 Okay, so it's a great honor and pleasure for me to come here for Samson's birthday. I mean, we have been friends for how many years? I don't even know. 28. 28, more than that, no? 29. 29, maybe. Okay, so I thought that it was a whole life anyway. So we have discussed about many things. I have listened to him on physics many, many times without being sure of understanding what he was saying. And I mean, I certainly remember that when he talks to me, he says, you're the unusual Frenchman. And I must say, I won't tell you what he refers to, but I'm very proud of this. Okay, so I will talk about Focke-Planck operators in the center of the enveloping algebra. So basically, I mean, I will explain certain things which are from physics point of view rather down to earth, but maybe with applications to questions in mathematics, which are of some interest. So I will basically explain how classical objects like Clape-Lations can be deformed naturally to another category of operators, which are in some sense operators well known in statistical physics, like Focke-Planck operators. And this deformation is legitimacy. I mean, it's not, you know, it's not for my own pleasure, but it was to connect with things like Zellberg's trace formula by an interpolation process. So the interpolation process I refer to is interpolation between the Laplacian on one hand and a dynamical system like the Judizic flow on the other hand. And Zellberg's trace formula is exactly a sort of formula in which on one hand you have spectral data like Laplacian's and on the other hand you have Judizic's. And we will see our certain understanding of this formula will lead to explicit evaluations of objects which were not known before, but using instruments which are in some sense taken from physics. So let me first of all start with extremely mental question, I mean, considerations. So first of all, the Euler characteristic, so you take X to be a compact Riemannian manifold, and you know that the Euler characteristic is a global invariant that the alternates some of the dimensions of the homology groups of X. Now if you take a diffiorpheism of X, so it acts naturally on the homology, so there is an action, linear action, and so you can find the left-shet number, which is again a global invariant which generalizes the Euler characteristic and which is just the alternates some of the traces of the diffiorpheism group acting on the homology. Now there is a way to compute explicitly this left-shet formula, which is the so-called fixed-point formula or in the case of the identity Chern-Gauss-Bernie theorem, but which is always based on the fact, I mean, can be based on the fact that you can actually to compute the homology you can just use the Durum complex, which homology is just the homology of X, and if you introduce a Riemannian metric, you introduce a former adjoint of the Durum operator and dx, which is the little dx plus little dx star to be the corresponding drug operator, and is square as a Hodge-Laplacian. So basically the starting point of this sort of computation is to say that you can re-express left-shet number, which is a global simple in some sense invariant in terms of super traces of operators, which are now calculated on an infinite dimensional object, and that the super trace of G exponential minus S times the Laplacian, and this formula is true for NES. So in some sense, if you just look at the right-hand side, there are many eigenvalues, the contribution of zero eigenvalues of nonzero eigenvalues disappear, which explains why the right-hand side does not depend on S, so that by making S tend to zero, you obtain, I mean, you can obtain at least that the way of proving the left-shed formula, the left-shed formula in terms of integral on fixed point set XG of the illiform of TXG. So if the points are isolated, you just get a sum of plus of minus one. So let me now move to the real subject of my talk. So I will consider now the heat operator on the manifold, on the Riemannian, compact Riemannian manifold, X. And I introduce again the Laplacian, delta X, and the corresponding heat current. So now G for T positive, it's the heat current on a time T. T will be fixed. So this is an operator which acts on smooth functions on X with values in R. And so I will ask four rhetorical questions. So the first question is, is a trace of the heat kernel a null characteristic? So can I think of this trace exactly in the way I was thinking of the trace of the difform of this before? Can I view this sort of topological invariant? So is this a generalized null characteristic? So the second question I will ask is, exactly as before when I started from the chronology and I replaced the chronology by the Durand complex, can I replace the smooth functions by a much bigger space? By a much bigger space R, such that I would write exactly a formula of the same time as before, re-expressing the trace of G in terms of a super trace of G but acting on a much bigger space equipped with a generalized Dirac operator, which is exactly the analog of the operator DX which I introduced before. And such that in a certain way I will, instead of getting, let's say, Chern-Gauss-Bonnet or the fixed point formulas when making B tend to infinity, I would obtain something like Zellberg's trace formula, which is an extension of Poisson formula. So basically the difference with what I did before is now our original space, which was the homology in the first example, is now the space of smooth functions. So let me just formalize the analogy again. So in the case of left-shed number, I had L of G, which is left-shed number, generalized Euler characteristic. On the right-hand side I had a local formula for it, which is a fixed point formula, or Chern-Gauss-Bonnet, which is local. And to prove the formula, we use an interpolation involving the Durand complex and basically the interpolation just tells you that the left-hand side is ultimately equal to the right-hand side. So this is exactly what we want to do. In some sense I start on the trace of the heat kernel, which is a global object. And now of course I know more or less what I want to get on the right-hand side, which is Zellberg's trace formula, but I want to find something in the middle which will do for us exactly what the Durand complex was doing before. So we are forced in some sense to consider the space of smooth functions in a manifold as the cormology of something. The question is to know what is it the cormology of? So before going on, but let me just remind you what Zellberg's trace formula is and a concrete example, though this was the original formula proved by Zellberg. So if X is a remanded surface of constant scalar curvature minus 2, we introduce a closed physics and their lengths L gamma. So Zellberg's trace formula in this case is an extension of Poisson formula. So it expresses on the left hand side the trace of the heat kernel in terms of a bunch of terms. The first term is a complicated one with an integral of r and with a volume factor in front of it, the volume of x, sorry it's not sigma is x, and then there are a number of terms which are indexed by the closed geodesics and with complicated factors. So you see exponential minus a length square, the usual thing. In the denominator you have two hyperbolic sign of length square over two and then you have this infinite sum. So in some sense what I will show is exactly finding what is a proper analytic and geometric structure which will interpolate between the two sides. So let me now move to the context on which I will effectively develop this theory. So this will be the context of the symmetric spaces. So I introduce G to be a real reductive group. So if you don't know what reductive groups are just think of SL2R if you wish and K a maximal compact subgroup which is the case of SL2Y, that's one, and x equal G mod K is the corresponding symmetric space. So I also introduce the carter splitting of the Lie algebra of the group. So typically the Lie algebra splits into two pieces P and K. There is a natural quadratic form of the Lie algebra which is positive on P and negative on K. That's an orthogonal splitting. And so this splitting of Lie algebras descends to a bundle of Lie algebras on the symmetric space, Tx plus n modeled on the splitting G equal P plus K. So again the basic example is G equal SL2R, K just a matrix, orthogonal matrix, 2 by 2 matrix, x is just a complex upper half plane and Tx plus n, there is this n, is a bundle of dimensions 3 over x. So the analysis that we shall do will be done on the product of the group by its Lie algebra and quotient by K. So quotient by K, K acts on the group on the right, it acts on the Lie algebra on the left and there is a standard procedure to do the quotient. So the analysis will be done on this and this subject is itself a geometric object. So it is at the same time a group theoretic object but that's also a geometric object that the total space x, x plus or hex hat of Tx plus n over x equal G mod K. So you have this if you like the upper half plane, you add this 3 dimensional bundle and so in the case of SL2 you will do the analysis, I mean differential operators will occur on a space of total dimension 2 plus 3 which is 5. How does the total bundle is trivial, right? The total bundle is trivial, yes. I mean the fact the bundle is trivial doesn't mean that the theory, I mean it doesn't imply at least directly that the theory I will develop is trivial because the, no, but I mean I want to make the point that there will be a non-trivial coupling between the base and the fiber. Okay, so we will do two separate constructions first of all on the group and on the other side, on the other hand on the Lie algebra. So for the moment we forget about the quotient and only explain the two constructions on one hand on the group and on the other side on the Lie algebra. So on the group side, Casimir and Kostard. So if you have a reductive group like an acompactly group, you have, I mean you have the extension of the Laplacian or minus Laplacian I would say I make it with a minus sign which is the Casimir operator. So that's an invariant differential operator on the group that takes into account the fact that there is this killing form which is positive in one direction negative on the other direction. So that's why Cg equal minus sum of e star i e i it takes into account the positivity or negativity of the form of the killing form B on the on the Lie algebra. So if you like that an extension of Laplacian but that's a good Laplacian, the p-direction and a bad Laplacian with the wrong side in the k-direction. So that's why this is a quadratic Casimir. Yes, yes, I will talk of after all about all the possible Casimir's. Yes, no, I mean that will be the end of the talk. Okay, so I introduce also the Clifford algebra of G equipped with a with an invariant form minus B and this Clifford algebra acts on the exterior algebra of G star. So basically Kostard introduced an operator which was square essentially is the Casimir. Okay, so to do this I just introduce now the enveloping algebra U of G that the algebra of left invariant differential operators on the group. So if you take the Lie algebra defines invariant vector fields and you look at the algebra of differential operators generated by these vector fields at U of G and so the Dirac operator is the first order differential operator on the group and it can be described as a section of the Clifford algebra tensed by U of G is actually differential operator order one of order one and it has first piece which looks like a Dirac operator the sum of C hat of EI star EI that in some sense the classical Dirac operator and there is a correction which is a cubic correction of lengths 3 in the Clifford algebra and which involves the fundamental three form three form anti-symmetric three form on the Lie algebra which is given by B of U bracket VW that's an anti-symmetric three form and so that's the way that Kostard constructed is Dirac operator and so the formula of Kostard I remember that that that I was told by you that it was already proved by physicists before but that the square of the Dirac operator of Kostard is essentially up to a constant given by minus a Kazimierz so this Kostard operator I mean doesn't act on smooth functions while the Kazimierz act on smooth on smooth real functions the Kazimierz acts on on on smooth functions on the group with values in the exterior algebra of G star because of the Clifford algebra while CG itself can be made to act on the smooth functions so this is such an identity we will use partly from an analytic point of view that is effectively try to reveal the hidden exterior algebra of G star now let me explain what we shall do on the Lie algebra side I told you the analysis is done on two pieces the group on one hand the Lie algebra and then we will couple them so on the Lie algebra G first of all we will make a weak rotation on the Lie algebra okay the Lie algebra G as I told you is not an Euclidean vector space it's a Lorentzian like vector space so we make a weak rotation to transform it into an Euclidean vector space so we replace P plus K by P plus I K and I introduce the harmonic oscillator on this GI so as you know the harmonic oscillator is given by this formula here where y is the generic is a generic point in GI and we will in some sense also on this GI introduce a Dirac operator a Dirac operator which is a written operator I mean that's that's connected with the widens idea of the twisting of the Dirac operator on a Euclidean vector space so we twist these on this Euclidean vector space the Dirac operator by the Gaussian by the Gaussian function so we get two new operators the Dirac modified Dirac which is D plus y wedge and the other one which is its formal joint and these two operators also act now on smooths forms on the Lie on on G with coefficients in the exterior algebra and so this operator here is a Dirac like operator acting on GI the sum of these two and the corresponding Laplacian that was a formula originally obtained by Witten tells you that up to a factor this Laplacian is given by the harmonic oscillator which is sort of bosonic number operator plus the freemianic number operator acting on the Lie algebra so we're going to couple these two constructions on one hand the costant operator acting on the group and on the other hand the Witten like operator acting on the Lie algebra so I will consider to construct now a new Dirac operator DB parameterized by B positive so this operator DB will be a combination of the Dirac operators on the group and on the Lie algebra so what does this mean this means that these new operator DB will act on smooths functions on the group times a Lie algebra with coefficients in the exterior algebra of G star so remember I told you that the costant operator was acting on functions on the group with values in the exterior algebra the Witten operator acts on functions on the Lie algebra with values in the same Lie algebra up to complexification so we can effectively combine the two so let me just give the formula so you see forget about the red factor so the first term in these DB that's a new Dirac operator acting on a big space the first term is a costant Dirac operator and the second one is just a scale version of the Witten like Dirac operator and there is a third mysterious term which is in red which is non-linear which is quadratic and who's I mean with presence plays an essential role because if it was not there I would not be there either and I mean it's introduced he tends to be forced because we will do the quotient thing by K remember that ultimately we will descend everything to the quotient by K so that's this operator DB so DB is a K invariant operator it's not yes sorry sorry excuse me yes yes yes yes yes absolutely yes yes so in the GI direction you have two components yp and yk and that's why you take you take the lig brackets so that's exactly this so the introduction of the quadratic term I won't have time to explain much about this is related you have to introduce these because you will eventually quotient things by K which introduces non-linearities by itself so what is this object that I will call the hyper elliptical aplation I will explain later what is hyper elliptic so it is just the operator LB you which is one half of DB square minus DH D hat K square so remember up to the quadratic factor I won't mention DB as just a sum of the Dirac costant operator on G plus the written operator on the Lie algebra you take the square of the sum and you subtract in some sense the square of the first term so you undo what you had done except that you have less tweets when you have a plus b square that a square plus a b plus b a plus b square so you subtract a square and you ultimately quotient the construction by K because all these objects are K invariant they're not G invariant the RK invariant so as I mentioned before the Lie algebra descends to a vector bundle Tx plus n which is sort of canonically trivial but not trivial as an Euclidean vector bundle so let me introduce again the space X hat that's a bundle over X over the symmetric space whose fibers are Tx plus n and so our operator LBX I explained before it will act on smooth sections over X script of the pullback of the exterior algebra of T star X plus n star so in other words G star is now replaced by T star X plus n star so using the Bergmann isomorphism I could put things in a more symmetric way instead of viewing things as a geometric object acting on this bigger space you could say I don't make the space bigger I just keep the space X down here but the price to pay is that you'll act on an infinite-dimensional vector bundle involving the symmetric algebra which appeared also in un-terminus talk this morning so it's the same is the same object except I just have one copied instead of having an infinite number of copies so let me give you first of all formula for this operator LBX so this operator LBX is supposed to be a deformation of the original scalar laplacian okay so this operator LBX if you just look at the red pieces it's essentially a forked plank operator so in other words this operator LBX it's harmonic oscillator in the fibers of TX plus n which I view as an Euclidean vector bundle after weak rotation there is this harmonic oscillator in red and there is this piece here which is what I call the judizic floor well if there was only TX if there was only TX you have a remanded manifold X you have the you have the fiber TX and on this big object of dimension twice the original dimension you have the judizic floor and that's what's represented by the nabla YTX so if you forget about the other geometric terms we just have essentially a forked plank operator which is a coupling of the harmonic oscillator in the fiber and the judizic floor in some sense which is a vector field on the base so this operator is non-self adjoint I mean it is not self adjoint obviously not the judizic floor is anti-symmetric it is it is not elliptic elliptic means that there is I mean it just differentiates once in the base direction not twice so it doesn't put it's not a Laplacian in the usual sense but what I'm saying is that from an analytic part of you it's a very good operator it's almost as good as the Laplacian it's why it's called hyper elliptic and what I'm saying besides is that this operator deforms naturally the original Laplacian so this actually you can prove in a rigorous mathematical way this operator LBX deforms in some sense the original Casimir operator or Laplacian if you like acting on the base X so that means that when B10 to 0 there is a collapsing of the space X script to the space X so let's try to understand a little bit this collapsing I mean what happens is that when B10 to 0 the harmonic oscillator gets scaled by a big factor and that in some sense when B10 to 0 you ignore everything of the eigen eigenspaces or eigenvalues of this operator except the zero eigenvalue that's exactly a collapsing phenomenon which is a sort of some sense exactly the same sort of collapsing that you have when you have look at Kaluza Klein theory in which you have very small circle the circle gets smaller and smaller because I mean in some sense you just concentrate on the fundamental on the ground state of the circle you have exactly the same phenomenon here you concentrate on the ground state of the harmonic oscillator except that there is here a non trivial coupling between the fiber and the base via the judizic floor which ultimately explains I won't prove that that in the proper sense the operator LBX deforms actually when B10 to 0 there is a language that is a theory in which you can say this is transport focal plan operator deforms the original Laplacian but it is not only that it deforms it because it still captures a lot of information on the original Laplacian in particular on its spectrum as long as it will work with compact cautions so as B10 to infinity after rescaling you see that it is the judizic floor which dominates so when B10 to 0 deform to the Laplacian when B10 to infinity the judizic floor dominates and if you think of taking traces of heat kernels you can see why when B10 to infinity you will ultimately select close to this X so let me apply these two locally symmetric spaces now so original symmetric space X is just Rn as a smooth space is just Rn so we need to build first of all a compact manifold from the space so basically what you do is that you take a discrete subgroup of the group G and gamma and I will assume that it is co-compact and torsion free so this is just complicated language to say that gamma acts freely on the in a discrete way on X so you can form the compact quotient gamma on X that's a compact manifold exactly like S1 is a quotient of R by the action of Z and so I will give my first identity the first identity says that the trace on the smooth functions on Z forget about this E just take E equals R for the moment on smooth functions of the heat kernel is equal to the super trace of the hyperliptic heat cut so we have exactly an identity which is similar to what I wanted to have before to deform the evaluation of the heat kernel into something more complicated involving more coordinates and actually if we think of it this idea of resolving of replacing this of saying the space of smooth functions is the comology of something is actually embedded in the construction I gave before through this written harmonic oscillator I won't give details on that but we have exactly what we want sorry what is small see oh see is just the B square I mean that the defect in in in in in costance formula that's just a defect in costance formula that the lengths of road G that's a square excuse me question about this this theorem yes so isn't that like the in physics you would say that you're super symmetric yes to sigma model and this is just localization yes so this is exactly so we have I mean in some sense we have constructed a super symmetry in a theory which is not super symmetric so that was yeah the algebra the algebra is where the fermions are and this is just localization the standard I mean localization see not no I mean you just we will localize like to rock okay we'll take B we'll take B to B and then we will then you get your physics yes yes that's all that this is what we will do but okay sorry wasn't that done in the physics literature long ago no I don't think no not for that no no because I mean the idea that focal plan cooperators to form the Laplacian you see what this contains in particular either the spectrum of the Laplacian is contained in the spectrum of these focal plan cooperators okay these focal plan cooperators have some real eigenvalues they have non real eigenvalues so ultimately when taking the super trace the non real eigenvalues disappear and what's left is just a real eigenvalues so you have had more eigenvalues but no okay so let me now say that there is a super symmetric interpretation involving involving the complication the complexification of the group but I won't mention it here so splitting the identity so each side I mean since you are in a compact on a copent quotient by a discrete group each side splits as an infinite sum indexed by the conjugacy classes gamma so this I mean exactly like Poisson original formula says that you did Colonel on s1 is just a sum over z you can exactly do the same thing and now the miracle is that this identity splits as an identity that is you can forget about the group gamma and you could say the identity remains true in some sense element of gamma by element of gamma it is not an identity of sums the identity remains true and that's an identity of objects which are called orbital integrals so let me briefly mention what semi-simple orbital integrals are orbital integrals are fundamental in in in the theory of representation automorphic forms and so on so let me just take the simplest case actually I take a semi-simple element gamma and I call bracket gamma conjugacy class so basically an orbital integrals that's its name is that you integrate the heat kernel on x I mean you need to introduce some notation for that on the orbit on the adjoint orbit of gamma by the action of g on the element gamma and the integral is an integral of the quotient of g quotient by z of gamma so these are the objects which are the ingredients which appear when you express the trace of the compact quotient in terms of the group gamma in some sense you re-express this trace as objects which are the orbital integrals so let me give a geometric description brief geometric description of what the orbital's integral are so I will just introduce this picture so this this is a flat space which is supposed to represent the symmetric space x equals Rn if you like so inside these are in or inside the symmetric space there is a sub manifold which is x of gamma which is also a symmetric space and it can be geometrically constructed by minimizing by minimizing the so-called displacement function d gamma of x which is d of x and gamma x where d as a remanian distance now the remanian distance on such symmetric spaces is convex in the sense that it is convex on judizik so the minimizing set is non-empty because gamma is semisimple and it is convex it's x of gamma the six of gamma is actually a symmetric space for the centralizer of gamma so how can you understand what the orbital integral is I just describe it geometrically what you do is that you take the heat kernel you take the heat kernel you take a normal you take a x0 in x of gamma you take a normal and you take a generic point why you look at its image bar gamma gamma is an azimetry it maps normal judizik to a normal judizik and you find that because of negative curvature the mutual distance of y and gamma y grows at least linearly when y tends to infinity and you use the ultimately the fact that the heat kernel decays at long distances like a Gaussian on the symmetric space so that the above integral exists so in other words you take the heat kernel at y and gamma y in integrated and normal directions you ultimately get a Gaussian integral which converges that's the orbital integral so the point of the hyperliptic theory will be to push ultimately what we will do is push the integral to x of gamma to the closed judiziks so the second fundamental identity is that effectively instead of having an identity of tracers you get an identity of the so-called orbital integrals which are now calculated using heat kernels over the symmetric space not over the compact coach this is true for any b so let me now take the limit as b tends to infinity so after rescaling this operator lb the original operator we had when b tends to infinity is dominated in a way by the judizik floor except that there is this extremely large term quartic term which dominates everything which means that localization will take place on commuting pairs y n and y tx and so ultimately the computation of the orbital integral localizes near the manifold judiziks this x of gamma associated with gamma as I explained before the point of this is to push the integral to x of gamma so let me try to explain what is the structure of the formula for these orbital integrals so I take a gamma to be a semi-simple element which means that it is in a certain way diagonalizable so I take its reduced form I write it as a product of a self-adjoint if you like a unitary element e to the a times k minus one where a is in p and k is in k so after conjugation you can always do that and these pairs are commuting so let me call z of gamma to be the centralize of gamma and with the algebra splits again as p and gamma plus k of gamma so z of gamma is still is still a semi-simple is still a reductive group sorry yes just on the previous slide yes so I think it's related to the previous offer question when you take the p to infinity yes this corresponds to the quarter of the flows right yes the geodesics yes so essentially it constitutes localization but your result comes from the deformed l right yes essentially this is not this is different for your plan plus your so so the localization will come from this deformed focal plot because you're taking the trace yeah but this result is not from the like conventional for the point this is already deformed one because your lb is defined as a constant squared minus constant yes but I told you the original result tells you that the original object you went to compute is equal to the deformed one okay this was the first thing the original object you went to compute is equal to the form one so if you localize a deformed one you get something which is equal to the original object okay so you started from the casimir you deform it to this focal point like operator but still you preserve the original object which means if you were working on the compact ocean it exactly reproduces in this limit the same the same contribution sorry that's so essentially I think that in this limit it reproduces a original result yes yes so it means like your deformation and yes there is a deformation but you do not change anything okay and and and now of course the result I have is no longer spectral result okay it's no longer the point is of this thing is if you can analyze now things judizik if you like close judizik by close judizik instead of having to localize on anything it number of judizik at the same time can you make a comment about yes the relationship to the physics yes because there is this paper on the realization that you do the formula by catechic or catechic yes yes so what's the relation to what you're doing I don't know the paper that you mentioned okay maybe we'll talk about this later okay so let me give the evaluation of the semisimple orbital integrals and this formula will extend the formula of zelberg I gave at the beginning so there is an explicit function j gamma which is a function on i k of gamma so let's fix the notation again you have gamma semisimple element you have it's the algebra z of gamma which splits into p and k of gamma and so this function here is defined on i k of gamma on the weak-rotated i k of gamma such that the original object the original object which is a trace gamma okay the orbital integral associated with the elliptic heat kernel and now let's look at the formula the formula should be looked at as an a tier singer like of index theorem in which integration on the manifold is replaced by integration on i k of gamma you have the first term which is j of gamma which will be an analog of the Todd or a roof genius but now it's a function now I have a churn character like if I need not introduce a twisting bundle so if you like you can ignore it if you're interested in sections of vector bundles you have to introduce it and you have a Gaussian integral here and so really the mystery lies in this function j of gamma so I told you it's a function it's not a characteristic class it's a function and so this function j of gamma it looks intimidating but in some sense it's a super symmetric version of the a roof genius on this manifold x you had these two bundle tx plus n but actually from the point of view of this point of view you have to look at this as tx minus n so the contribution will be the ratio of the corresponding a roof gene genera or a roof functions so I give the function here I give the formula which is every time a ratio of functions evaluated on the p and the k part the p part always appears in numerator the k part in denominator and these functions take into account find splitting of the Lie algebra under the action under the adjoint action of the representation of k of gamma the the detail for the moment is not important here so the center of the enveloping algebra so again let me remind you that u of g is the enveloping algebra that's a non commutative algebra and inside there is a center which is a commutative algebra so I will introduce s of g which is the algebra of polynomials on g star and i of g which are the invariant polynomials on g star so the group the group there is an action let's say of the of the group or of the Lie algebra on these polynomials and look at these invariance so the Duflo isomorphism tells you that these two algebras are properly isomorphic so the isomorphism is not revealed you cannot guess what it is just by looking at these two algebras that it tells you that there is a canonical isomorphism of these two algebras there's a center of the Lie algebra is indeed an algebra of polynomials on g star so in particular for instance when you want to look at what corresponds to the Casimir see g in this algebra i of g that's minus b star plus b star of row g row g b star being the du of b that is it's a polynomial of degree two on g star so center and enveloping algebras so and i take l to be an element of the center z of g and i take its counterpart in the algebra of polynomials i of g so what i'm saying is that this element 2d minus 1 of e this polynomial restricts to an element of i of z of gamma so the restriction is not entirely is not entirely natural but there is a natural projection from the Lie algebra on z of gamma so that you can define in some sense the proper sense the restriction of this element so from a polynomial you get a polynomial on the Lie algebra of the center and again from this polynomial now you rebuild a differential operator you rebuild a differential operator on z of gamma with constant coefficients so in other words you take the element of the center you look at the corresponding polynomial you restrict it to z of gamma and you rebuild a differential operator acting on z of gamma with constant coefficients i would say even acting on the complexification of z of gamma sorry the center yes z of gamma is just a Lie algebra z of the centralizer of gamma yeah the Lie algebra okay so the formula for general orbital integrals now instead of writing it for the heat kernel i take any i take any kernel in some sense of the casimir so this is a formula that i proved with Xu Shen in 2019 which says that the orbital integral of l so in other words a new ingredient now is this l this element of the center which actually probably involves all the possible casimiers that you were mentioning times a kernel an arbitrary kernel of the of the casimir well i mean it's given by the action of a new kernel you take a new kernel which now acts on the Lie algebra on i times k of gamma okay so you need to introduce to to understand this this formula is not too complicated you have the action on the right-hand side of the differential operator l z of gamma which acts on z of gamma and you compose it with a kernel of a corresponding Laplacian on z of gamma and you make it act on this distribution j gamma times in the case we introduce a twisting bundle the corresponding character for the twisting bundle we evaluate this kernel at zero so in other words you make it act on this distribution you have a smooth kernel you make it act on the distribution we evaluate it at zero and the formula is that the original orbital integral is equal to this new evaluation of smooth kernels on the Lie algebra of z of gamma okay so for l equals one that's essentially the original formula so the principle of the proof we start from the formula for this l equals one so when gamma is regular that is when it's regular that is if you like when all its eigenvalues are distinct so it centralizes a Carter subgroup the formula i gave before for the heat kernel simplifies dramatically and can be differentiated in gamma and then ultimately we use fundamental results of Arish Chandra to evaluate the action of a differential operator on our original formula the formula without the l to obtain a formula for l for gamma regular and so if gamma is semi-seable or arbitrary we use a previous result for gamma regular and limit results of Arish Chandra to obtain the general formula so let me just remind you what the function j gamma is so that for general semi-simple element gamma that sort of quotients of determinants evaluated in the p and k part of the Lie algebra of z of gamma now it turns out that when gamma is regular they are sort of cancellations i would say supersymmetric cancellations between the contributions of p and the contributions of k so that the formula simplifies dramatically so more precisely you can re-express the function of j of gamma in terms of the roots of root system but essentially of the purely imaginary roots of the reductive group so the imaginary roots are the ones we just correspond to the action of k on the p and k part so that means that there is exactly as before where the eigenvalues were disappearing here it's a p and k factors which cancel each other and so this allows us to treat in some sense to prove that this function in the proper sense is smooth as a function of gamma as long as gamma is regular that is if it's eigenvalues if you like are distinct so I will complete this by reminding you something that you all know that's a Langevin equation that in 1908 Langevin introduced a Langevin equation mx double dot equal minus x dot plus u double u dot to reconcile Bröhnian motion and classical mechanics so Bröhnian motion for m equals zero and classical mechanics for m equals infinity so in this theory of the interpolating hypoleptic Laplacian m equals b square plays a role of a mass so this hypoleptic interpolation plays exactly I mean in some sense uses the tools all the objects introduced by Langevin at the time of course with no group theoretic I mean interpretation so here is the interpretation is a paper by by Langevin in 1908 where you find exactly the equation written either what it before and with an interpretation that you all know okay I think I will just give the references here and happy birthday Samsung thank you very much yes please yeah I always have that main question about the constants Dirac so like if you took some other Dirac on the group what would go wrong or what would go right I mean you see there is the the the key thing which I've not explained is that the chasmier lies in the center okay so in the equality of the trace is equal to the super trace the fact that it lies in the center plays a fundamental role because basically you need to prove that you know to write the super trace of some super commutator vanishes and the fact that the bad term disappears is related to the fact that is a chasmier so there is only one operator which will do what you want all the other ones will always contain something which makes them outside of the center that's somewhat complicated whatever book yes yes yes so that's why you need absolutely the cost okay so that's I did not explain this but the chasmier is in the center I mean and that plays a fundamental role in the fact the definition yes yes two questions could you give an example of something that you know you know businesses like myself can understand that you can do with these wonderful formulas yes the application that's question one question two is if you think of G as being for example the group of gauge transformations yes can you use this and hide in this idea yes okay so so let me try to to answer to this okay so first of all about the role of the judizic flaw why is your judizic flaw there and then I will give an explanation I think which is if you think of the judizic flaw and you think you write in in coordinates y d over dx if you take a symbol that you should make a Fourier transform with the x variable what you get is square root of minus 1 y psi if you exponentiate this you see that this is Fourier transform okay so the ultimate reason why the judizic flaw appears it's because in some sense it does for us mathematicians what it does is a Fourier transform okay so we are geometrically trying to implement the idea of the Fourier transform in a context of course if you think of Fokker Planck operators I think it's fair to say nobody thinks of Fokker Planck operator of doing Fourier transform of anything okay it is a dynamical statistical physics system that you try to describe but the way we use it is this idea of Fourier transform the second thing is that in the case where the face is flat that is where you work in the case of r or s1 everything can be can be worked out explicitly in some sense it is not so simple even that case to understand mathematically what what happens so let me try to say it in words so you have the question that you know that you can quantize the group of simplectomorphisms we can quantize there is a quantization to operators you pass from simplectomorphism to unitary operators okay let me say roughly what happens in this theory is that what you're quantizing is not simplectomorphism real ones you're quantizing complex ones okay so the quantization of complex simplectomorphism introduces a face I mean if you look at the the corresponding operator which appears on the other side it will no longer be unitary but the worst is that it will be badly unbounded okay however if the consequences of this theory are true that would mean that two operators which look very different are ultimately the same because they are conjugate by something okay this something does not exist because it doesn't act but still the operators behave they behave properly that is they still have the same spectrum okay so if you were doing this forgetting about about this you would just remain in the realm of classical schrodinger like operators if you just quantize these non complex transformations you pass to a completely different category of operators yeah so I think that the best explanation I can give for the moment other questions cannot put in certain functions representations yes yes yes you can put representation and that's why you have this extra contribution of pairing of the a roof function I mean with a churn character function which in this case is just a character corresponding to this representation yes so that's why it was there yes other questions if not thanks again okay thank you