Gelfand-Tsetlin theory and Coulomb branches
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Webster, Ben
The algebra U (gln) contains a famous and beautiful commutative subalgebra, called the Gelfand-Tsetlin subalgebra. One problem which has attracted great attention over the recent decades is to classify the simple modules on which this subalgebra acts locally finitely (the Gelfand-Tsetlin modules). Ininvestigating this question, Futorny and Ovsienko expanded attention to a generalization of these algebras, saddled with the unfortunate name of“ principal Galois orders”. I’llexplain how all interesting known examples of these (and some unknown ones, such as the rational Cherednik algebras of G (l,p,n)!) are the Coulomb branches of N=43 Dgauge theories, and how this perspective allows us to classify the simple Gelfand-Tsetlin modules for U (gln) and Cherednik algebras and explain the Koszulduality between Higgs and Coulomb categories O.
 So my talk is going to be kind of dual in some sense to Joel's talk, so it's also going to be about the question of how to understand the representation theory of Coulomb branches. But it's going to be dual in the sense of it's going to be looking for more things like projectives rather than things like symbols. So let me sort of introduce a general algebraic setup, which I think it's good to think about Coulomb branches in terms of. So let A be a k algebra, of course here k is a field, and let R inside it be a commutative subalgebra. We can also have a fun discussion about what non-commutative algebras you could put here instead, but commutative is a good start. So whenever you have a setup like this, there's a natural set of modules, category of modules you can study, which is quite interesting. So A module is Gelfand Settlin, and I mean this is in the grand tradition of naming things after people who had extremely little to do with them or at least had some very distant connection in the past. So some people might prefer to call these harsh chandr, which I think would follow the same tradition or weight modules, which I think would be confusing. So I'm just going to use this terminology, and I hope people will forgive me. And of course, that's a lot to write, so I'm going to write Gt. If it's locally finite for B, it is R locally finite. So that means that for all elements of this module, if I look at R times it, and then I look at the dimension of that as a k vector space, it's finite. So of course, this includes all finite dimensional modules, but will also include things like this category O that Joel mentioned. All right, so how do we study modules like this? The one reason I like this definition is that there's an obvious way to break this module up into pieces. If I have a locally finite module over a commutative algebra, then it naturally breaks up into a sum of things that are killed by powers of maximal ideals. So given lambda in the set of maximal ideals of R, I can consider sort of the quote unquote weight space, which is all the things in V that are killed by some power of the maximal ideal. And for whatever reason, I like to think of lambda as something floating off in space, and then it has some particular maximal ideal attached to it. You could sort of say, like, oh no, this maximal ideal, that's the point in max spec. All right, and a very easy lemma. V is R-Gelfan-Setlin, if and only if V is the direct sum of these subspaces. Do I want some finiteness condition on R? Oh, oh god, I always get stuck on these stupid things. Oh, right, I want R to be finite type. And if R is finite type, then this quotient is a finite extension of k. And so it's, yeah, anyways. All right, you're up. Yeah, finitely generated as a k module, as a k algebra. Yeah. All right, so what are examples that might be interesting to think about? So you might choose A to be U of G and R to be U of H. In that case, an R-Gelfan-Setlin module, it's tempting to say weight modules, but that's not quite right. I'm taking a power of this maximal ideal. So it's generalized weight modules. All right, another very interesting case to consider is A being U of GLN and then R being the Gelfan-Setlin subalgebra, which is what this whole setup is named after. So that's the subalgebra generated by the center of the universal enveloping algebra of GLK for k equals 1 up through. All right, and one final example to consider is, well, A is the Coulomb wrench for some group and some representation. And unfortunately, Joel and I didn't coordinate notation before our talk. I'm using the notation of BFN's paper that N is a module over this group G, and I'm using V for something else. So I apologize about that, because Joel, of course, used the notation from my paper. Anyways. So there's a natural commutative subalgebra to consider in here. OK, yeah, well, this is a pretty shitty setup, so I apologize. So A is the Coulomb branch, and I can take R to be the commutative subalgebra given by the equivariate parameters for the group. So R is the equivariate homology of a point for the group, which includes inside the Coulomb branch, of course, because it's the equivariate homology of something else. All right. So a basic question, what are the simple GT modules for a given A and R? So this one of what are the different weight modules for a semi-simple lead group is one that's elicited a lot of study. And in part, people started thinking about this example because this one was too hard. And they said, well, let's maybe add this extra condition, the whole Galfan set when subalgebra acts locally finitely. And that was a problem that also attracted a lot of attention over recent years. But it's one where people got a little bit stuck. And let me try to explain where they got a little bit stuck and how I think we can get unstuck now. So there's a general approach to understanding this category. And I think part of the reason this is such a nice category to consider is we have such a nice approach. And this approach, so definition, we say A R is R is Chandra. If for all elements of A, the bimodule RAR is finitely generated. Well, it's obviously finitely generated as a bimodule. But this is that it's finitely generated as a left or right module. So this is sort of a size criterion that A is non-commutative, but the way it not commutes past things in R is not too bad. There's sort of only finite amount of extra information when you multiply in the left as well as on the right. Look, that's what it says in the paper. This is the definition from the paper. What am I supposed to do? And given this theorem, droste futone and osienko, is that for a Harchandra pair, there's sort of a nice way of describing these guys. And let me just say what it is in words. You look at these, you think of them as functors, and then you say, what are the natural transformations between these functors? So somehow, Galphons-Setlin module is the same thing as saying what all these weight spaces are, and then an action of all the natural transformation between the functors of having weight spaces. So the category of A modules, which are Galphons-Setlin, is equivalent to the modules over some other category, and those should be discrete, so this other category is going to have its apology. And the other category is you just think about, well, what kind of elements of my algebra would give me natural transformations between these weight functors? Yes. Yes. So the sort of important consequence this Harchandra hypothesis has is that if you are generated by a finite dimensional R invariant subspace, then you are a Galphons-Setlin module. So that's sort of the important thing you need to make this theorem work. So here C is a category where the objects are just this max spec of R, and the hams are given by an inverse limit. So the problem from lambda to mu is I take the inverse limit of A mod, let me get this right, M mu to the k plus M lambda to the kA. So I'm multiplying by one of these, oh no, I did the wrong one, multiplying by one of these maximal ideals on the left and by the other one on the right. So this quotient, this is the natural way to get a natural transformation between a kind of truncated version of this weight functor, where you require a particular power to act by zero, and as I let k go to infinity, I get something that's defined on the whole weight space. So as an inverse limit, this has a topology, and I only want representations where that action is continuous when I give the representation the discrete topology. I'll just write discrete, and that's what I'll be doing. All right, so this is a very nice theorem, but it's only as nice as your understanding of what these homomorphism spaces are. So let me tell you there's sort of a slight generalization of this theorem, which is that if I want the simple Galphons-Setlin modules such that a particular one of these weight spaces is not zero, then those are actually in bijection with simple discrete modules over what I'm going to call a hat sub lambda, that's from lambda to lambda. All of the examples I said in the beginning are harsh on the. So for example, if you wanted to understand all weight modules, like in the usual sense over Li algebra, simple ones, where a given weight space was not zero, this tells you there is some algebra out there, but those are the same as simple modules over. But that algebra is horrible and infinite dimensional. It's very large. In particular, its finite dimensional modules are all the different weight spaces of all the different finite dimensional guys, plus maybe some more. So it's going to be enormous. In particular, it should have infinitely many simple representations of all kinds of different dimensions, going to be a very complicated algebra. It's not clear you gain anything from thinking this way. But for Coulomb branches, there's a very nice answer to this question. So I should just pause for a moment and ask, are there any questions? So for the carton inside of U of G, yeah, I don't know anything interesting about this algebra that you couldn't figure out some other way. And I mean, I would say somehow what's going wrong there is that U of H is too small. In particular, it's not a maximal commutative subalgebra. Whereas the Gelfand-Setlin subalgebra is, and this guy is also a maximal commutative subalgebra. Maybe we should find out that two is indeed a special case of three. Sorry? Yes. Two is indeed a special case of three. But you learn very interesting things. If you only care about two, it's worth thinking about three. So I'm going to tell you a theorem about Coulomb branches that tells you something interesting and new about U of GLN. Sir, a random string from last lesson, is it just a bijection of sets? No, it's, I mean, so it is a bijection of sets, but it's a particular one. The lambda weight space of this symbol is a simple module over this guy, and that gives you the bijection. So I mean, you can strengthen this if you looked at all modules over a lambda hat. You would get the category of Gelfand-Setlin modules modulo those that have trivial lambda weight space. So you do learn something about extensions, but when I only talk about simple modules, yeah, the only structure there is a set. But it's a bijection which is very natural. So you just look at this guy acting on the lambda weight space. Here? Discrete. All right, cool. Sorry, this equivalence is you look at all the weight spaces, and these are the natural transformations between the functors of taking weight spaces. So I didn't want to get down to the details of it. You can go read their paper, but I mean, this guy naturally acts on the weight space, and it's that action. I'm not sure if you answered the question. The question is, are the C modules, are they continuous? They're discrete. There's a word discrete here. So yes, they're continuous, and they have the discrete topology. Right? I mean, this is just some way of saying that some power of this ideal is zero, but I don't want to tell you which one. So I don't want to fix any particular one, but I want some power of it to be zero, and that's the same as saying that the inverse limit acts on it with the discrete topology. All right, anything else I screwed up? All right, very good. OK, right. So for Coulomb branches, there's a very nice general answer. So one thing to note for Coulomb branches is this max spec of r. I can really think of as semi-simple elements of g times c. So this is the Lie algebra of the group g times c star, or this is the rotation c star. And this is modulo conjugation. So sorry, this is for the Coulomb branch. So I'm going to specialize from now on and only talk about Coulomb branches. And all other interesting examples I know are actually subcases of this one. So here I'm just writing out for you what is max spec of the equivariant co-amology of a point for g, right? That's the torus of the group, mod the vial group, which is actually the same as semi-simple elements mod conjugation, and I forgot here that I should have put a c star, because I also want the loop rotation. So when I think of the Coulomb branch as equivariant co-amology, I have this equivariant parameter h for c star. But when I get an actual non-commutative algebra like the universal enveloping algebra, I specialize h to be 1. So that means my, that's the same as saying that my semi-simple element is of the form x, 1. So when I exponentiate it, I get the usual rotation action of c star, and then I've somehow adjusted that by some co-character in the group. All right, and out of this data, I'm going to get a description of this algebra for the Coulomb branch. Yeah, yeah, the adjoining action. How do you get the element of the speck of r from that? Element of the speck of r. Yeah, so I'm looking at, you know, lambda plays a role here and it plays a role here. This is a theorem for, yeah, it's the whole category. So for every maximal ideal, I assign a r module to it. And the discrete thing is just that that r module factors through some power of the maximal ideal, but I don't want to tell you which one it is. All right, so defining this requires a little bit of notation. So remember, I have this representation n that I fixed from the start. This is part of the data of how I construct the Coulomb branch. So having chosen an element of the Lie algebra, I have a subspace in here where this is the subspace where x has integral weights. And I have a further subspace in there. So I'm going to call n lambda minus, and this is where x has non-positive weights. No, I was doing that right. So lambda is the result of the same thing? Well, not quite, but yes. So x is lambda thought of as an element of the group. But you know, x is only well-defined up to conjugation, but if I replace x by its image under the adjoint, then I'm just acting by that element of the group on n lambda. All right, and similarly, if I do this in the adjoint representation, I'll get a subgroup which is exponentiate the subalgebra that has z weights under the adjoint action. And then inside here, well, if I was going to be really consistent, I would call this g lambda minus, but I'm actually going to call it p lambda. So this is a parabolic that, again, comes from looking at the adjoint action of this x and looking at where it has non-positive weights. All right. So let me then make one more definition, let y lambda be the sort of the associated vector bundle to g lambda crossed over p lambda with n lambda minus. So let me just remind you what this is. This is I take a coset of p lambda. I take an element of n. So the condition I need is that n is in g acting on n lambda minus. All right. What was the point you were making with the max vector a star g plus b? What was that we were saying about the max vector b? So those are the weights that the commutative subalgebra inside the Coulomb branch could act by. So when I want to analyze a representation of the Coulomb branch, I want to look at this commutative subalgebra acting on it. And if it's locally finite, then that's going to break up into some sum over maximal ideals. And the different options for the maximal ideals are exactly these conjugacy classes. And if I've specialized h equals 1, then these conjugacy classes where the part along, so the component along the loop direction tells you, oh, when I mod out by that maximal ideal, what scalar does h get specialized to? And I want to look at the case where h is equal to 1. All right. So theorem. So I should give some credit for this to Hiraku because he sort of explained this way of stating it to me. You will search in vain for the paper of his that proves this, but it is in my paper. This algebra A hat lambda is the Borel-Mor homology, equivalent with respect to g lambda of the fiber product y lambda over m lambda of y lambda. And of course, more generally, if I take comm from lambda to mu, that's what's tempting to think that it's sort of always this guy where I just take this product. But you'll note there's some weird asymmetry. This g lambda and n lambda, well, shouldn't I take g mu and n mu? But this is only true if lambda and mu are conjugate under the affine-vile group. All right. So that is to say I can choose them in a maximal torus such that they differ by an honest co-character in that maximal torus. I just get 0 if lambda and mu are not conjugate under the affine-vile group. And if they're conjugate under the affine-vile group, well, these weights change, but they change by an integral amount. So actually, this g lambda is equal to g mu and this n lambda is equal to n mu. All right. And of course, there's something missing here. I need a hat here. And that just means I complete with respect to grading. OK. So all questions about modules over the Coulomb branch that have this Gelfand-Setland property are actually questions about this finite dimensional convolution algebra. Yes? Does that have something to do with the multiple terms of the formula? Not obviously. So I mean, an exercise to the audience is figure out how Joel's talk was secretly using this. So I suspect this complaint about the fact that there was non-equivariant homology instead of equivariant homology is that, in fact, of course, there's an obvious quotient of this algebra. Right? If I remove the equivalence, then I will get a homomorphism from this guy to just the usual Borrel-Mor homology of this fiber product. And it's easy to work out. Every simple factors through this map. So I wasn't able to confirm this since I haven't had much time since I saw Joel's talk. But I suspect very strongly what was happening is you guys were defining a module over this and then sort of pulling back and going through this equivalent without realizing that was what you were doing. Sorry? So you are completing after some factor like that. Well, this is by the fact that it's a discrete module. So if I have a module over this completion, which has the discrete topology, then that means some power of the obvious maximal ideal in the completed equivalence homology of this g lambda has to act trivially. And if you're simple, that means that it has to be the maximal ideal itself. So I mean, there's secretly some shift here. So that the obvious maximal ideal over the homology goes to the maximal ideal that corresponds to lambda here. What is g lambda? G lambda. So it's the subgroup whose Lie algebra is the space where x has integral weights under the adjoint action. If you have a co-character, you can think of that as a point in spec of the equivariate co-homology for a point. So the sort of important thing here is, for example, let's see, what can I safely delete? Not much. This is getting pretty tight. So yeah, that's the best way to say this. So for example, if lambda is generic, then this is trivial. So if you have sort of a totally generic central point in spec, then this guy becomes trivial and this statement just says that you get a copy of c. And one interesting corollary. No, no, no, not if lambda is zero. If lambda is generic, then g lambda is trivial. If I look at every root eating lambda and I don't get an integer, then this means that this guy is just, sorry, g lambda isn't trivial. It's the torus. And so I just get the equivariate co-homology of the torus and I complete with respect to the obvious maximal idea. I mean, x is, yeah, x lambda are the same thing. So no, lambda is x1. But these live in the same space thought of in a slightly different way. This guy lives in spec of h star g of a point. And this guy lives in g modulo the adjoint action, or I should say g semisimple mod the adjoint action. And it's left as an exercise to the viewer to remember how these things are isomorphic. Yes, no, no, it's not the centralizer of lambda. It's the stuff that lambdas integral with respect. Oh, maybe you're right. Wait. If you take the exponential of that element of the Lie algebra, then yes. But if you think about it as a co-character, then no. So exactly why I was, oh, this is already trashed. All right, so somehow, I mean, if you don't understand Coulomb branches, you're not going to magically understand them from me saying this. The important point here is that the weights that you can get in a Coulomb branch essentially correspond to elements of the torus mod the vile group. And if you want to understand the representations where that maximal ideal appears, there's a sort of simple topological way to do this. And again, since I'm running low on time, I leave as an exercise to the audience to think through, for example, what thinking about perversives on n lambda tells you about the representations of this algebra. There's a very natural interpretation of this, familiar to anyone who's read Ginsburg and Chris's book. All right, so a very important special case will be interesting to people who just like representation theory and have never heard of Coulomb branches before, which is u of gln itself. So in terms of these pictures for quiver gauge theories, what this means is we take 1, 2, 3 in circles and then n as a square. So that means my g is gl1 times up through gln minus 1. I am stopping at n minus 1. I don't mean to go to n. The gln is flavor. And my representation n is 1 by 2 matrices times 2 by 3 matrices dot, dot, dot up through m by n minus 1 by n matrices, where this group acts in the usual way by pre-n post composition. All right, so the Coulomb branch here, and again, exactly the answer depends a little bit on your conventions about flavor. So I believe I'm following what Joel did by fixing the flavor to be scalars. So this is going to be u of gln mod a maximal ideal of the center. So there's sort of a way to do this where you incorporate the equivariant cohomology of gln as well, and that becomes the center of C. Sorry, the center of u of gln. Let me, since I'm talking about individual weights, each individual weight, the center acts through some character, so I might as well fix that already. And if you think through what's written on this board, let me not quite erase it yet. The space is y lambda. I get out of this. So I can think of a point in here as a quiver representation for this quiver. And I'm almost getting the quiver flag variety, but I'm not acting by gln on here, so I get something a little bit funny. So if I had included the gln there, I could just say, oh, by paper of Eric and Michaela's, exactly what I'm getting is the type an kLR algebra. But that's not quite right. So what's the a lambda I get, or actually, let me be a little more careful about this. So there exist integral weights such that when I sum together, so this collection x, this is some set of integral weights. When I sum over pairs in here, I take this bimodule a lambda mu. I get. Yeah, oh, sorry, yes. So this is almost the type a kLR algebra. So what's different? Well, because I have this square here, my strands with label n aren't allowed to move. So let me just tell you the type an kLR algebra, I'm definitely running out of time, that we want here. And this is with one strand label one, two strands label two, etc. So this is an algebra where the elements are string diagrams like this with dots on them and labels from this set of nodes. Maybe I have a one here and a three here and a seven here and a two here, but with property that strands with label n don't cross. So in terms of quiver flag varieties, somehow putting this box here means I should just choose the standard flag on this guy and only look at flags of this quiver representation that agree with the standard flag on this guy. And because I can never change that flag, here I have a strand label n, here I have a strand with label n, they're not allowed to cross. Other guys are allowed to cross them, but they can't cross each other. And to emphasize the difference here, I'm going to draw these strands with label n as red. It's exactly that on the nose. So the interesting question is, do you want dots on here? And the answer is it depends whether you keep the center or not. All right. So these things, they're all not just for the quiver and quiver. Yes, Joel is pointing out that this is an interesting special case of a much more general theorem by Joel Peter Tingley, myself, Alex Weeks, and Oded Yacoby. This works for all quivers. Another exercise to the audience is to, well, OK, if you want, I can change it to my name. This works for all, but I certainly should say symmetric. So a very interesting question is, what happens when you mash together this talk and Herakus and try to figure out, well, how does this apply to these slightly generalized versions of Coulomb branches? And what one might hope is that you would get KLR algebras for non-simply lace type. And I will tell you that that is surely a vain hope. And what you will actually get is KLR algebras of this bigger diagram that you got your interesting Dink and diagram by folding up, but not folding in the dual way, which I think I've now decided should be called furling. So for a quiver gauge theory, yes, you always get some version of a KLR algebra, suitably generalized. And essentially, these guys corresponding to tensor products, but without the sort of cyclotomic quotient relation. All right, running very low on time. So I think I'm going to skip the actual correspondence here. So the way these match up is that when you have a weight, you diagonalize it. You look at the weights that occur along the diagonal. You put those on the real number line, and then you label them with which node they came from. So I'll just say that. That's the correspondence you guys can work it out. One interesting corollary simple Galvan-Settlin modules for U of GLN are more generally for these quiver guys, which includes orthogonal Galvan-Settlin algebras, et cetera, correspond to a dual canonical basis. And the obvious question is dual canonical basis in which space? And let me just say, in this case, you take the negative part of U of SLN. So the negative Unipotent Radical take its universal enveloping algebra, and then you tensor with N copies of the standard representation and look at a particular weight space in there. And for the other orthogonal Galvan-Settlin algebras, you'll get the other weight spaces in here, et cetera. And the generalization to other quivers involves, Joel talked about this representation. He was using lambda for a highest weight here. So here you tensor together the fundamental representations corresponding to lambda. So the answer is no. No, I mean, I can explain to you what's going on. But somehow the point is that, yeah, so there is a sort of spherical thing that happens, which is, yes, for, so which algebra you get depends on this choice of central character. And somehow the central character is telling you where these red lines are. And this doesn't actually happen for U of GLN because all the regular blocks are the same. But for some other quiver, you know, other representations where you have kind of more general highest weight, yes, there are some situations where essentially these red lines get close enough together that there are sort of some item potents that don't occur. And in this quiver case, we call these parity KLR algebras. But in general, yeah, it can sort of happen that you can't, you get sort of a KLR algebra, but maybe you can't get all the item potents. There's some sort of obstruction. Yes. Oh, well, I mean, if you have the parabolic ones, you get something Merida equivalent, right? So I can choose, yes, there's some, some work, I'm running out of time. So we can discuss this later. Believe me, I have thought about this issue. I'm not completely screwing this up. I'm just being a little, you know, skimming over a few details. Is that what you're asking about? I don't, wait. But by spherical, like, you're thinking that there should be some item potent. But you mean as opposed to the complete flag variety. Yes, but that's not the issue. We can discuss it later. So there is something you have to be a little bit careful about here. But I mean, for most weights of even the spherical guy, you get this. Really that, so right, the issue you're digging into here is I had a parabolic P lambda and I'll only get the KLR algebra. It's kind of on the nose if that parabolic is a borrel. And of course, usually it is, right? Like most, most co-characters have a borrel attached to them, not a more complicated parabolic. Really that more, somehow that more complicated parabolic isn't the issue. All right, because of course if you replace it with a borrel, you get a merida equivalent algebra. So like, obviously that's not the issue. All right, but I'm totally running out of time, so let's not get totally sidetracked with this. Let me just say another important example is if you take G to be GLN and N to be the adjoint representation plus L copies of CN, then you will get the rational Terednik algebra of GL1N. And there you should be concerned about spherical things. Again it doesn't make that big of a difference, but let me sort of, right, let me just add spherical here. And under this isomorphism, where does the equivariant co-amology of GLN for a point go? It goes to the algebra of symmetric polynomials in dunkel-optam operators. So now a Gelfand-Setlin module for the spherical rational Terednik algebra is one where these dunkel-optams act locally, finitely. You could call that a dunkel-optam module if you wanted to. And it's a known theorem due in part to many people in this room that the simple module in category O over these rational Terednik algebras correspond to dual canonical basis vectors for a twisted fox base. And you can actually extend this theorem to dunkel-optams. So simple dunkel-optam modules are going to correspond to a dual canonical basis in the twisted fox base tensored with u minus of GLN hat. And since I'm low on time, let me try not try to explain anything about this, but let me just say these algebras you get a lambda or more generally the sum come from what are called weighted KLRs. Yeah, twisted fox base. I mean, this is your conjecture, so you better know what it means. Yes. So the correct statement depends on what your maximal ideal looks like, but it's always whatever sort of you want to fix a maximal ideal and look at its orbit under the affine file group. And whatever you got in category O, you take that and you tensor it with u minus of GLN hat. So you always get sort of the same amount extra. It's always kind of category O plus this guy that's fixed and doesn't depend on the parameters other than kappa. Right, I should say like GLE hat where E is the denominator of kappa. And in particular, if E is not rational, it's GL infinity. Or do, and I should say that if you do GLPN, this seems to be very closely related but a tiny bit different. But this would be weighted KLR of the same GLP. Yes. Yes. I'll just mention that this sort of seems to fit into the same bucket, but it's a little bit more complicated. It definitely has this kind of correct form. It's a rational Galois order. But this is something I'm currently working on with a grad student at PI. All right, so, okay. Oh, damn. Maybe I should just stop. That's too bad. Well, if you wanted to know about duality, you shouldn't have asked so many questions. So this all explains why symplectic duality works, but I guess I'll leave it as an exercise to you to figure out why that's true. All right, so I'll just stop now and if you have more questions, feel free to bother me. Thank you again and have a great evening. personalized rodz показывucken.
 So my talk is going to be kind of dual in some sense to Joel's talk, so it's also going  to be about the question of how to understand the representation theory of Coulomb branches.  But it's going to be dual in the sense of it's going to be looking for more things like  projectives rather than things like symbols.  So let me sort of introduce a general algebraic setup, which I think it's good to think about  Coulomb branches in terms of.  So let A be a k algebra, of course here k is a field, and let R inside it be a commutative  subalgebra.  We can also have a fun discussion about what non-commutative algebras you could put here  instead, but commutative is a good start.  So whenever you have a setup like this, there's a natural set of modules, category of modules  you can study, which is quite interesting.  So A module is Gelfand Settlin, and I mean this is in the grand tradition of naming  things after people who had extremely little to do with them or at least had some very  distant connection in the past.  So some people might prefer to call these harsh chandr, which I think would follow the same  tradition or weight modules, which I think would be confusing.  So I'm just going to use this terminology, and I hope people will forgive me.  And of course, that's a lot to write, so I'm going to write Gt.  If it's locally finite for B, it is R locally finite.  So that means that for all elements of this module, if I look at R times it, and then  I look at the dimension of that as a k vector space, it's finite.  So of course, this includes all finite dimensional modules, but will also include things like  this category O that Joel mentioned.  All right, so how do we study modules like this?  The one reason I like this definition is that there's an obvious way to break this module  up into pieces.  If I have a locally finite module over a commutative algebra, then it naturally breaks up into  a sum of things that are killed by powers of maximal ideals.  So given lambda in the set of maximal ideals of R, I can consider sort of the quote unquote  weight space, which is all the things in V that are killed by some power of the maximal  ideal.  And for whatever reason, I like to think of lambda as something floating off in space,  and then it has some particular maximal ideal attached to it.  You could sort of say, like, oh no, this maximal ideal, that's the point in max spec.  All right, and a very easy lemma.  V is R-Gelfan-Setlin, if and only if V is the direct sum of these subspaces.  Do I want some finiteness condition on R?  Oh, oh god, I always get stuck on these stupid things.  Oh, right, I want R to be finite type.  And if R is finite type, then this quotient is a finite extension of k.  And so it's, yeah, anyways.  All right, you're up.  Yeah, finitely generated as a k module, as a k algebra.  Yeah.  All right, so what are examples that might be interesting to think about?  So you might choose A to be U of G and R to be U of H. In that case, an R-Gelfan-Setlin  module, it's tempting to say weight modules, but that's not quite right.  I'm taking a power of this maximal ideal.  So it's generalized weight modules.  All right, another very interesting case to consider is A being U of GLN and then R being  the Gelfan-Setlin subalgebra, which is what this whole setup is named after.  So that's the subalgebra generated by the center of the universal enveloping algebra  of GLK for k equals 1 up through.  All right, and one final example to consider is, well, A is the Coulomb wrench for some  group and some representation.  And unfortunately, Joel and I didn't coordinate notation before our talk.  I'm using the notation of BFN's paper that N is a module over this group G, and I'm using  V for something else.  So I apologize about that, because Joel, of course, used the notation from my paper.  Anyways.  So there's a natural commutative subalgebra to consider in here.  OK, yeah, well, this is a pretty shitty setup, so I apologize.  So A is the Coulomb branch, and I can take R to be the commutative subalgebra given by  the equivariate parameters for the group.  So R is the equivariate homology of a point for the group, which includes inside the Coulomb  branch, of course, because it's the equivariate homology of something else.  All right.  So a basic question, what are the simple GT modules  for a given A and R?  So this one of what are the different weight modules for a semi-simple lead group is one  that's elicited a lot of study.  And in part, people started thinking about this example because this one was too hard.  And they said, well, let's maybe add this extra condition, the whole Galfan set when  subalgebra acts locally finitely.  And that was a problem that also attracted a lot of attention over recent years.  But it's one where people got a little bit stuck.  And let me try to explain where they got a little bit stuck and how I think we can get  unstuck now.  So there's a general approach to understanding this category.  And I think part of the reason this is such a nice category to consider is we have such  a nice approach.  And this approach, so definition, we say A R is R is Chandra.  If for all elements of A, the bimodule RAR is finitely generated.  Well, it's obviously finitely generated as a bimodule.  But this is that it's finitely generated as a left or right module.  So this is sort of a size criterion that A is non-commutative, but the way it not commutes  past things in R is not too bad.  There's sort of only finite amount of extra information when you multiply in the left  as well as on the right.  Look, that's what it says in the paper.  This is the definition from the paper.  What am I supposed to do?  And given this theorem, droste futone and osienko, is that for a Harchandra pair, there's  sort of a nice way of describing these guys.  And let me just say what it is in words.  You look at these, you think of them as functors, and then you say, what are the natural transformations  between these functors?  So somehow, Galphons-Setlin module is the same thing as saying what all these weight  spaces are, and then an action of all the natural transformation between the functors  of having weight spaces.  So the category of A modules, which are Galphons-Setlin, is equivalent to the modules over some other  category, and those should be discrete, so this other category is going to have its apology.  And the other category is you just think about, well, what kind of elements of my algebra  would give me natural transformations between these weight functors?  Yes.  Yes.  So the sort of important consequence this Harchandra hypothesis has is that if you are  generated by a finite dimensional R invariant subspace, then you are a Galphons-Setlin module.  So that's sort of the important thing you need to make this theorem work.  So here C is a category where the objects are just this max spec of R, and the hams  are given by an inverse limit.  So the problem from lambda to mu is I take the inverse limit of A mod, let me get this  right, M mu to the k plus M lambda to the kA.  So I'm multiplying by one of these, oh no, I did the wrong one, multiplying by one of  these maximal ideals on the left and by the other one on the right.  So this quotient, this is the natural way to get a natural transformation between a  kind of truncated version of this weight functor, where you require a particular power to act  by zero, and as I let k go to infinity, I get something that's defined on the whole  weight space.  So as an inverse limit, this has a topology, and I only want representations where that  action is continuous when I give the representation the discrete topology.  I'll just write discrete, and that's what I'll be doing.  All right, so this is a very nice theorem, but it's only as nice as your understanding  of what these homomorphism spaces are.  So let me tell you there's sort of a slight generalization of this theorem, which is that  if I want the simple Galphons-Setlin modules such that a particular one of these weight  spaces is not zero, then those are actually in bijection with simple discrete modules  over what I'm going to call a hat sub lambda, that's from lambda to lambda.  All of the examples I said in the beginning are harsh on the.  So for example, if you wanted to understand all weight modules, like in the usual sense  over Li algebra, simple ones, where a given weight space was not zero, this tells you  there is some algebra out there, but those are the same as simple modules over.  But that algebra is horrible and infinite dimensional.  It's very large.  In particular, its finite dimensional modules are all the different weight spaces of all  the different finite dimensional guys, plus maybe some more.  So it's going to be enormous.  In particular, it should have infinitely many simple representations of all kinds of different  dimensions, going to be a very complicated algebra.  It's not clear you gain anything from thinking this way.  But for Coulomb branches, there's a very nice answer to this question.  So I should just pause for a moment and ask, are there any questions?  So for the carton inside of U of G, yeah, I don't know anything interesting about this  algebra that you couldn't figure out some other way.  And I mean, I would say somehow what's going wrong there is that U of H is too small.  In particular, it's not a maximal commutative subalgebra.  Whereas the Gelfand-Setlin subalgebra is, and this guy is also a maximal commutative  subalgebra.  Maybe we should find out that two is indeed a special case of three.  Sorry?  Yes.  Two is indeed a special case of three.  But you learn very interesting things.  If you only care about two, it's worth thinking about three.  So I'm going to tell you a theorem about Coulomb branches that tells you something interesting  and new about U of GLN.  Sir, a random string from last lesson, is it just a bijection of sets?  No, it's, I mean, so it is a bijection of sets, but it's a particular one.  The lambda weight space of this symbol is a simple module over this guy, and that gives  you the bijection.  So I mean, you can strengthen this if you looked at all modules over a lambda hat.  You would get the category of Gelfand-Setlin modules modulo those that have trivial lambda  weight space.  So you do learn something about extensions, but when I only talk about simple modules,  yeah, the only structure there is a set.  But it's a bijection which is very natural.  So you just look at this guy acting on the lambda weight space.  Here?  Discrete.  All right, cool.  Sorry, this equivalence is you look at all the weight spaces, and these are the natural  transformations between the functors of taking weight spaces.  So I didn't want to get down to the details of it.  You can go read their paper, but I mean, this guy naturally acts on the weight space, and  it's that action.  I'm not sure if you answered the question.  The question is, are the C modules, are they continuous?  They're discrete.  There's a word discrete here.  So yes, they're continuous, and they have the discrete topology.  Right?  I mean, this is just some way of saying that some power of this ideal is zero, but I don't  want to tell you which one.  So I don't want to fix any particular one, but I want some power of it to be zero, and  that's the same as saying that the inverse limit acts on it with the discrete topology.  All right, anything else I screwed up?  All right, very good.  OK, right.  So for Coulomb branches, there's a very nice general answer.  So one thing to note for Coulomb branches is this max spec of r.  I can really think of as semi-simple elements of g times c.  So this is the Lie algebra of the group g times c star, or this is the rotation c star.  And this is modulo conjugation.  So sorry, this is for the Coulomb branch.  So I'm going to specialize from now on and only talk about Coulomb branches.  And all other interesting examples I know are actually subcases of this one.  So here I'm just writing out for you what is max spec of the equivariant co-amology  of a point for g, right?  That's the torus of the group, mod the vial group, which is actually the same as semi-simple  elements mod conjugation, and I forgot here that I should have put a c star, because I  also want the loop rotation.  So when I think of the Coulomb branch as equivariant co-amology, I have this equivariant parameter  h for c star.  But when I get an actual non-commutative algebra like the universal enveloping algebra, I specialize  h to be 1.  So that means my, that's the same as saying that my semi-simple element is of the form  x, 1.  So when I exponentiate it, I get the usual rotation action of c star, and then I've somehow  adjusted that by some co-character in the group.  All right, and out of this data, I'm going to get a description of this algebra for the  Coulomb branch.  Yeah, yeah, the adjoining action.  How do you get the element of the speck of r from that?  Element of the speck of r.  Yeah, so I'm looking at, you know, lambda plays a role here and it plays a role here.  This is a theorem for, yeah, it's the whole category.  So for every maximal ideal, I assign a r module to it.  And the discrete thing is just that that r module factors through some power of the maximal  ideal, but I don't want to tell you which one it is.  All right, so defining this requires a little bit of notation.  So remember, I have this representation n that I fixed from the start.  This is part of the data of how I construct the Coulomb branch.  So having chosen an element of the Lie algebra, I have a subspace in here where this is the  subspace where x has integral weights.  And I have a further subspace in there.  So I'm going to call n lambda minus, and this is where x has non-positive weights.  No, I was doing that right.  So lambda is the result of the same thing?  Well, not quite, but yes.  So x is lambda thought of as an element of the group.  But you know, x is only well-defined up to conjugation, but if I replace x by its image  under the adjoint, then I'm just acting by that element of the group on n lambda.  All right, and similarly, if I do this in the adjoint representation, I'll get a subgroup  which is exponentiate the subalgebra that has z weights under the adjoint action.  And then inside here, well, if I was going to be really consistent, I would call this  g lambda minus, but I'm actually going to call it p lambda.  So this is a parabolic that, again, comes from looking at the adjoint action of this  x and looking at where it has non-positive weights.  All right.  So let me then make one more definition, let y lambda be the sort of the associated vector  bundle to g lambda crossed over p lambda with n lambda minus.  So let me just remind you what this is.  This is I take a coset of p lambda.  I take an element of n.  So the condition I need is that n is in g acting on n lambda minus.  All right.  What was the point you were making with the max vector a star g plus b?  What was that we were saying about the max vector b?  So those are the weights that the commutative subalgebra inside the Coulomb branch could  act by.  So when I want to analyze a representation of the Coulomb branch, I want to look at  this commutative subalgebra acting on it.  And if it's locally finite, then that's going to break up into some sum over maximal ideals.  And the different options for the maximal ideals are exactly these conjugacy classes.  And if I've specialized h equals 1, then these conjugacy classes where the part along, so  the component along the loop direction tells you, oh, when I mod out by that maximal ideal,  what scalar does h get specialized to?  And I want to look at the case where h is equal to 1.  All right.  So theorem.  So I should give some credit for this to Hiraku because he sort of explained this way of stating  it to me.  You will search in vain for the paper of his that proves this, but it is in my paper.  This algebra A hat lambda is the Borel-Mor homology, equivalent with respect to g lambda  of the fiber product y lambda over m lambda of y lambda.  And of course, more generally, if I take comm from lambda to mu, that's what's tempting  to think that it's sort of always this guy where I just take this product.  But you'll note there's some weird asymmetry.  This g lambda and n lambda, well, shouldn't I take g mu and n mu?  But this is only true if lambda and mu are conjugate under the affine-vile group.  All right.  So that is to say I can choose them in a maximal torus such that they differ by an honest co-character  in that maximal torus.  I just get 0 if lambda and mu are not conjugate under the affine-vile group.  And if they're conjugate under the affine-vile group, well, these weights change, but they  change by an integral amount.  So actually, this g lambda is equal to g mu and this n lambda is equal to n mu.  All right.  And of course, there's something missing here.  I need a hat here.  And that just means I complete with respect to grading.  OK.  So all questions about modules over the Coulomb branch that have this Gelfand-Setland property  are actually questions about this finite dimensional convolution algebra.  Yes?  Does that have something to do with the multiple terms of the formula?  Not obviously.  So I mean, an exercise to the audience is figure out how Joel's talk was secretly using this.  So I suspect this complaint about the fact that there was non-equivariant homology instead  of equivariant homology is that, in fact, of course, there's an obvious quotient of this  algebra.  Right?  If I remove the equivalence, then I will get a homomorphism from this guy to just the  usual Borrel-Mor homology of this fiber product.  And it's easy to work out.  Every simple factors through this map.  So I wasn't able to confirm this since I haven't had much time since I saw Joel's talk.  But I suspect very strongly what was happening is you guys were defining a module over this  and then sort of pulling back and going through this equivalent without realizing that was  what you were doing.  Sorry?  So you are completing after some factor like that.  Well, this is by the fact that it's a discrete module.  So if I have a module over this completion, which has the discrete topology, then that  means some power of the obvious maximal ideal in the completed equivalence homology of this  g lambda has to act trivially.  And if you're simple, that means that it has to be the maximal ideal itself.  So I mean, there's secretly some shift here.  So that the obvious maximal ideal over the homology goes to the maximal ideal that corresponds  to lambda here.  What is g lambda?  G lambda.  So it's the subgroup whose Lie algebra is the space where x has integral weights under  the adjoint action.  If you have a co-character, you can think of that as a point in spec of the equivariate  co-homology for a point.  So the sort of important thing here is, for example, let's see, what can I safely delete?  Not much.  This is getting pretty tight.  So yeah, that's the best way to say this.  So for example, if lambda is generic, then this is trivial.  So if you have sort of a totally generic central point in spec, then this guy becomes trivial  and this statement just says that you get a copy of c.  And one interesting corollary.  No, no, no, not if lambda is zero.  If lambda is generic, then g lambda is trivial.  If I look at every root eating lambda and I don't get an integer, then this means that  this guy is just, sorry, g lambda isn't trivial.  It's the torus.  And so I just get the equivariate co-homology of the torus and I complete with respect to  the obvious maximal idea.  I mean, x is, yeah, x lambda are the same thing.  So no, lambda is x1.  But these live in the same space thought of in a slightly different way.  This guy lives in spec of h star g of a point.  And this guy lives in g modulo the adjoint action, or I should say g semisimple mod the  adjoint action.  And it's left as an exercise to the viewer to remember how these things are isomorphic.  Yes, no, no, it's not the centralizer of lambda.  It's the stuff that lambdas integral with respect.  Oh, maybe you're right.  Wait.  If you take the exponential of that element of the Lie algebra, then yes.  But if you think about it as a co-character, then no.  So exactly why I was, oh, this is already trashed.  All right, so somehow, I mean, if you don't understand Coulomb branches, you're not going  to magically understand them from me saying this.  The important point here is that the weights that you can get in a Coulomb branch essentially  correspond to elements of the torus mod the vile group.  And if you want to understand the representations where that maximal ideal appears, there's  a sort of simple topological way to do this.  And again, since I'm running low on time, I leave as an exercise to the audience to  think through, for example, what thinking about perversives on n lambda tells you about  the representations of this algebra.  There's a very natural interpretation of this, familiar to anyone who's read Ginsburg  and Chris's book.  All right, so a very important special case will be interesting to people who just like  representation theory and have never heard of Coulomb branches before, which is u of  gln itself.  So in terms of these pictures for quiver gauge theories, what this means is we take 1, 2,  3 in circles and then n as a square.  So that means my g is gl1 times up through gln minus 1.  I am stopping at n minus 1.  I don't mean to go to n.  The gln is flavor.  And my representation n is 1 by 2 matrices times 2 by 3 matrices dot, dot, dot up through  m by n minus 1 by n matrices, where this group acts in the usual way by pre-n post composition.  All right, so the Coulomb branch here, and again, exactly the answer depends a little  bit on your conventions about flavor.  So I believe I'm following what Joel did by fixing the flavor to be scalars.  So this is going to be u of gln mod a maximal ideal of the center.  So there's sort of a way to do this where you incorporate the equivariant cohomology  of gln as well, and that becomes the center of C. Sorry, the center of u of gln.  Let me, since I'm talking about individual weights, each individual weight, the center  acts through some character, so I might as well fix that already.  And if you think through what's written on this board, let me not quite erase it yet.  The space is y lambda.  I get out of this.  So I can think of a point in here as a quiver representation for this quiver.  And I'm almost getting the quiver flag variety, but I'm not acting by gln on here, so I get  something a little bit funny.  So if I had included the gln there, I could just say, oh, by paper of Eric and Michaela's,  exactly what I'm getting is the type an kLR algebra.  But that's not quite right.  So what's the a lambda I get, or actually, let me be a little more careful about this.  So there exist integral weights such that when I sum together, so this collection x, this  is some set of integral weights.  When I sum over pairs in here, I take this bimodule a lambda mu.  I get.  Yeah, oh, sorry, yes.  So this is almost the type a kLR algebra.  So what's different?  Well, because I have this square here, my strands with label n aren't allowed to move.  So let me just tell you the type an kLR algebra, I'm definitely running out of time, that we  want here.  And this is with one strand label one, two strands label two, etc.  So this is an algebra where the elements are string diagrams like this with dots on them  and labels from this set of nodes.  Maybe I have a one here and a three here and a seven here and a two here, but with property  that strands with label n don't cross.  So in terms of quiver flag varieties, somehow putting this box here means I should just  choose the standard flag on this guy and only look at flags of this quiver representation  that agree with the standard flag on this guy.  And because I can never change that flag, here I have a strand label n, here I have  a strand with label n, they're not allowed to cross.  Other guys are allowed to cross them, but they can't cross each other.  And to emphasize the difference here, I'm going to draw these strands with label n as  red.  It's exactly that on the nose.  So the interesting question is, do you want dots on here?  And the answer is it depends whether you keep the center or not.  All right.  So these things, they're all not just for the quiver and quiver.  Yes, Joel is pointing out that this is an interesting special case of a much more general  theorem by Joel Peter Tingley, myself, Alex Weeks, and Oded Yacoby.  This works for all quivers.  Another exercise to the audience is to, well, OK, if you want, I can change it to my name.  This works for all, but I certainly should say symmetric.  So a very interesting question is, what happens when you mash together this talk and Herakus  and try to figure out, well, how does this apply to these slightly generalized versions  of Coulomb branches?  And what one might hope is that you would get KLR algebras for non-simply lace type.  And I will tell you that that is surely a vain hope.  And what you will actually get is KLR algebras of this bigger diagram that you got your interesting  Dink and diagram by folding up, but not folding in the dual way, which I think I've now decided  should be called furling.  So for a quiver gauge theory, yes, you always get some version of a KLR algebra, suitably  generalized.  And essentially, these guys corresponding to tensor products, but without the sort of cyclotomic  quotient relation.  All right, running very low on time.  So I think I'm going to skip the actual correspondence here.  So the way these match up is that when you have a weight, you diagonalize it.  You look at the weights that occur along the diagonal.  You put those on the real number line, and then you label them with which node they came  from.  So I'll just say that.  That's the correspondence you guys can work it out.  One interesting corollary simple Galvan-Settlin modules for U of GLN are more generally for  these quiver guys, which includes orthogonal Galvan-Settlin algebras, et cetera, correspond  to a dual canonical basis.  And the obvious question is dual canonical basis in which space?  And let me just say, in this case, you take the negative part of U of SLN.  So the negative Unipotent Radical take its universal enveloping algebra, and then you  tensor with N copies of the standard representation and look at a particular weight space in there.  And for the other orthogonal Galvan-Settlin algebras, you'll get the other weight spaces  in here, et cetera.  And the generalization to other quivers involves, Joel talked about this representation.  He was using lambda for a highest weight here.  So here you tensor together the fundamental representations corresponding to lambda.  So the answer is no.  No, I mean, I can explain to you what's going on.  But somehow the point is that, yeah, so there is a sort of spherical thing that happens,  which is, yes, for, so which algebra you get depends on this choice of central character.  And somehow the central character is telling you where these red lines are.  And this doesn't actually happen for U of GLN because all the regular blocks are the  same.  But for some other quiver, you know, other representations where you have kind of more  general highest weight, yes, there are some situations where essentially these red lines  get close enough together that there are sort of some item potents that don't occur.  And in this quiver case, we call these parity KLR algebras.  But in general, yeah, it can sort of happen that you can't, you get sort of a KLR algebra,  but maybe you can't get all the item potents.  There's some sort of obstruction.  Yes.  Oh, well, I mean, if you have the parabolic ones, you get something Merida equivalent,  right?  So I can choose, yes, there's some, some work, I'm running out of time.  So we can discuss this later.  Believe me, I have thought about this issue.  I'm not completely screwing this up.  I'm just being a little, you know, skimming over a few details.  Is that what you're asking about?  I don't, wait.  But by spherical, like, you're thinking that there should be some item potent.  But you mean as opposed to the complete flag variety.  Yes, but that's not the issue.  We can discuss it later.  So there is something you have to be a little bit careful about here.  But I mean, for most weights of even the spherical guy, you get this.  Really that, so right, the issue you're digging into here is I had a parabolic P lambda and  I'll only get the KLR algebra.  It's kind of on the nose if that parabolic is a borrel.  And of course, usually it is, right?  Like most, most co-characters have a borrel attached to them, not a more complicated parabolic.  Really that more, somehow that more complicated parabolic isn't the issue.  All right, because of course if you replace it with a borrel, you get a merida equivalent  algebra.  So like, obviously that's not the issue.  All right, but I'm totally running out of time, so let's not get totally sidetracked  with this.  Let me just say another important example is if you take G to be GLN and N to be the  adjoint representation plus L copies of CN, then you will get the rational Terednik algebra  of GL1N.  And there you should be concerned about spherical things.  Again it doesn't make that big of a difference, but let me sort of, right, let me just add  spherical here.  And under this isomorphism, where does the equivariant co-amology of GLN for a point go?  It goes to the algebra of symmetric polynomials in dunkel-optam operators.  So now a Gelfand-Setlin module for the spherical rational Terednik algebra is one where these  dunkel-optams act locally, finitely.  You could call that a dunkel-optam module if you wanted to.  And it's a known theorem due in part to many people in this room  that the simple module in category O over these rational Terednik algebras correspond  to dual canonical basis vectors for a twisted fox base.  And you can actually extend this theorem to dunkel-optams.  So simple dunkel-optam modules are going to correspond to a dual canonical basis in the  twisted fox base tensored with u minus of GLN hat.  And since I'm low on time, let me try not try to explain anything about this, but let  me just say these algebras you get a lambda or more generally the sum come from what are  called weighted KLRs.  Yeah, twisted fox base.  I mean, this is your conjecture, so you better know what it means.  Yes.  So the correct statement depends on what your maximal ideal looks like, but it's always  whatever sort of you want to fix a maximal ideal and look at its orbit under the affine  file group.  And whatever you got in category O, you take that and you tensor it with u minus of GLN  hat.  So you always get sort of the same amount extra.  It's always kind of category O plus this guy that's fixed and doesn't depend on the  parameters other than kappa.  Right, I should say like GLE hat where E is the denominator of kappa.  And in particular, if E is not rational, it's GL infinity.  Or do, and I should say that if you do GLPN, this seems to be very closely related but  a tiny bit different.  But this would be weighted KLR of the same GLP.  Yes.  Yes.  I'll just mention that this sort of seems to fit into the same bucket, but it's a little  bit more complicated.  It definitely has this kind of correct form.  It's a rational Galois order.  But this is something I'm currently working on with a grad student at PI.  All right, so, okay.  Oh, damn.  Maybe I should just stop.  That's too bad.  Well, if you wanted to know about duality, you shouldn't have asked so many questions.  So this all explains why symplectic duality works, but I guess I'll leave it as an exercise  to you to figure out why that's true.  All right, so I'll just stop now and if you have more questions, feel free to bother me.  Thank you again and have a great evening.  personalized rodz показывucken.
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