分野別セミナー

アブストラクト： Any equivariant irreducible vector bundle for the conformal group SO_0(4,1) on the 3-sphere S^3 is parametrized by an odd number (=2N+1) and a complex number \lambda. On the other hand, any equivariant irreducible vector bundle for the conformal group SO_0(3,1) on the 2-sphere S^2 is a line bundle, and is parametrized by an integer number m and a complex number \nu. In the present talk, we consider the problems of construction and classification of all the differential operators, that are symmetry breaking operators with respect to the conformal pair (SO_0(4,1), SO_0(3,1)) between the spaces of smooth sections of a vector bundle V_\lambda^{2N+1} over the 3-sphere S^3 and a line bundle L_{m, \nu} over the 2-sphere S^2. In particular, we solve these problems when the rank of the vector bundle is less than or equal to 7 (i.e., for N = 1,2,3), and for |m| > N, and propose a strategy to solve them for N > 3. The method we use is the F-method of T. Kobayashi, which in our setting allows us to reduce the problem of constructing differential symmetry breaking operators to the problem of solving an overdetermined system of 2(2N+1) ordinary differential equations on 2N+1 unknown polynomials.