July 10, 2003. Pavel Etingof (Massachusetts Institute of Technology). When is the Fourier transform of an elementary function elementary?

By an elemementary function on an n-dimensional complex vector space V we will mean a complex-valued function of the form f(x,h)=A(x)*exp(i*Re Q(x)/h), where Q(x) is a rational function, h is a positive parameter, and A(x) is a product of (complex) powers of polynomials of x and powers their complex conjugates. Such a function can be considered as a distribution on C^n. We will study the following question: when is the Fourier transform of f(x,h) also elementary for all values of h? The simplest example of this is the Gaussian f(x)=exp(i*Re Q(x)/h), where Q is a nondegenerate quadratic form, and we will be interested in studying other possible examples. First, by sending h to zero and using the stationary phase method, one can derive a necessary condition of this ("the semiclassical condition"): the differential dQ is a birational isomorphism between V and V^* (i.e. the inverse map to dQ is rational). This is equivalent to saying that the Legendre transform of Q is rational. It is clear that a homogeneous function Q satisfying the semiclassical condition must be of degree 0 or 2; but this is certainly not sufficient. In fact, the problem of classification of homogeneous functions Q satisfying the semiclassical condition is very interesting. For example, assume that V=W+C (so x=(y,t)) and Q(x)=f(y)/t, where f is an irreducible cubic polynomial on W. Then W is the complexified space of 3 by 3 hermitian matrices over division rings R, C, H or O (so dim W is 6, 9, 15, or 27), and f is proportional to the the determinant polynomial. This is proved using a beautiful theorem of F.L.Zak on the classification of Severi varieties. On the other hand, one can show that any (Laurent) monomial in variables x_1,...,x_n of degree 0 or 2 satisfies the semiclassical condition. So one may wonder which of such monomials Q give rise to elementary functions f with elementary Fourier transform, (assuming that A is a product of powers of coordinates and powers of conjugare coordinates). In other words, which Q satisfy the "quantum condition"?. The answer turns out to be unexpectedly interesting. For example, suppose that Q=x_n^m/y_1^{m_1}...y_{n-1}^{m_{n-1}}, where m=m_1+...+m_{n-1}+2. Then functions f (i.e. choices of A) for which the Fourier transform of f is elementary are (up to scaling) in bijection with exact covering systems of type (m_1,...,m_{n-1},1,1); we recall that an exact covering system of type (p_1,...,p_k) is a covering of the group Z/pZ, where p=p_1+...+p_k, by cosets of Z/p_kZ, one copy of each (so p_k should divide p for such a system to exist). Thus, the monomial x^3/y satisfies the quantum condition, while the monomial x^5/y^3 does not.

List of talks at previous sessions of the seminar.