 Yes, today it's a pleasure to introduce Bill Graham from the University of Georgia who will be speaking about Pavings of Generalized Springer Fibers. Thank you for the invitation. This is actually a nice seminar and I only learned about it rather recently and I, as I said, unfortunately conflicts with one of our seminars here but you have to sort of get some way to deal with that anyway let me, let me share my screen and then I can accept I need to be made a host or something to share it's the host can share. Oh, can I do that. Let's see. Right now. I'm doing it right now. Okay, good. Let me try to share. It still says only the house can share. We have to. Oh, I bad I bad. Yeah, I had me. Sorry. Okay, so, so today what I'm going to talk about is, as I guess, an artist that is, you know, pavings of generalized Springer Fibers. And what's new in this talk will be, will be joined is joint work with Amber Russell and Martha premium so so. Okay, so let me begin with a course many I'm sure many of you know what I'm going to start off talking about that. But let me I want to begin by talking about the springer resolution and Springer Fibers. So, so in this setup, as usual, I mean I hesitate to write down our notation, we will have as usual G will contain be will contain tea and also contain, or maybe I'll say be equals T times you. So, in this seven are probably don't want to waste too much time to tell you what these things are but of course this is in this context how I will say this is complex semi simple. This is of course a Burrell and a maximal tourists and it maximally put the input radical. I'm going to let you know, so this I'm always working over C here that's actually not really essential I don't think but let's just do that anyway. I'm going to let Z be the center of G. Now, the main example in this talk will be that will be just G equals SLN. So we'll be in the type a situation. Although a certain amount I mean good if it's a very start of the talk especially everything works for all types. But then as I go to some of the more combinatorial stuff that's always worked out so far at least for Taipei. So the start this is forces is completely general. So, most of you probably know the springer resolution is, is okay. I'll write it this way, until the, it's, I can already as G cross upper be with you. So this maps down to it. So, this maps down to and which is the set of no put it's the no potent cone G says the set of no put all of some we algebra G and that's of course just by the action map. I'll just write it as G and G dot X. And then there's also the map to the to be which is the flag variety which is G mod B. And that's isomorphic of course to the set of B one where B one is a Burrell sub algebra. And so then, given X and N, the inverse image new inverse of X, I'll tell I'll call it for them, I'll give it a call it a couple of things in this talk when you call it for the moment I'll call it until the X is called the springer fiber. And it's often identified with its image BX in B and then in fact BX can just is then can be identified with the set of well sub algebras be one such that be one contains X. So probably most of you know this already but. And so then of course in in type a as again I think most of most people know right now. So in type a G orbits on N, in other words no potent orbits are just in bijection with partitions lambda of N. So we can choose, you know, X lambda in the orbit of so X landed always denote here some element in the orbit of length corresponding to lambda. And then we can form, you know, say I'll be explained or something like that. Okay. Usually people talk about be explained and not until the next one, but for reasons that will become very, you know, really want to talk about until the next one. And so. So, so, let me recall a definition that again, I feel like most people in this audience are probably familiar with but a paving and by Afines of a variety or reduce scheme. Is a filtration by closed sub varieties. You know, M zero containing M one such that MI less MI minus one is the distraught union of varieties isomorphic to the sea or not all of the same dimension I miss our category but I'm not going to let me not try and introduce any more complicated location so we can have multiple varieties of different dimensions. Now, in this talk actually what we're going to be, we're not we're going to need to consider pavings by things that are not just CR. So, so I'm going to still cause I'm going to consider a little later in this talk will also be considered by, I'll say CR mod a finite abelian group. And we'll still call this paving by Afine so maybe you can call if you want to call you can call it in order to paving or a quasi paving or something like this. So, this is this is the same definition as above except that I'm not going to be able to quite get my piece the pieces my varieties isomorphic to CR. The finite abelian group. So, if you as I said if you want to eat I don't know what to call them so for the purposes of this talk me will call them or the pavings. So the theorem that sort of gets the ball rolling. And I, I'm not actually sure who this is due to my be spot on Stein or or Steinberg is that, I mean I know spot on Stein has done a number of things related to this, and I'm pretty sure he's one of the names associated to this but I'm not sure that he's the only one I think Steinberg also. But the theorem is that is it. Is it the springer fiber BX has a paving by Afines. So, this is nice for several reasons. One is it enables you to compete the Betty numbers. Another is it tells you something about the vanishing of odd co homology groups which is useful for various reasons. So, because it's going to appear in later in my talk, let me give you some a little bit of details about how sort of become a tour to this paving so. So, but the details I'm going to give are not I believe the way spot on Stein originally did it. So, the way that what I'm going to describe this is taken from a paper by Caleb G and Martha pre cup. But they were dealing actually with Hessenberg varieties of certain types of which sprayer fibers were an example. And I think for sprayer fibers that their construction was was basically in some work of Tomas code so I don't really want to try and you know get the references right but, but I think anyway I think that's it that's as far as I know that the situation. So, let me just try and explain this sort of, you know briefly so what I want to do is I want to, I want to start off with the tableau of shape lamp so start off so some some rotation having to do with with tableau of shape lambda. And I'll just, and I'll just, I'll just basically illustrate this kind of with some examples and pictures. So, let me start off with some tableau of shape when the end one just draw one here sort of not too systematically. So, I'll call this tableau sigma and I'm going to do this another, but that's this tableau sigma and I want to start off with a tableau of the same shape which I fill in this kind of, you know, this filling order so you have to watch this part of the talk is. But I'm the filling order is just basically I'm going to fill in this by starting from the bottom and going up and going from left to right. Okay. And so then what I can do is to this to the tableau sigma I can associate a permutation, which for reasons, you know, slightly mysterious maybe I'm going to call, there's a permutation here which I can call w inverse. The w inverse in this picture will be the permutation that takes w inverse of one is three because the threes in the box where one was w inverse of two is one, etc. So that's, and then I'm going to call Sigma, we give Sigma name depending on this permutation it's going to, I'm going to call it rw. So that so so then, you know, basically, obviously, permutations are going to be in bijection for with this tableau on the shape, you know, just because I can fill in the numbers anyway I want. And then in this setup, you know, the inversions. These are actually versions of w inverse in our in bijection with cares, K and L, where K is bigger than L in K comes before L in the filling order prepares. What's I say, you know, in our w. So for example, three one is a is an inversion here, because three comes before one in the filling order, but you know three four is not in version. Now, the, the, the, the, this, the number of inversions. It's the dimension of the Schubert cell was called x w zero, which is the dot WB in the flight variety. So, now the interest so what's so this of course is all, you know, very well known, but maybe what's a little bit less well known, but I think it's very nice is that the intersection. So I, let's say I should say you have to choose, you know, X lambda thoughtfully so I won't let me not write down what that is but but let's just let's assume you've chosen X lambda thoughtfully. It's actually not too hard to say what it is but the intersection of X naught of w with the springer fiber X B of X lambda is non empty, if and only if the tableau sigma equals R of W is rose strict, which means increasing along rose. So you notice I did write down that's rose one that's rose strict. So in this case we can denote the intersection so call this. So if not empty, call this intersection C sigma. And then the C sigma form the cells of an affine paving of the excellent. And then finally the dimension. I'll just tell you this just to say just I mean everything in this setup can be calculated combinatorially. And so the dimension of C sigma. It's, it's can be calculated in terms of the tableau sigma. And it's the number of K and L pairs such that K is bigger than L, and K comes before L in the filling order. And if there is an entry to the right of L, then are in K is less than R. So this is an example here so this is a subset of those inversions that I talked about before which is what you would expect the dimension of this intersection is in general going to be smaller than the intersection dimension of the shoe itself. If I go back up to my example here. I can see for example that that seven, seven, four isn't well seven five is a is one of these inversions that gets counted because there's nothing to the right of five. Seven four doesn't count because even though seven is bigger than four and seven comes before four in the filling order. Seven is not less than six in the filling in so that that gets excluded, but seven six is an inversion that's happening that dimension so basically where you throw out a certain number of the inversions and you get that. So this is easier sometimes called or could be called springer inversions. They don't count on the dimension. Okay, so there's a question in the chat. Yes, from Edward in Taipei do you need not a finite group. Not for the not for these but for the generalized springer fibers you do. So for these things that you don't it's that. So it's only. You don't need it model finding for any of the springer fibers and it's only for these generalized springer fibers that are. So that was kind of a preview of, of what is to come. Okay, so. So, any other questions. Okay. So anyway, hence, you know, pt of the plunkery pollen will be excellent, which I'm actually just I'll write it down just so I want to reindex it a little bit. I'm going to divide by two in my power so I'll just say it's the dimension of each two I of the x lambda times t to the I so I'm just dividing through my x points. That's just going to be the sum of. So Sigma in rst lambda that means it's a real strict table of shape lambda of t to the size of land. And that's maybe that's just some. That's the plunk rate. Okay, so next step. So, next I want to talk about the, this a generalized springer resolution. So, in general, so in contains a dense orbit, a dense geo read I'll call it script of PR for the, this is the principal or regular. And in fact, and can be realized just in terms of that already it's actually spec of the functions on our so that's basically a theorem of constant in a sense. So our so, so our, it's just the regular algebraic functions on X. So, the principal orbit is not simply connected, however, in fact, the central group of the of this is equal to the center is or is I can be identified with the center of G, if G is simply connected, which is in the case of SLN. So, it's natural to look at the variety, the universal cover of the principal no potent orbit that. So this is the universal cover. So that maps down to OPR and then sort of sitting in here you have N. Well, you can actually make a variety and here, which is spec of the functions on O till the PR and it's a, it's sort of a result of I guess for when skiing constant. So that this O till the PR actually sits inside of N. And so then, so you get this map here and these maps are actually quotients by the horizontal maps are quotients by Z. And so, so what you would like. Let me pause for a second in some ways the variety M is better than N. And that at least, here's one way at least in which it's better. If you decompose the functions on the principal orbit, you only see into as a representation of the group G, you get it's a direct sum of finite dimensional representations. You only see the representations whose, whose weights lie in the root lattice. So you don't see all the other representations of G sitting there whereas if you look at M and sees all of the representations of G not just the ones whose weight is lying in the glass. So, at least that's some argument that M is, you know, better in some ways and then in. So what we would like is that, you know, a resolution. So we'd like to have something that it's like the springer resolution, making this diagram commute where the horizontal maps. So what we would like is, you know, the horizontal maps ocean spicy. And we'd like that we'd like, we'd also like, since this is going to be a resolution, we'd actually like, you know, until the to M to be an isomorphism over. Just because that's, that's, that's what's true in the spring resolution case that until the end of my smart system over the, over the principle. And so the theorem is that. So, is it. You can construct such a variety and tell that. But in fact the best, the best until the get is really one which is, which is an orbital rather than actually, you know, a smooth thing so it's it's not quite a resolution singularities, but it's, it's, it's pretty good nevertheless. So, let me explain briefly how you, how you get this because it's, it's, I think it's sort of, I mean, in the end what I wanted is I wanted to describe. So, maybe before I tell you a little bit about how you get this. So, so our ultimate goal will be, will be to describe even of until the x lambda which will be the inverse image x lambda under the composition. Okay, so. So, are there any questions that Leonardo instructed me that we should have a brief break at some point I don't know if we want to have our break now or I can maybe I'll tell you how to construct this until then we can have a brief break if there's going to be a break. Are there any questions at this point I'll. Is it but no it's not a fiery square, this is a commutative diagram. So. Okay, so, so let's let me explain how you, how you can construct until look. So, let me just draw some a couple of pictures so pictures here so. So basically what I'm going to do is I'll just explain this rather quickly here. So this is going to, let me just draw some pictures in type a so so you will be the, say the upper triangular matrices. And sort of sitting inside you want to look at the part that's sitting right above the diagonal. I'm going to call this part here. I'm going to call it sort of an opening. I'll call it V add the ad. And then there's the stuff that's, that's further above the diagonal. Whatever this is, I'm going to call this stuff here. I'll call this you bigger than four for some reason that I want to explain right now. So that so the stuff right above the diagonal and you is I'm going to give call the name V add the rest of the stuff I'm going to call you beginning before. And more intrinsically V add. It's the span of the simple root spaces. And there's a be a covariant projection from you to the ad which I can identify as you mod you bigger than before so that you didn't before is put stable under you so that that that projection is be invariant not be a covariant. Now the key thing to notice is that the ad. This is an affine torque variety for the tour for the tourists for the ad joint which is just T add which is T mod the center of the. And so an affine torque variety, what, what's the date of an affine toward right we can do this is having a couple of kinds of data, it corresponds to the character lattice q which is the root lattice, which is the character lattice of the of the vector lattice T add. And then there's the real vector space. V which will be q tensor over z with R. And then the cone it corresponds to a cone in the generated by the simple roots, which which are elements of q. So, and I find torque variety can be described by basically a cone in this in this vector space V, as well as the lattice of characters of the tourists. And so, you can make a torque variety for T by keeping the same cone, you keep the in the cone the same, but you change the lattice to P, which is the weight lattice. You know, in other words characters of tea. And so then we let. So then we let B, T you act on the by having by having you actually, then you let you tell the. So I want to, I want to, I want to soup up instead of just sweeping up the. And what I want to do basically is I want, I want to get some space where the center acts faithfully. So to get the space where the center acts faithfully, I mean I've I want to replace V add by V. In fact what I know is that is that is that V goes to V mod Z which is V add. So these are torque varieties V contains T and this goes, this is contains T and here, which is team on Z. So what I've done here is I've, I want to, I want to create a variety until the where I replace V add by this space V, where the center acts faithfully. And I have to do this in a be a covariant way so the way I can just do this to ensure that it's be a covariant is I just, I take the, the fiber product here, the add the view. And as a space, it's just the cross you bigger than four but giving that other description it shows it has a V action, and then I define until the equals G cross B of you tell the net maps to know and tell the which is G cross the view. And the, the, the thing one has to prove is that this Matt is it is it and tell the maps on to M that which actually requires a little bit of proof so that's the, that's the, that that's the construction there. But let me just say that. In fact, V is not smooth, except in type a one. And the reason is that is that cow is the cone by defining the or V add. Well if I take cow intersect the lattice queue that's generated by the simple roots, which are on the edges of the cone, and they're in queue. But how intersect P is not generated by elements on the edges of the cone. And the picture in SL three is something like this. So this is like alpha one and alpha two and you can form your cone like this, but what you're missing here is that you're missing sort of the fundamental dominant roots, lambda one or lambda two, lambda one. So in it so for in SL three. And then you can form elements to generate the column counter second P. So that so it's not smooth, but however, the is isomorphic to see if you know see to something module a finite group. And the reason is that we have, if we just we can just do the same game with lattices again, if we have the add the add is just a very simple space is just seem to the R, and that corresponds to the lattice queue. Well, that we have V that corresponds to the lattice P. And what we can do is we can take, I could define some V till here and say this corresponds to the lattice one over N times Q. And if I do something like this, then the lattice one over N times Q will be generated by that elements, the simple roots over N. And so this V till is just CN. And this is the this one here is the quotient by Z. And this is the quotient by by some finite group H. And in fact what happens is that V till the mod Z to the Z to the R something like this is isomorphic to the add. V till the sub group H is isomorphic to V, where H is the sub group of the center to the R. And then actually what happens is I guess the center to the R, my H is isomorphic to Z. So that's the sizes of things so each is like that for dimension one subject. Okay, so maybe I will, I can take a break now, sorry about having our breakable for the halfway point, but maybe I'll stop and share questions or otherwise we can just take a break for a couple of minutes. Okay, thanks very much. Any questions.