 נראה, wonderful deal. Just wonderful. Thank you for inviting me, it's a great honor for me to be here and to give this course. The title of the course is about, this course is about algebraic presentation, ארגודיק אקטיינס, שאני חושב שזה נראה איריאל, וזה נמצא, בבקשה, עם אלכס פורמן. ועכשיו, אני אעשה על ארגודיק תיאורי, על נלגוברייק דברים. אז, בלעות של האנגרדיטות, אנחנו עושה על תיאוריEh... סמירות בורמה. ואנחנו שגיעים על תיאורי, ואנחנו עושה על תיאורי而דרדיטות, ולכן נעשה על אלכס פורמן. גם נת öğrire עכשיו על אלכס פורמן, רמנות, אבל הרבה פעלות של אלכס פורמן, אלכס פורמן, דו-לב אובר טופולוגיקל פילטס, ובאותה כאלה, הם קריאו טופולוגיקל סטרקטר, בין זו זריסקי טופולוגי, וכמובן, וכמובן יכולה להגיד משחק סטרקטי בגלל כאלה כאלה. ואנחנו תגידים את זה. אז, כשאנחנו עושים, כמה כאלה כאלה כאלה כאלה, ‫הוא אופסיק כרטריסטיק, ‫אני חושב שבכל שזה zero. ‫כדי להפסיל את המדעים, ‫כל מה שאני אעשה ‫הוא נתקדם גם עם ‫פוסטי כרטריסטיק, ‫אבל זה אולי נמצא ‫שכחה ובסגיע, ‫שבכלל שזה zero, ‫אז אני לא חושב על זה. Also I want to say that most of what I'll say holds also in a more general setting, where K is not necessarily a local field, but a field with an absolute value, which is complete under this absolute value. But again, I will not care about it much now. I will have G, a K-algebra group. I will denote by G the K point. I promise to confuse the two, and many times regard capital G as boldface G, and don't mind too much about the difference in structures, but of course at some point it is important to have this, distinguish the two. And I will have G act on V, a K-algebraic action on a K, a K-variety, and associated with this of course we have the action of capital G on capital V, which is the K point of boldface V. And I want to emphasize that this is an action of second countable, locally compact group, second countable, locally compact space. And of course here I'm taking G and V with the topology associated with the K structure, thinking of K as a topological space. So K, as its analytic topology, K to the N as its analytic topology, affine subspace. Zeroes of polynomials in K to the N have their subspace topology, and general K-varieties, or rather K points of K-varieties, have the structure, I mean the associated topological structure. Also they have the associated K-analytic structure, which is somewhat related. Okay, now let me give an example, let me set of examples, consider SL2R acting on R2. So obviously this action has two orbits, the origin and the rest. Now this is a closed orbit, and this is open. So the orbit structure is very easy to understand. Now let me also study dynamics not of SL2R itself, but of various subgroups in it. Let's consider the groups K, P, A, U inside SL2R. And I suppose most of us understand just by denotation what I mean by these symbols, but nevertheless K now is SO2, P is this group, A is the diagonal group, and U, well I guess isomorphic to R star, U is the unipotent group, isomorphic to R. So I'll draw here the space of orbits associated with each such action, let me take some colors. K-orbits on R2 are just circles of various sizes, P-orbits. Now here we have three ones, we have the origin, and we have the rest of the x-axis and everything else for A, again the origin. Now we have the x-axis and the y-axis, as well as various hyperbolas like these ones, and also U-orbits. Here, actually every point on the x-axis is fixed, and the other orbits are just horizontal lines, whether your action is given by shifting. Now you see that here, I mean this is typical for unipotent actions, every orbit is closed. The orbit space is not housed off, because we cannot separate these points when putting the quotient topology on the orbit space, but still every orbit is closed, so every point in the orbit space is closed. Here this is not the case anymore, not every orbit is closed, those hyperbolas are closed, the origin is closed, but the x-axis for example is not closed, it's almost closed, I mean its closure also contains the origin. And similarly in the other examples, so now I want to make, I mean I want to state the general fact that the g-orbits on v locally closed, well locally closed means the intersection of an open and a closed set, or equivalently, a locally closed is an adjective describing a subset of a topological space. This subset might be, I mean one way to define this is that this subset is an intersection of an open set and a closed set, equivalently this subset is open in its closure. So this fact regards either the Zoritzky topology, the K-topology of the G-action on v. Now since every g-orbit is locally closed, I get points in different orbits could be separated in gene variant open set. So for example this point here could be separated from this point here, because well this point is contained in the complement of that point. This point here could be separated by this point here, because I can take the complement of that line, which is gene variant open set that contains that one but not that one, et cetera, et cetera, et cetera. So equivalently, let me restate this, if I take v more g as a topological space, again I'm talking about v with the analytic topology, with the k-topology, and I'm looking now on the space of g-orbits. I mean if you're an algebraic geometry you might be a bit worried about just looking at the space of orbits, this is a complicated, sometimes ugly space from an algebraic geometric point of view, but for us this is just, I'm just now thinking about the k-points in v and the k-points in g, and truly I'm looking at the space of orbits and I'm putting on it the quotient topology. So the topology that makes the map from v to v more g open. And this topological space is t0, so topology separates points, this is what we see here, and second countable, because v is second countable. So this is, I mean maybe not an ideal situation, it's not Hausdorff, but it's a nice one, and Borel theoretic point of view, it is ideal. So let me write the following, if you take now the Borel sigma algebra associated with this topology, then it is countably generating, countably separating, excuse me. I need to explain what that means, but I will do it momentarily, but I'm just drawing a conclusion of this, which I will explain again. There exists a Borel map from v more g 201. Okay, so I need to explain a few things here. So let me discuss Borel spaces for a minute. So a Borel space for me, a set is a set with a sigma algebra. And now I want to state a theorem that should be called the fundamental theorem of descriptive theory, and goes as follows, all uncountable Polish spaces isomorphic as Borel spaces. Let me recall that a topological space is Polish, so this is an adjective describing topological spaces, if it is separable, now meaning that there exists a countable dense subset, and it is completely metrizable. So it admit a metric, which gives the given topology under which the space is complete. So again, this is not a metric property, it's a topological property, it's the existence of this metric that you care about. These spaces are called Polish spaces. Whenever I have a Polish space, I can take the Borel sigma algebra associated with topology and care about it as Borel space. And this theorem that I quoted says that those spaces that are obtained by this, I mean, if my space was uncountable, are all the same. And this is really, this is truly remarkable. I mean, the interval 0, 1, the counter set, separable Hilbert space, separable Banach space, a manifold, they are all the same from Borel point of view. They are isomorphic, this isomorphism is by no means natural or unique. There are many isomorphisms, but they do exist. So, just to give you an example, some examples, or maybe four. Let me now say what a standout Borel space. This is the underlying Borel space of a Polish space, a Polish topological space. So now I can give you a list of examples. In fact, an exclusive list of examples up to isomorphisms. I have finite spaces, countable spaces, all finite spaces of the same cardinality. With the SBS structure that is the discrete structure, they are all isomorphic. All countable spaces are all isomorphic and maybe 0, 1. Or if you wish, the counter set, or a separable phase, or whatever example you have in mind. They are all the same. I care about Borel spaces. I have this discussion, I will go back to it. But again, we discuss Borel spaces and these are the best kind. The interval 0, 1 is a typical standard Borel space. But I mentioned the notion of countable separation. So a countable, countably separated Borel space is one admitting a countable collection of Borel subsets who separate the points. Equivalently, if I'm having such a countable collection, every Borel subset of a Borel space gives me a map to 0, 1, the characteristic map. If I have a countable collection of Borel sets, I am given a map to 0, 1 to the power of this collection. And if this collection separates the point, the map will be injective. And the map will be Borel. So this property is actually equivalent to the following. There exists an injective, that separation, Borel map 0, 1 to the n. Because again, of course, if there exists an injective Borel map into 0, 1 to the n, then definitely the Borel sets separate the points. They separate the points already here. And the other implication is the one I already explained. So we have the following, but also by this isomorphism, I get the following equivalent definition. So a space is countably separated, if and only if it admits an injective Borel map into the interval 0, 1. So I did explain what countably separating is, and obviously you see that this implication holds, because a countable basis for the T0 topology of Vmo G definitely give us a countably separating collection on Vmo G. And then we do have that one, as I argued here. So this is nice. This is pretty much what I want to say about the Borel structure of V and Vmo G, and as you see, it is quite simple. Now what comes next is studying measures on V. So let me give the following definition. Two measures on a Borel set, equivalent, they have the same ideal of null sets. Whenever I have a measure on a Borel space, I get the notion of what is a null set, and different measure might give me different notion of null sets, and if they give the same, I would say that they are equivalent, and a measure class, that is a class of equivalent measures, then could be just be given by specifying the corresponding ideal of null sets of such a sigma ideal. I want to make the remark, every sigma ideal is associated with a measure class. For example, if you consider the sigma ideal of meager sets, meager in the sense of a Borel category, of Borel category, excuse me, then this is a sigma ideal, but it is not associated with any measure class, so by such I mean sigma ideals that do correspond to measure classes. Now, so let me give some examples of measure classes on spaces. First of all, our guy V is a k-analytic manifold. I don't intend to use much of its k-analytic structure, so I'm just mentioning it. It is a k-analytic manifold, and as such it has a canonical measure class, the volume. For any LCSE group S, we have the Haar measure. Maybe I'll take the left Haar measure on S, so I have the right Haar measure on S, I have the left Haar measure on S as measures that are equivalent. Typically, they are infinite measure, but they canonically give measure classes. It is equivalent as a measure class to the right Haar measure. So again, it gives a canonical measure class, and I'll just give a warning. This is an invariant measure class. It gives a finite invariant measure for the left action if and only if S is compact. S is a compact group, which, by the way, gives a very nice criterion for compactness that will be used later. So this is one example, but now I want to stress that if T is a closed subgroup, then I can consider the S action on S mod T on the corset space, and S mod T has a unique invariant measure class as well, but typically not finite invariant measure. Measure here as opposed to measure class. But maybe I'll give a concrete example. Take S ln k and let it act on the associated projective space. On this one, the volume form is the same as the Haar class. I mean, the volume form as a measure class is the same as the Haar measure class, and it is a gene invariant, gene invariant measure class. But again, there is no finite invariant measure on this space. More generally, if G is semi-simple and Q is k-parabolic, the same holds for the action of G on G mod Q. The Haar class, which is also the volume class regarding this guy as a k-analytic manifold, is gene-variant, but there will be no actual finite invariant measure. Now, maybe continue this. I said that S has left-ar measure, which is always invariant, but it is typically infinite measure. Infinite measure, I mean that the measure of the whole space is infinite. It is finite if and only if S is compact. So it's nice, it also gives me some sort of compactness criterion, which is definitely not obvious. So if T, also if S mod T, if T is normal in S, and S mod T is invariant finite measure, then if and only if T is co-compact in S. That's for a normal sub-book, normal closed. Okay. Now, another set of definition, I'm about to define ergodicity. I know that I'm being very basic in this first talk, but I think I need to set the things right. So now I'm about to briefly recall the notion of ergodicity. I hope this is not too much. I'm not insulting anybody. Let me discuss this in the setting of the G-action on V, though it is more general. A gene-variant measure class on V is called ergodic if every gene-variant subset is null or conal or full. These are, again, I think measure class, they have the ideal of null sets, and so these notions are meaningful. And of course, I'm assuming that everyone saw this definition before, but let me just remind. This is, of course, the same. Almost everywhere defined Borel G-map to 01. Instead of looking at subsets, I can look at maps, characteristic maps. It's almost everywhere constant. It's essentially constant. But now I claim this is slightly less trivial and this is the same as having every Borel G-map into the interval 01 to be essentially constant. Why so? Let me briefly explain this. This is easy, but nevertheless take your measure class and push it down to 01 by the map. I have a map from V to 01, to the interval 01, assuming I have such. Actually, if you don't want to work with measure classes, just pick a measure in it, a probability measure, and push it down. What do I mean by push down? Whenever I have a map from X to Y and I have a measure on X, I can push it down to Y because how do I measure things on Y? By the pushed measure, I pull sets or functions back to X and measure them there. So I have the notion of pushing measures. So I'm having a measure on V, I'm pushing it to 01. If it was essentially constant, fine. If it was not, then it has a support which is not just one point. So I can take a point separating the support and look at the preimage of things below the point and preimage of things above the points and these are two invariant subsets. G, this supposed to be G-invariant map. It's clear to everyone so you didn't stop me, but. How is this accountability-separatability concerning part of what is less? I'm I will be about to connect the dots here in a minute. Now where am I? And exactly, this is my last equivalence. Now I can replace 01 by any counterbly separated space So a space, an action of g on v is ergodic if and only if every almost ever defined Borel gene variant map to any countably separated space is almost ever, it is essentially a constant. And this is clear because whenever I have a countably separated space I can map it into zero one. So it is clear from that criterion over here, which was clear from this one, right? And this is nice because my spaces, spaces of orbits are countably separated. Okay, so now I'm getting the following corollary, every georgodic major class is supported on a single orbit. Moreover, and maybe this is not a direct corollary of what I said before, but take my word for this. And coincides with the hard class on this orbit. Maybe I'll put this in parenthesis, because this is not directly, logically speaking a corollary, but let me explain how I think about it diagrammatically and this is the point of view that will become useful later on. So I'm having v and g act on it, and I'm having a major class on v. Somehow let me think about this major class as a separated action, separated from v. So think about instead of a major class on v, think about georgodic space, not necessarily an energy break variety or manifold. And think about a GMAP, a Borel, GMAP from x to v. So having an energetic major class here, I can push it over here. And this is how I want to think about my given gene variant, georgodic major class on v. And the claim is that if I go farther to the space of orbit, this map become constant, because this is a countably separated space. So there must be an orbit, G mod H, in here, such that all my measure is pushed to this orbit. And the thing in parenthesis is basically saying that, moreover, since the G action here on v has locally closed orbit, the orbit, the topology on orbit basically is the same as the natural topology or the unique gene variant topology, sorry, not the unique, the natural locally compact topology on G mod H. This is homeomorphism, and in particular Borel isomorphism, because the action of G on v is so nice, as a locally closed orbit. So again, this is just a Borel space, I mean that's the dramatic point of view that I will discuss later on more formally. But now I just want to have it on the board just to think in these terms, but this is a Borel space, it has no a priori topological structure, but it carries an ergodic measure class. And maybe you want to think about it as just v with the ergodic measure class that you have, but in a different category somehow. And because of this, I want to separate somehow this picture. And the claim is that the map from x to v pushes an ergodic measure class here, that must be mapped to one single point in the space of orbits. And this means that the image of x is actually contained in one single orbit. And because the dynamics of G on v is so nice, this single orbit, every orbit is just topologically the same as G-mode age, again G-mode age has a natural topological structure, which is defined by the being a coefficient of G, or the cosetopology. And I claim that this is a Borel isomorphism. This is a Borel isomorphism. So actually whenever I have an invariant measure class on v, it is the invariant measure class of one single orbit, if it is ergodic. So this is a very important picture to have in mind. Are there any further questions about it? So let me then, before taking a break, let me now discuss what happens if my measure class is not ergodic. Now take new a G invariant measure class on v. Actually, it need not be G invariant just to start with. So we have v, and we have v more G. Again, if you're an algebraic geometry, you might think of this as a terrible space. This is not a categorical quotient or anything. This is just the space of orbit, but from Borel point of view, it is countably separated, and it is very nice. And I have new a measure on v, and I can push it down into a measure on this space, which is nice enough. It is nice enough because it injects into 01, remember? So in my mind, actually I think of this map from v to 01, if you wish, this space is nice. And the claim is now, that's the general statement of ergodic decomposition, or I guess what I'm about to say is called the theory of disintegration of measures. Whenever I have a measure on a nice space, which is mapped into a nice space, and then this measure could be read as an integration of measures on fibers. Let me put another schematic picture here. This is the space v, and it's this square, and it is mapped into the one-dimensional space here, which is supposed to be the space of orbits, and I have the fibers, and the claim is that, okay, I have new here, a new bar here, and these are well defined, but now for a point t in here, I can find, that's what I'm about to say, a measure on, which is supported on a single fiber, new t. And this process, this is a trans-dimensional process. It is not everywhere defined, but it is almost ever defined, and you can carry it for any measure. So, the symbolic way we write it usually is that we write a nu as an integration over the base space of the measures nu sub t with respect to nu bar, the measure on the base, and for, and the claim is that for almost every with respect to nu bar point, or for any t inside v mod g exists nu sub t such that this formula holds. And what is the meaning of this formula? It's a recipe to define the measure nu out of this data over here. If I have a set in v, I can integrate its intersection with every fiber and get a number, and these numbers give me a well defined function over here, and I can now integrate this function with respect to the measure nu bar over here. That will be the result of this expression, and the claim is that this expression reconstructs nu for me. Okay. Yes, so we get a map from v mod g called disintegration map to probability. Let's say that nu is probability measure here in this discussion just for preciseness. I mean, really I care for measure class a priori, but I just will pick a representative here. Let's say it's probability for to facilitate this writing down this formula. Then I get a map from v mod g to probe v called disintegration, and this is a Borel map, almost ever defined with respect to nu bar. Yeah, I mean, there is a canonical Borel space, Borel structure on probe v. Yes, having a choice of Borel structure on v, you have a canonical Borel structure on probe v, which is the weak one that comes from pairing, integrating Borel subsets or functions. Okay, I put this in parentheses, so let me explain. If nu is a gene variant class, this is if and only if nu t is gene variant class, and this is if and only if is the how on the orbit. The fiber of this map is, the fibers are just orbits now. I mean, part of what I said here was a general nonsense about push-down measures and disintegration, but now here I specify it for this setting of hours of taking place of orbits, and I'm getting this easily. So what is the moral of this discussion? I mean, after believing this, I understand ergodic measure classes for G-action algebraic varieties, and these are just hard measures on orbits. In fact, I understand by ergodic decomposition all possible gene variant measure classes, and they are just integration of that. So this theory of G-invariant classes on varieties is dull, is well understood. There is another theory which I will explore after the break, and this is what happens if I discuss invariant measure, invariant probability measures, not just measure classes. This is an interesting theory, but we will see that also this is dull. I mean, we have full solution of everything, but then later on I will, okay, I will discuss about it later on, but I will somehow open up the horizons not just to define gene variant measure classes, but also ergodic actions not of G itself, but of subsets, sorry, subgroups of G. But this will come later. Let's now take a break. Oh yes, please. Does it imply that gene variant measure classes upstairs correspond to measure classes upstairs? Yes, it does. Exactly. So this measure class no is completely determined by this one. Thank you. Yes. So we come back at 12. Excellent. We come back at 12. My next subject. So first of all, I want to make the following observation already made. If n in G is a normal K-algebraic subgroup, so in particular it is closed, but also the risk is closed. And when I say closed, I mean for the usual topology, otherwise the risk is closed. If is co-compact, then there exists the hard measure on G mod n which is finite and also could be normalized to be a probability measure. Now, the nice thing now is that when I look at the collection of all such ends, I can find a minimal one. Is it a subgroup generated by a unipotent radical on the split turret? So it depends now. G is not assumed to be a semi-simple... No, but a unipotent radical plus the split turret, K-speed turret. Yes. You need... Okay, so things get more complicated in characteristic P, but I'm ignoring this. You take the unipotent radical, you mod it out, you take all non-compact semi-simple factors. No, I mean you have SON for example. I mean G itself could be compact. And yes. So it's an anisotropic reductive group. But I don't need a fine structure of this. I'm just arguing for existence and it's convenient for me not to know too much. So I can look at the space. I mean if you want to be formal, I will do a bold face, a normal subgroup here, all the actual K-algebraic subgroups such that on the level of K-points they are co-compact. And this is a non-empty collection. G is in here. I have the netherian property so I can find a minimal element over here. Netherianity. I don't know if I spelled it right. OE. Maybe it is spelled correctly in my notes. Maybe not. It's OE. Netherian. Is it good enough approximation? It is complicated. Okay. So I'm looking at this collection and it has a minimal object by netherianity and this minimal object a priori need not be unique. In fact I'm arguing now, I will argue in a minute that there is a unique minimal object. How do you call it? A least element in this poset. Of course I'm looking at it as a poset under inclusion. Like this minimal element call it n0 in here. And in fact a claim n0 is a least element. So it's smaller than everything else in this collection. Why? Because if g mod n compact I can look at the product group n time g mod n0. I can map g in it by the, I mean just take the product homomorphism into these two groups and the image is closed. The image of an algebraic group a homomorphism into another algebraic group is always closed hence compact. So it will be, I mean the orbit of the identity here will be a closed thing and but the kernel will be the intersection of n with n0. So it will be something which is contained in n0 and co-compact. So an element here which is smaller than n0 by minimality of n0 it must equal n0 and this means that n0 is in n. The kernel is n0 equals n0 by minimality. So this shows that n0 is inside n. So this is a sketch for the proof that n0 is actually a least element in this poset. And so this means if I care about classification of a gene variant probability measures I can look at all compact group quotient and there is a some biggest one. G mod n0 which is, which by itself carries a probability measure, the R measure, a gene variant probability measure. Somehow what I'm about to argue in the next few minutes is that this is really the reason for any gene variant probability measure on, I mean this n0 is the cause of any gene variant probability measure on any algebraic variety. That's what I want to say, you know. So in particular I want to, before stating the theorem, I will state a result in particular if, we'll see that if G has no compact algebraic factor, algebraic stands here for K-algebraic, that is n0 is G. Every gene variant is supported on fixed points. So let me put it in a symbolic fashion. I can look at V and I can take the fixed points in V. That's a subset of V itself. And now I can take probability measures which are supported here. These are all, of course, gene variant probability measure. I view them as probability measures on V. So they are here. So always have a natural injection like this, just a minute. And this claim here is saying that this is an isomorphism, or equality, provided G has no compact factor. Yes, question. You're a general working isotope G, is that connected? Yes, it's not a working definition, but it's definitely implied by this. I mean, if G was not connected, not the risky connected, then definitely I will have the group of connected components of G. So I don't, which is a compact factor, it's a finite factor. So it's not a general working definition of this discussion, but definitely it is implied by this specific setting. But this is very, very important. I forgot and left this on the board, but maybe this is good, because this is, remember, the picture of orbit structure of various group actions on R2. And you see here maybe for SL2 itself, when it's act on R2, we have two orbits, the point, I mean the origin, and I can put delta measures on this, and this will be a G-invariant measure, and on the rest, I cannot put a G-invariant measure. I mean I can put a G-invariant measure, the Lebesgue measure, but it will not be finite. And if I look at the U-actions on R2, on each blue line, I mean this is just the copy of the reals, and there is a unique invariant measure on it, which is the Lebesgue measure, the Har measure, and it is not finite. I cannot put any finite measure, but I can put whatever measure I want on the x-axis, which is a U-invariant. And the same picture goes here, I mean for A, the only thing I can do is having a delta measure at the fixed point, and same for P. So you see that the whole picture here is very, very degenerate, but when my group itself, SO2, which is another algebraic group, acts, then I have many invariant measures. I can take whatever Har measure on sphere, and any combination of these, like integration, as now it is erased, but you remember the ergodic decomposition, I can take whatever measure I want on the space of orbits, which is a ray, and I can integrate the Har measures on spheres, on circles with respect to this measure on the base, and get whatever invariant measure in here. And this is the full picture of classification of measures, and you see that there is a difference between, I mean compactness is an issue, compactness of the acting group. So this general formula explains much of what we've seen there. Maybe I will give later, in a few minutes I will give a more general proof of this, but let me now just give a, shall I do this? I'll just sketch without writing anything, a proof of this, just based on this picture and this reasoning that I said that on, when I took a one-dimensional group and discussed invariant measures for this, orbits were either one-dimensional, infinite, and do not have finite invariant measures, or single points. I mean again, if you was a connected, one-dimensional, non-compact group, then every orbit is either non-compact and then you have the Har measure, which is not finite on it, or a singleton, and then it is you fixed. Now whenever I have a G-invariant probability measure on V, then I can take all the one-dimensional connected subgroup non-compact ones that generate G, that are in G, and if G is non-compact factor they will generate G, and the space of, the space of invariant measures will be invariant under all of these, and exactly will be the measures that are invariant under every one-dimensional connected non-compact subgroup, again, because they generate. So I just want to, it is enough for me to consider such guys, and for such guys I can discuss invariant measures, I can take ergodic decomposition, and by this picture the invariant measure must be on the fixed points, by this picture and the previous explanation. And this is almost a full proof for this fact, and again you can make out of it a full proof, at least in characteristic zero, later on I will argue about something which is a bit of a generalization of this, and I will give another proof. So this is why I allowed myself to be so brief now, what I want to say now. Ah, now in general every G invariant probability measure is supported, sorry, here I discuss the case when I don't have any compact factor, in general I always have a canonical maximal compact factor, which is given by N0, and the claim that I will make in a minute is that any G invariant probability measure on V is supported on the N0 fixed points. But now I realize that I forgot to say something, I will come back to this, I forgot to give a very nice corollary of this picture, so maybe I'll put it here, it is the Borel density theorem, if G has no compact factor and gamma in G is a lattice, so this means that gamma is a discrete subgroup such that G mod gamma has a G invariant probability measure, something that I told you that it is quite rare. Now gamma is not an algebraic group by itself here, of course, then gamma is a risky Gens in G, so this Borel proved this in a quite intricate proof, first and foremost gave a very neat proof of this later on using a power carrier occurrence, and basically if you think about it first and foremost proof enough, it is nothing but this proving this general statement, though we never made it. Let me explain this, I can look at G mod gamma and I have a map to G mod the group, I guess this is the K points of this risk closure of gamma here, so I have a G invariant measure here, probability measure, I can push it down, push the G invariant probability measure, and the push must be contained in a G fix point here, but this is a transitive space, if I have a G fix point then all this space is just a point, and this means that the stabilizer is just everything, so this is a very neat explanation of this fact, and I'm back to my discussion, so maybe I'll forgive me for being a little sloppy, let me restate it as a theorem, so let us read it again. Given G, now general G, I am allowing co-compact factors, co-compact normal subgroups, co-compact K-algebraic subgroup, and in particular I have N0 which is the minimal one, and the claim is that whenever I have an action of G on any V algebraic, K-algebraic action, and I'm looking at a G invariant measure, it is supported on the N0 fix point, and N0 is normal, so the N0 fix point is a G invariant subset on which the action of G, I'll write it down because it's trivial, but important, is G invariant subvariety on which G acts via its compact factor G mod N0, so again this explains what I said that all invariant probability measure dynamics is happening via a compact factor, and there are no really interesting dynamic groups by themselves or non-compact ones. Any question about the statement? I'm about to prove it. Yes, I guess I did some logical maneuvers here. This theorem here in particular implies that one because this decays N0 is G and then you get this equation, and this Borel density theorem follows from that one, but maybe I confused you because also I sketched by heart another proof for this one, and now I'm going to give another proof which is different from the one I sketched for this more general theorem. The method of proof is very important for me, it somehow, it starts like this one, this observation, and I will take a minimal object in a collection of subgroups, and this is a very very useful trick that appears all over in this theory, so let's start with giving the right collection. So we look at a collection of all h in G k-algebraic subgroup such that G mod H is a good candidate for V. Again, I want to discuss all probability measures on all possible varieties, all G invariant probability measures on all possible varieties. I already know that when I act on a variety, really from the point of view of ergodic theory, I'm only interested in orbits, so instead of looking at all varieties, I will look at on all possible orbits, all possible coset spaces, so I'm looking at all ages, all possible stabilizers of points, all age, k-algebraic subgroup, such that on G mod H I can find a non-trivial probability measure. Okay, and of course this collection is not empty because I have G inside it, I just have the one point with the delta measure, that's the fixed points that you see over there. Now, so again, by notoriety, and I will not attempt to write it again, this word, I have a minimal element, I'll denote it H0, and then I can have this variety V0, and I can find, by definition of H0, I can find some invariant probability measure on this. I will note that the risky support of mu0 is everything. So whenever I have a measure, I could take the support of this measure, so the minimal closed subset which carries the full mass, and I can do it also in the risk topology if you want, or I can just take the risk closure, which is the same of the standard support, and the measure must be everywhere supported, because the support is G invariant subset by itself. I mean the measure is invariant, so everything which I just constructed canonically from it must be G invariant, and there is no G invariant subset in this transitive space. Okay, so now I have, if I was really explaining, I mean there is a little fast that I should do about group of, I mean connective component is not a problem, the problem might be that I have a, גלוע קומולוגי might be a problem, I have the action of, take the action of r star on itself by taking, multiplying, by multiplying by x square, I mean x takes y to x square y, I mean then we have, or take just, there is a little issue, let me not go into it, okay, just, sorry, okay, it could be easily solved, that's the answer, and then I want to give ideas here, okay, now let me now fix another space, so this is v0, I've chosen it, and now let me discuss a generic G-variety v, and invariant measure on v, and let me assume, without loss of generality, that the risky support of mu is everything, the risky support of mu is a G invariant sub-variety, let me just focus on that one, so I'm discussing this guy, and now, sorry, I'm proving the theorem, I'm now, yes, but yeah, bear with me, now I'm, so I put myself, I gave myself a setting, I discussed a certain space I constructed, v0, and I discussed a generic other space, v, and I'm about to make a claim, for every x comma y inside v times v0, the stabilizer in G of y is contained in the stabilizer of x, that's a claim I'm proving now, I'll do it by negation, so I will look at v inside v0, and I will look at the set of bad points, so take u to be all x, y's such that this does not happen, by assumption u is not empty, and you can check, this is an open condition, I mean this is a closed condition, not this, is an open condition, and the risky open, now since mu0 and mu are fully as risky supported, then if I look at mu times mu0 of u, this is not zero, because mu times mu0 measures positively every open set, so I can take, define the new measure to be mu times mu0 restricted to u and normalized, I just constructed now a probability measure on v times v0, which is supported on u, now unfortunately I erased the picture that was here about ergodic decomposition, but now you remember that there must be a new sub t, a probability measure, a gene variant supported on a single orbit in u, maybe I didn't say, I didn't emphasize it, u is gene variant, it need not be transitive, I have a measure new on it, I can decompose, this measure is probability gene variant measure, I can decompose it by disintegration into measures on orbits, and for generic orbit I will have a well-defined gene variant probability measure on a single orbit, and maybe you already see where I'm going, this is about to contradict the minimality of h0, so maybe supported on a single orbit in u, the orbit of, let me just give it a name, x, x is a point in v, and v0 remember that this is gene or the h0, so let me denote it like this, now the orbit, what is a stabilizer of this point, the stabilizer is g, modulo the intersection of these two, of stub x intersecting with conjugate of h0, and by changing coordinates this is the same of g modulo, let me stub x g inverse intersecting h0, I just now took another point, I conjugated that point, so basically I'm looking at the point which is intersecting, but now this piece over here, this stabilizing group, this is contained in h0, h0 intersecting something, so this is one orbit, I mean this stabilizing group now is an element in this collection over there, it's a k group that supports an invariant probability measure, but h0 was supposed to be minimal, so minimality of h0 implies that this is actually equal, and this means that h0 is inside stub x, and I think this is what I want to, and this implies what I want to say, so any question about it? Okay maybe I guess I didn't say, minimality says that there must be an equality here, and this means that this group is included in that one, equivalently this group is included in that one, and this is exactly the equation I wanted to prove, that for every x, y, and this is my x, y, the stabilizer of x is contained in the stabilizer of y, okay, sorry for not saying it in the right order, but I think I said it all correctly, now I didn't yet prove the theorem, I just justified the claim, any question about it? I'll use that claim to explain the theorem, to prove the theorem, so now I will take, I will specialize for v being v0, I will take x to be h0, and y to be gh0, the stabilizer of x, so x is the base point in g mod h0, and y is the, well g shift of it, the stabilizer of x is contained, sorry, the stabilizer of y is contained in the stabilizer of x, right, so this shows that h0g is contained in h0, by this I'm getting that h0 inside g is normal, okay, so the conjugation of every stabilizer is included in another stabilizer, okay, so in particular g mod h0 is a group carrying an invariant measure, h0 is co-compact, it shows now that h0 contains h0, contained n0, because n0, remember, was minimal with respect to this property, this is normal and co-compact, but now the claim also implies that h0 is inside the stabilizer of x for every x in, so the claim I proved is that the stabilizer of y, y was typically some gh0, the stabilizer of it is a conjugation of h0, the stabilizer of y is contained in stub x, but now I know that h0 is normal, so h0 itself is contained in stub x, for every x in every v provided that, I mean mu is fully supported, in general I need to reduce myself to the support, so this is what I got now from the claim, h0 is the stabilizer of x for every point in the risk support of mu, for every such guy, so if you want also sub mu is h0 fixed, but now, once more, now I will take v to be gmod n0, and this is another candidate, and I see now that h0 fixes a point in gmod n0, and get that h0 is inside n0, and combine the two, I'm getting that h0 is n0, and I conclude that all along I was playing, I didn't know it until now, but all along h0 was n0, and what I got here is that really whenever I have a gene variant probability measure mu, every point in the support is invariant under n0, and this is exactly what the theorem said, okay, so this is a nice theorem, any question about it or the proof? so the proof, you can argue that it was a bit complicated, for example I gave you for this almost, this good approximation of the theorem, I gave you briefly a very short proof, and I could make out, I could made out of it another proof for this one, but I gave you this proof because it generalizes, and now I want to give you a corollary, not of the theorem, but of the proof, which is important, is important for the setting I will study, nice, so now I'm adding a generalization, but maybe you have a time that I want, you want me to finish, okay, generalizations, so this generalization is very important for the things I will lecture in my next talks, fix now a group gamma, this need not be an algebraic group, but it has an algebraic representation, consider the action, so just a representation in an algebraic group, exactly, no, no, just take any, I mean if gamma was topological group then I would ask my representation to be continuous, but so maybe this is locally compact second countable group, and this is continuous on morphism, say two, and consider the action of gamma on V, now V is supposed to be a G-space and I'm considering the action of gamma on V, VR, its representation into V, the orbit space now is complicated, might get really ugly, but still I have this nice gamma invariant map, so I can disintegrate things over it, so same reasoning as before implies the following theorem, oh sorry not yet, implies sorry every gamma invariant, ergodic measure is supported on a unique G orbit, so I have this picture if X, X is a gamma space, I'll repeat the picture, and I have V here, then this map splits like that somehow, and now the theorem there exists a minimal normal K-subgroup N inside G, such that the map from gamma to G to G mod N has a pre-compact image, that's first fact, I mean this by itself now, this is easy to prove, you prove it the same, I'll be the same reasoning basically that we prove that N0 exists before, and now next imitating this proof that we gave, you can prove that every gamma invariant measure a G algebraic variety is supported on the N fixed point, so again I mean okay here I just call it N, I mean it's not, it's that N that is given by the first part of the theorem, again the first part of the theorem is I mean it's the same proof as the one I gave you before basically, and then you have the second which is a translation of this and the proof is just the same, I'm not claiming now that G mod N by itself is a compact group, but the closure of gamma in it is, yes please, I'm about now to say something about support and dynamics on the probability measure and then that will be the end of this talk, so this is important theorem and I want you guys to remember it if you want to attend further talk also of mine, but of course I will recall, but now here is the corollary, assume G acts on V and I'm having a measure on V which is not necessarily G invariant, then I can consider the stabilizer of this measure, that's a subgroup of G and I think of it as my gamma with the inclusion map, so corollary now consider gamma to be the stabilizer in G of mu when mu is some probability measure on V and take gamma in G, just inclusion again and apply the theorem and the stabilizer of a measure mu in probability I claim is compact modulo, now it's actually compact because the stabilizer is closed, modulo, the fixator of the Zariski support of mu, now take mu, it is supported on the certain algebraic variety, the minimal one is the Zariski support of mu, the fixator of it is an algebraic subgroup, it's an algebraic subgroup that certainly contains this n by the defining property of n, so modulo it, the image of gamma must be compact and if gamma is a stabilizer then you just get it and basically if you add and it must be that your group, your measure is actually some hard measure with respect to some compact group, did you obtain modding out trash, modding out some noise that the measure doesn't see, so this is an important thing, so is there any question about it because because gamma is closed in this, yes correct, so I'm a bit rushing because I want to end up this talk with having these two pieces of information on the board, so that's the first and the next fact I will not prove to you but you can just elaborating on the same kind of tools, you can prove the following fact by Zimmer, the action of G on probe V as also locally closed orbits, so we said that whenever n algebraic groups action an algebraic variety, then the action is nice, I mean the space of orbit is nice in a certain sense and that was locally closedness of the orbit, also it goes when I'm discussing the action of G not on V itself but on the space of probability measures on V and moreover when I act on an algebraic variety then stabilizer of points are of course algebraic by themselves, here this is not the case but it is almost that case and that's this corollary that I said, the stabilizer of points is algebraic by compact, so it's a good enough approximation, over the reels remember that for real algebraic groups compact subgroup is always algebraic by itself, so over the reels all this fast is not, I will not write it down but if you care more about the reels you should write it down in your notes, over the reels a stabilizer of measures are just algebraic, in general you have this compact noise and that will be the end of my first talk, thank you for listening. Is there questions, locally closed I guess for the weak start topology of the, for the weak start topology on probe V correct, I mean there is a natural topology here which I didn't discuss at this point of the day, I mean there is a, maybe I should say probe V, there is a norm, this is a subset of a known space of all signed measures and on this you have the total variation norm for example, this is a very strong topology and but also this is a dual space by Ritz, STORM for continuous functions on V, at least if V was compact but this could make sense in any case and I mean for locally compact it's a dual space and it has a natural topology as a dual space called the weak start topology and with respect to this topology you have this property, yes thank you for. Maybe I'm missing perfectly the point but do you not need gamma to be large in some sense and see what the image of gamma is? No here, here I didn't assume, I mean in this discussion on the board here I didn't assume that the image of gamma is a risky dense in any case, it was arbitrary and in fact I use it for groups which are typically not a risky dense, I mean later on when I will discuss things I will discuss algebraic representation and it will be typical for me to assume that the image is a risky dense so you see things but for this classification theorem you don't need it.