 It's a great honor and a great pleasure also to talk here, to Alan's birthday. So I have been knowing him for approximately 30 years. I mean, I think the first time we met, it was in Oba Bolfa, 88, no, don't spoil what I'm going to say. Okay. Okay. I hope I will correct myself also later. 88, no, it was something special because I met you there with Pierre at the same time and it was a really lucky event, scientifically speaking for me. So we collaborated on a regular basis from time to time. I mean, I collaborated with you and with Pierre and sometimes together, all three. And actually you're right, I mean, the first time it was reminded to me, it was recorded to me by Gerhard Racher who is here, that it was actually in Luxembourg in 87. And I don't remember the talk you gave, I mean, I hope you excuse me. But I think it was related on something I'm going to mention later in my talk, but it could be something different because as you all know, Alan has a broad range of interests so it could be something different. But you would tell me what, I mean, I guess it's related to what I'm going to mention. By the way, at the same meeting there was also George Mackey there and it's strange but what I'm going to talk about is strongly based on part of the work of Mackey. So I will start talking about dual groups, actually, the title could have been dual groups of infinite groups. So I would just, here is a plan, I mean, I would talk very quickly on finite group and then I move to representation, that's the main focus, unitary representation of infinite groups and then dual groups which one can associate to discrete groups, discrete infinite groups and then characters. I would explain this. So just a reminder, if you have a finite group, G, then you have a finite dimension representation, you can decompose it as a direct sum of irreducible presentation. One fact is that this decomposition is not unique, I mean, if you have a multiple of an irreducible presentation, you can decompose this in several different ways but you can get uniqueness if you actually, if you put together multiples of irreducible presentations. Then you get to unique decomposition, this is what's called the central isotypical decomposition. So you have uniqueness if you do this and, okay, so you cannot hope for more. Now, there is a way to study representation of groups, finite groups. That was, as I noticed, first considered by Amy Nutter in 1929, you go to the group algebra and then you extend every representation and you get the representation of the group algebra and then you have, I mean, bijection, you can translate everything, you can translate the representation theory of G, the representation theory of this ring and it preserves everything, I mean, reducibility equivalence and so on. And then you have something, you can actually parameterize the reducible representation by their kernels, the kernels in the group algebra. These are maximal ideals, it happens to be maximal ideas but these are also primitive ideas in the sense that a kernel of reducible representation are called primitive ideals, so it happens to be the same for finite groups. Now, this is a way to study representation of a group. Another way, which was actually not 2,000, I mean, not 1990 as you may think but 1890 by Frobenius, who actually introduced the subject, I mean, systematic way of representation theory, representation of finite groups. And actually, he already noticed that you may consider characters of representation and this is a more, of course, more concrete object, it has function, simply function on the group and then every representation is determined by its character. And you get to bijection actually between, you can parameterize the representation by their characters. The characters in this case is the space, every character is a trace function, is a central function, class function, a function which is invariant under conjugation and special ones, I mean, there's our minimal central idempotence of the group algebra. So this is a way, I mean, without knowing that's what you learn already, I think in the first year at university, sorry. You need a certain scholar to make it an idempotent, I think. You have a card? Yeah, yeah, yeah, normalized. Yeah, we have to, we have to normalize, you're right. We have a normalization central idempotent normalized in the right way. Okay, this, you learn that it's possible to determine actually, sometimes it's easier to determine this class function than determine the irreducible presentation. That's what already Frobenius did. Now, we move on to general groups. I mean, if you have a locally compact group, and you may consider or I will consider only unity representation in here by space. You can also, of course, take Banach spaces, but then I mean less is known in general. So we have a notion of equivalence, reducibility, natural notion. And then we look at the unitary dual, which is the space of irreducible presentation, of equivalence classes of irreducible presentation. And this is, I mean, the aim is usually to, at least the first aim is to determine this set. And but the second one is connected with the composition of repetition. I mean, determining irreducible representation is not the whole story. You have also to, for instance, I mean, maybe the most important representation that already for finite group is the regular presentation. So you want to decompose given natural representation in terms of irreducible. Actually, the first thing is that, the first thing which happens is that you direct some, do not suffice. You have to consider something, I mean, more liberated, but similar. I mean, no problem. I mean, that you have to look at the direct integral of representation, of irreducible presentation in this sense. And this already happens in the, for Rn or Zn if you want. I mean, the regular presentation is actually not to direct some of the irreducible one. Reducible representation of Rn are simply the character here, one dimensional. There is no invariant one dimensional subspace of the regular presentation of Rn. So if you, the Fourier transform is actually gives you, you may interpret this as the composition direct integral of irreducible presentation. So this already happened. I mean, there is nothing, nothing strange about this. Okay. Now, some group, for some group actually, we are going to define them a little bit later, which are called a group of type one. You have indeed, I mean, satisfactory decomposition theory. You can decompose every representation as a direct integral of multiple of irreducible one, exactly as the case of finite group. And this composition is unique exactly in the same sense as the, for finite group. But something new happened here for other groups which happen to exist. You may have the composition of a representation in irreducible, which are completely disjoint to each other. Nothing to do one with the other. And it means that you have the composition of a set of irreducible presentation. Another one about set of disjoint set of irreducible presentation. So that's really strange and but you can see this already for free group in 60 generators. It works also, of course, for two generators. If you take the copy of Z generated by one of the generators, you induce this and you take a direct, sorry, you take characters, which are simply parameterized by this circle. You induce these characters to the group and you take direct integral. So this thing is irreducible. This is by Mackey's theory. And they all are pairwise non-equivalent. This is also part of Mackey's theory. So you get really for every eye a different, so 60, decompositions. So that's not satisfactory, of course. It tells you that already that the irreducible representation are maybe not the right object to consider for groups like free groups. And that's the point of my talk actually. You have to replace the reducible representation, the space of reducible representation by some other spaces. Okay, first we saw already in the center of the composition you have to look at multiples of the irreducible representation. These are factorial representation. So if you have a representation, you can look at the closure for the weak topology of the linear span of the operators given by the group, by the representation. This is a fundamental algebra. And it's called a factor. These are the building blocks, actually, a fundamental algebra. If the center is trivial. And the representation is said to be factorial if it's the center of this degenerated fundamental algebra of pi of g is trivial. So you have examples. Already the building block we saw for finite groups multiple of reducible representation or factorial representation. But the thing is that you have other, there are other factorial representation which are not of this type. And this was discovered by von Neumann and Murray. And this is actually the reason why there is a theory of von Neumann algebra. If you look at the regular representation of the free group, two generators, or PGLDQ, then this is a factorial representation but it's not a multiple of an irreducible one. So there is something which is happening here. And this actually one could think that the factorial representation are maybe the right, you should replace the irreducible representation with the dual space by the factorial representation. But be aware that the irreducible representation are, of course, factorial. So you are getting more. So it's not the right space. So you are not going to take all factorial representation coming to this point. And so, again, if you look now, if you replace irreducible the composition, sorry, the composition irreducible pieces by the composition in factorial pieces, then you have a satisfactory decomposition theory. The free representation decomposes as the direct integral of factorial representation. And this is somehow unique if you, okay, in the precise sense which is given here. So factorial representation are better, but of course, I mean, there is still a problem of determining them. And here I'm coming back to a group of type one which I didn't define that we introduced already. So a group of type one is simply defined as a group for which you don't have factorial representation besides the obvious one. So the one which are multiple of irreducible representation. And there is a natural borrow structure on the space of unitary irreducible representation which is called Machiborid structure. And here is a basic fact about this group. And actually a negative fact. It tells you that if the group is not of type one, then you have no chance to determine the space of all irreducible representations up to equivalence. And said in another way, I mean, you have this space is a standard borrow space if and only if the group is of type one. So for a group which are not of type one, you have really to do something else. To look for other new spaces. And here are some examples. I mean, a group of type one, a billion groups are a billion, for instance, a billion groups, semi-simple lay groups, need-potently groups. But there are already among the lay groups some groups which are not of type one, some solvable lay groups. And nearly all interesting infinite discrete groups are not type one. There is a theorem by Atoma in 1964 that unless G is virtually a billion, G is not type one if G is discrete. And if it's virtually a billion actually, presentation theory is something you can determine. So there is no problem. So interesting infinite discrete group are not type one. That's the moral of the story. Now I'm coming back to this idea of Emy Nutter which you can perform for infinite groups. Now you have to replace the group algebra by the c star algebra of the group which is a completion of this norm here of the group algebra. So I'm taking G discrete, but we can do the same in a similar way, this construction for any locally compact group. And then you have, as we saw in the for finite groups, you can extend every representation, unitary representation of G to the representation of this c star algebra. And you have a bijection between the representation theory of G and of the c star algebra. Now we can look as we did to space of kernels of this extended representation, so this kernel in the c star algebra. And okay, this is a new dual space. For finite group we saw we had the bijection. And the question is, I mean, is this a reasonable dual space? There is another way to introduce this space. I mean, this space is simply, you can look at it as simply, forgetting about the c star algebra. At the group level, you can simply say this is the, it classifies the reducible representation not up to equivalence as G hat, as does G hat, but up to weak equivalent. So weak equivalent means that every matrix coefficient of pi can be approximated by matrix coefficient of sigma and vice versa. So this is space of reducible representation, quotient space, modulo, stronger or weaker, I don't know, I mean, equivalence relation. So with bigger classes. And it has a natural topology. This is, I mean, Jacobson topology. And when G is type one, you don't get anything new in the sense that this map is a bijection. So that's really type one in this sense. I mean, several senses now, as we saw, similar to finite groups. With some technical difficulties you have to replace direct sum by direct integrals and so on. So actually, the things are working very well. Now, when G is not type one, what does happen? There isn't a remarkable result by Efros 63. This prime G is always a standard borrowed space. Probably one should have a second countable, but my groups are always second countable. So for this talk. So it's a standard borrowed space. So it means that this space is, at least there is no abstraction, no obvious abstraction to classify this space. So this is something one could do for some groups, discrete groups, maybe others. Now we, so this part of one program, I mean, was to translate Amy Nurture's approach to unitary representation of a infinite group. We have still, what's missing is the part by Frobenius with the characters. And that's something you can do. I mean, okay. There are some problems. I mean, we are going to define characters. It's something more complicated as you see. Characters are something which live on the group C-cell algebra. These are linear functional on an ideal here, which is, I mean, they are finite value. It should be something like a measure, but it could be infinite. I mean, we are, of course, used of measure which could be infinite. I mean, and that's the same thing. There is a part where it's finite. This is an ideal. And it could be, I mean, on the other elements of the C-cell algebra, it could be infinite. And there is a condition of semi-finateness. You can approximate every element in the C-cell algebra by elements with finite trace. And you have this trace property here. And it's a character if it's somehow, this condition somehow tells you that it's extreme in the set of all traces. And this is the space of your characters. And sometimes, or you are lucky and the character is finite, so it means that it takes only finite value on all the C-cell algebra, the whole C-cell algebra. This corresponds somehow to probability measure. And we denote by E of G the set of finite characters. So in your condition, the second condition is the point for every x there exists a y. Now the plus notation is the positive definite elements, yes? Yes, yes, the positive element in the C-cell algebra. But then it is 0, it's also a. So every x is approximated somehow or dominates something which is non-zero. Like every non-zero x. Non-zero, of course. Okay, so these are characters. So it's difficult to digest and we will see some examples. Here are a few examples. I mean characters of type 1. Here how you get characters of type 1 over what's called type 1 characters. Assume that the image of the C-cell algebra you know, I recall that you can extend pi to the C-cell algebra. If it contains all compact operator it's irreducible. I mean for this it's the fastest to contain one compact operator. You look at the ideal of the elements of the C-cell algebra which are mapped to compact operators. This is a two-sided ideal and you get a trace on this. Simply the natural trace gives you a character. So this is one way of obtaining characters on the group. Okay, we will see examples of this. Actually finite characters are easy to understand. Finite characters are simply given our continuous class function which are of positive type and normalized and extremal with this property. So finite characters are actually easy to understand. These are simply the class function with special property. Infinite characters are actually more mysterious. For lead groups for instance, usually but there is no general theorem about this given by conjugation invariant measure or distribution. So these are still more complicated objects but somehow familiar objects. Now I come back and try to relate characters to primitive ideal space. So every character gives you a factor representation. This is the extremality property of the character. It gives you, if you look at the, there is a representation associated to this and you look at the the fundamental algebra, it's a factor and the kernel, it happens that the kernel is a primitive ideal. It's the kernel of an irreducible representation. So you start with, we have a factor representation but the kernel is always the kernel of an irreducible one. So this irreducible one doesn't interest you really. I mean you just know that it's, there is one and there are usually a huge bundle of irreducible representation which have the same, the same kernel. Actually what you can do is simply if you have a factor representation you can decompose this as a direct integral of irreducible representation. That's, you can always do. What you don't have is just uniqueness but you can always decompose an irreducible and all the irreducible pieces will have the same kernel if the representation, factor representation is the same kernel as the original representation. But they are not interesting. I mean because they are not unique there is nothing special about them. And okay we have the same thing about the finite, finite character. But I'm going, what I'm trying to convey to you is the idea first that prime of g is a good dual. It's worth to be determined for some discrete groups. But you have also this, the space of characters here and maybe the finite character. These are also good objects. Maybe they suffice to parameterize prime of g, maybe not. We will see some example. So when g is type one there is nothing new. Although it's interesting but in this case to determine or to look at the characters that's what Gelfand Harishandra did for semi-simple regroups so they determined sometimes without determining the irreducible representation these are type one groups they determine the characters so as propinus did. So even for type one it's interesting to look at the characters. I'll just talk on groups which are not type one. Here are two results on characters relation between characters and this primitive idea space. If g connects a solvable group we saw that I told you that some of them are not type one. But still this is a remarkable result by Pukansky. There is a you can parameterize the kernels of irreducible representation by the characters and he has even more than that he has a procedure to determine these characters as distribution of the group. And here is something interesting. If your group is nilpotent I mean discrete group polynomial growth of a virtually nilpotent group then you don't have infinite characters. So things are easier here. And already the primitive idea space is determined by this set this class function. At this set you can I mean for concrete if we are given a concrete group g nilpotent group you can really I mean there is an algorithm to determine this u of g. And here is just a remark probably I mean Pukansky's result is probably true for any d group but I'm not sure I don't know how to prove it probably it's just I mean you have to put pieces together but anyway here are about finite characters so these are the characters which are less mysterious as I told you simply class function on the group which are of positive type and extremal with this property. Already Toma who was advocating actually this space of finite characters as a dual for discrete groups he determined for s infinity so the inductive limit of the symmetric groups finite symmetric groups there is a result by Kirilloff of glnk for the infinite fields I'm coming back to this which was completed for sl2k by Petters and Tom in 2016 For g nilpotent I mean there is I mean a bunch of results and a way to determine this u of g and so on there is no I mean no compact formula but you know how to proceed to determine u of g I some years ago I did it for slnz for n bigger or 3 so 2 is special as you may expect because it's essentially the free group and then probably no way to I don't know it seems to be difficult to determine u of g for this case and there is a result which didn't appear I mean in publication by Petterson but lattice is in higher rank simply good here is just a remark characters you can view characters as generalization of normal subgroup because the indicator function of normal subgroup is a trace and a character something more general but in fact it's in some sense it generalizes invariant random subgroups but ok I'm not going to talk about this here is an example quickly lamplight a group you are all familiar with this I guess it's given by transformation if you look at the transformation of the dual group sorry the transformation given by shifting a coordinate on the dual group which is 2 to z so characters as here this was shown by Guichard in 63 and this is essentially a kind of mecha theory which you can do not only for reducible representation but also for factorial representation so for instance orbits gives you reducible representation so here you are shifting axis 2 to the z and t is the shift operator and orbits give you reducible representation quasi orbit I mean closure of orbits I mean 2 elements are equivalent of the same quasi orbit closure of the orbit is the same they give you primitive ideal invariant probability measure gives you a finite characters and then if you look at infinite invariant measure you have to be careful here because you're not interested just on representation by finite by semi-finite factors but which should be on the non or finite on some non-zero element on the c-star algebra that makes the thing I mean they're called normal factorial representation or traceable as Dixie says traceable representation so that means that you are not going to take any boreal measure it has to be regular in order to get really infinite character but you have infinite characters here so there's nothing strange about or I mean that you get some familiar object so characters are connected finite or infinite are connected to some to some familiar object and you see the difference between between irreducible representation which are given by orbit if you want by atomic measures and characters which are given by probability measure invariant probability measure so you see what's happening here I'm moving to GLNK I could talk about similar group but I will restrict to GLNK if you take GLNK the finite character this is killer of the results I told you about there are only two actually the Dirac view that function the Dirac function at the group unit and the function equal one and if K is an algebraic extension of finite field then there is still no new character so all characters finite and this was shown by Rosenberg 89 and moreover this map here is actually bijection so everything is fine and Rosenberg asked whether there are infinite characters say on GLNQ excuse me so do you have any restriction for the value of L? sorry? no no no and yeah I mean different from 1 2 is okay right? yeah 2 is okay, 3 is okay 60 is okay yeah yeah no just 1 it's 1 is probably true but anyway no is it true? no I don't know it's probably I mean that's the there is no restriction I look first at GL2Q as you know GL2Q acts on a tree actually in bad GL2Q in GL2QP for some P and then you have a the action of GL you have the Brouhatt tree associated to this and so you have an action on a tree and here is a wonderful construction by Pierre and Pierre-Luc Alain if you have a group acting on a tree they constructed I mean they proved for these groups K a manability they're constructing a Fredor module but we don't we don't have to know what this is precisely but they constructed a family of representation Pi t all defined on the space of vertices L2 of vertices with the following property I mean you start P0 is a natural representation on L2 of the vertices and you end with the natural representation on the space of edges plus the trivial representation copy of one-dimensional space which is fixed and you have kind of the formation of this coming from P0 to P1 to the trivial representation and along I mean the whole thing at this information that's the crucial property for me it's just P0 and Pt differed just by a finite rank operator and moreover this is a continuous deformation so you can read this in the original paper by Pierre but there is a nice account about this by Pierre Juge very nice, very short I mean I liked it very much and okay but this is a wonderful construction I mean I use it to if I look at it to Q here what I can say if you are close to one starting from some T0 to 1 you can get the representation of this Pt maybe I guess this Pt are not irreducible and you get the irreducible representation which is infinite-dimensional and such that it remains in its range the algebra of compact operator and they are not equally equivalent to each other so this is actually an easy application of Juge Valet's construction because how do you get this compact operator this is essentially I'm coming back maybe to this slide here what you have here what I don't have is okay what you have here is that actually this difference is compact on the space L2 of the vertices for every element in the sister algebra of G now because of this fact I mean you are approaching here at P1 contains the trigger representation and P0 is somehow the regular representation multiple at least weakly contained so this pi of Ts they weakly what I'm going to say sorry this pi of T don't weakly contain pi0 because they are approaching the trigger representation and G is not amenable so you have you find the kernel of pi0 you find A which is nonzero of pi T and this is exactly this is the compact operator you are looking at so this is an easy application of results but it tells you that you have uncountably many infinite character of GL2 Q they are all of type 1 I'm not able to contract infinite character of other types maybe they are not that they are all infinite character of type 1 it tells you also that prime G is incountable this was not the case for let's say PGL2 over a break extension of finite field this was just a finite set of the prime of G now if we I move now to Q for bigger and bigger than 3 here is a different way to obtain representation with infinite character I take this H which is kind of maximal parabolic subgroup of G and I take the quasi-irregular representation and I claim this is irreducible infinite dimensional of course and it contains in its range the compact operators and here what I'm using is just the following fact the crucial fact is the for this H is what I called a normal subgroup maybe it's a bad terminology H is a normal subgroup of G equal GLM Q which means that if you intersect H with its conjugates so malnormal I guess is if this is trivial if this is amenable that's the crucial fact at least for 3 I have actually for bigger than 3 another argument that's what's used here and actually the crucial thing which makes this already implies that the irreducible but the crucial fact is that the trivial representation if restrict the regular representation to H it has a fixed point by H that's obvious that's one fact and the other fact is that actually because of this probability it's not weakly contained in the rest let's say in zero which is the orthogonal of the constant to H to this space so these are the two facts which I mean I mean contained but only only once only once and that means that you you're splitting your representation into two pieces the space H into two pieces one which is finite dimensional and the orthogonal complement which are invariant under H and here the representation here is not weakly contained in the representation given by here and that's already suffice that's a general fact for c star algebra to produce a non-zero compact operator once you have one because it's irreducible you have all so this gives you yeah I mean a representation with an infinite character of type one and actually you can instead of looking at the regular representation you can look at you can take a character of age there are uncountably many of them you can induce this and you get something the same thing and you produce in this way uncountably many representation which are which have infinite character okay I could replace of course I mean what I did for gl and q you can do this for algebraic group or a q point of an algebraic group of a q with q rank bigger than that's you can do okay and I guess it's the end of the story thank you Allah yeah I would have to be reminded that Belgium is not part of France I mean that's thank you Meshir you have a question or comments yeah the Julia element is there an analog for this you should ask Alain and Pierre maybe I think for this Pierre Jules contraction for group acting on cube complexes it's a project I mean that's it what do you think Alain there should be something like no I mean you need somehow yeah maybe for the element but this is not only the element it's the homotopy that's the homotopy you're right I'm using the homotopy in the case of buildings for example you can do this not the homotopy and the homotopy is not I mean for me it's not I needed the homotopy because I have to go through to I have in mind the paper I think by what Hickson and Gendner and Hickson where they do something in finite dimension of cube complex it's way more difficult and way more subtle than for trees because of the many directions and I'm not sure that you get case class operators in particular but there is a point I mean your talk in 87 was it about this Pierre Jules Jules Ballet construction it was a proof of the Zellbef principle for group acting on trees and the point is that from this federal module you extract a trace and the funny thing is that out of something which is not equivalent you get something which is which is really a trace and this trace is not positive but it gives you what you need for the Zellbef principle just to answer your question yeah yeah yeah my memory is not what it used to be any question? can you say something special for the primitive idea space of circus group? yeah something but not I cannot tell you I mean here already maybe this is the whole story although I'm not sure you see where was it for GL2 we stick to GL2Q I mean a good thing as we saw I mean there is no no abstraction actually to describe prime of G it could be but okay you can say maybe I mean these infinite characters may give you already parameterization of prime of G that's not impossible so for you can do for the actually for surface group you can do something similar with using SL2R and produce also infinite character but always of type 1 actually I'm not able to construct one single example for GL2Q or GL3 of a character of type 2 to infinity right to 1 I know there are no only 2 but to infinity maybe there are not they don't exist I don't know and so for surface group you can do something similar probably there must be some construction similar construction to Jules Valet no I mean further module with this with the hot topic probably I mean that would work for any no you can do something for SL2R also probably why should you see maybe yes yes but it's more it's more complicated and it's okay not with finite rank operator place plus I don't know yeah why you think this is going not to question is something can be done probably to which extent that's not clear okay so there is no question let's thank you again