okay but um 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 uh uh finitely many non-zero coefficient while here you don't uh and in particular config 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 columns you just have c okay um okay maybe uh 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 modern setting like if you take a ring here that contains some z brackets sigma sigma prime with sigma and sigma prime mean parameters for heck algebra then it's really working well too so you don't have to stick to complex theory so here the the question was okay is this uh this result just say that in some sense this e-worry hacker algebra is too small um but if you think a bit about it it's not that surprising because as I said earlier in this mazur well you have infinitely many chambers and this is very good being infinite you can well say that maybe finiteness is too strong and maybe you want almost finiteness and this is uh whatever yeah still have to whatever we discuss now so I I said that I will tell you what is this so I will do it this now okay so um 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 uh let me call it j such that and I'll have to write the condition correctly can I ask a very stupid question what what is c of y and double square that's what I will define that the point of this part is to define this object here it is loy jenga algebra so it's in a paper of long jang yes here you mean yes so that one I will define it here that the point of all this section is to tell you what is this is and then to explain why we made a bigger algebra to see this happening okay um so this is this condition here which is that if you take an element of this subset then uh you can always dominate it by an element of a finite set so here this domination thing um it's a preorder 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 coroutes and what you want is that all the coefficients are positive okay and having this so there is a lemma bearing it's a technical lemma but it helps to make sense of the next definition if you take e and almost finite subset of y then for any e prime so here no condition and e prime there is a finite part g of y such that the intersection of e and e prime is included in finitely beneath translate of this kuchach plus here this is so this um 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 uh seedable bracket here so so a priori is just the set so uh it's formal series so index 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 okay so when i say this it's it's support like the set of lambda such that a lambda is non-zero is almost finite okay and this e lambda is just a family of symbols that are multiplicative symbols like what you want for basis you want that a lambda times a mu is a lambda plus mu for any lambda and the lujanga 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 um this is an algebra a complex algebra here but again you don't really need to be overseas you can be over any kind of beginner frame okay and um similarly one can define so following the same model lujanga 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 tits 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 the reductive case here you have an essential inessential 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 will just give you the statement that interests me that uh so we proved that um so here now you just look at the action of w v on y and you look at elements that are invariant under this action what means just that here if you take a lambda or aw.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 tits cone which was not obvious at all at first so uh if you take the w v invariant in this algebra it's contained so actually we proved something a bit stronger but i just need this here okay because actually what we did is that we said that okay we know how to describe this in terms of support in terms of these coefficients and in particular this prove that this is true but having this we can define this combinated algebra contained or there is a monomorphism so there is there is a monomorphism actually i mean it's really so actually we have a bijection which is compatible to the algebra structure and in particular this is but here i phrase 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 couldn't say the completed e-woaric algebra because so far we just have one e-woaric algebra um so uh in spite with this almost finite subset things we defined a notion of almost finiteness in y times w v okay so a subset e w v 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 state of w's that may show up in the in the in this subset uh so so you want this one to be finite and for such 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 okay so almost finite uh 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 not be at the set okay maybe i should write it like this of elements um so it's formal series really really formal here so here just a notation for now i would just say what on this later uh so with the set here being almost finite as which has been defined here okay so here a pre-regis symbol but uh in reality what we do here is that uh we see this in the again a bigger algebra Bernstein-Loustich completed Bernstein-Loustich algebra uh where we have bases defined from such symbols and we say okay let's look at just this formal thing and what we prove and it's well one of the hot part of the thing okay maybe um yes it's just this one is just the first thing this h-hot is is actually a convolutional algebra in which this e-wall-hacker algebra defined by Bardi-Pence-Gousson and also naturally embeds so it's a for thing 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 okay check that it's well defined and then check that everything's all fine and well it's a bit tricky and then the uh uh maybe i can say something further here so it naturally embeds as the sub-algebra of elements with finite support okay so this kind of 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 hacker algebra so we really have this analog of what is happening in the reductive case and well just as a by-product the center of this e-wall-hacker algebra that was built earlier is just that it's really just a trace of the center of this completed algebra in the e-wall-hacker algebra okay so just two consequences and then i 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 i mean 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 think you know that the e-wall-hacker algebra is a finite type as a model of its center here if you look at this completed e-wall-hacker algebra then it's kind of obvious that it will not be of finite type of over its center so it's something really much more ugly in some sense but you really need all this element in there if you want to really generalize what's happening on the reductive case okay now thank you for your attention you have some application of the existence of this some applications not yet not yet i mean we just work this out but i mean now the idea is trying to understand what are the models 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 models for instance but it's really in progress so if you allow can you have vanishing from this i'm not sure i mean when you compute this convolution project you may have a theory but no maybe because of this um maybe because of this condition i mean so how do you prove that this is well defined that you have kind of an argument like the argument Zoggle 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 arguments we could have some integrity here but uh have nothing about it so does it have an interpretation as a that's the hope that's really hope so the okay my motivation behind all this is that trying to have some compact induced thing like saying okay it's the undermorphism 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