[SOUND] [MUSIC] Dear colleague, hi, we're starting our 11th lecture. Its title is Modular Differential Operators for Jacobi Modular Forms. So we discussed modular differential operators for usual modular forms, but now we would like to construct similar operators for Jacobi modular forms. Let me fix the standard setup. First of all let M, Be a quadratic lattice. Of rank n. Then a complex variable, a complex vector, from the complexication of this quadratic space we can represent as a sum z e times the vectors of the basis ei where z e is a complex number. Then we define The vector d/dz, this is by definition the co-vector and we write co-vector in the dual basis. Means that scalar product of ei and ej* is equal to Kronecker delta symbol. So you can check then if we take the quadratic differential operator dz, d z. Let denoted by Delta this differential operator of order two. Then I would like to apply this operator to the following standard function e to the power 2 pi i (l, z). Where l is a vector, l is a vector from real, let's see, real lattice, real space M and that is a complex vector. Then you can calculate, you'll get 2 pi i to the square, l to the square, e to the power 2 pi i (l,z). Please check this. This is a standard formula, or related to Laplacian, but in our case I don't fix any signature of the matrix M. So we have a differential operator of order two for any non-degenerate quadratic lattice M and you see that we can apply this differential operator to the standard exponential function. But now I would like to write down this operator in more concrete context. Now we take a positive-definite lattice L of rank n0. Then we take The following hyperbolic lattice U plus L renormalized by 1. Then the signature of this lattice L1 = (1, n0+1). I remind you that by U, we denote the hyperbolic plane with the following Gram matrix, so this is a unimodular hyperbolic plane of signature 1,1. Now, we can take an arbitrary vector, that from the complexification of L1, And I write down this vector as a vector, (tau, omega) in the natural basis of the unimodal hyperbolic plane + we have orthogonal sum here also, by plus I denote the orthogonal sum. U and L- 1 are orthogonal to each other, then the vector z (tau, omega) is in U and z is in L under C. Certainly, for U also we're taking its complexification. Let's write down the differential operator Delta for the lattice L1. It would equal, in this particular basis, to 2 delta/delta tau times delta/delta omega- Delta over delta calligraphic z, its square. We have minus here because we renormalized the positive-definite lattice by minus 1. So this is the differential operator of order two and I would like to understand its action on the space of Jacobi form. Now, in the first part of our lectures, we explained the functional equation of Jacobi form of weight k and index m using the function f tilde on the Siegel upper half plane. Where we added the third formal variable in the following way. We add the factor, e to the power 2 pi i, m omega, where m is the index of the corresponding Jacobi form. The similar procedure works in this case of modular form in many variables. If we consider a Jacobi form of weight k with respect to a lattice L, then we can construct the corresponding function depending on the additional variable omega e to the power of two pi i omega. And in this case, we use a Siegel half plane of genus two and the fact that the Jacobi group is a parabolic subgroup of Siegel modular group of genus two. Totally in the same way, we can consider the second function as a function on homogeneous domain of type IV and Jacobi group will be subgroup of special orthogonal group of signature (2, n0+2). This really is a full analogy of the first construction. You can find this construction in my papers, and I'll give you explicit reference in the PDF file related to this lecture. Now I would like to apply the differential operator Delta L1 and I would like to consider its action On this function in three variables, tau, z, and omega. What we get? We get 4 pi i, I guess now I would like to analyze first of all the action with respect to this variable. Then we get 2 pi i from this factor, an additional factor 2 from this coefficient. Then, we have, The differential operator with respect to tau- the square (d/dz, d/dz). Its action phi tau z e to the power 2 to the power 2 pi i omega. Where this vector now, we can to put as coefficient here, so, we get the following definition. I can define the operator, H, which acts on Jacobi form of weight k and index. And with respect to the lattice L and its action up to the factor -1 over 8 pi to the square is exactly the operator, which we have in this line, (4 pi i d/dtau- (d/dz, d/dz) Action on function phi. What do we have? And why I denote this operator by H because this is the classical heat operator, The heat operator related to positive dividend, lattice L. So we can rewrite, certainly this is the rater, after r normalization, as follows. 1/2 pi i, d over d tau, minus the Laplacian related to the positive-definite lattice with coefficient, I'm sorry, here we'll have plus, let me change it. With our additional factor, we've got + 1/8 pi to the square and we act by this operator phi(tau, z). This is our main object for this lecture. [SOUND] [MUSIC]
