日時:2024年5月7日(火),15:00-16:30
場所:理学部A201号室
講師:Víctor Pérez-Valdés 氏 (龍谷大学)
題目:Construction and classification of vector-valued differential symmetry breaking operators from $S^3$ to $S^2$
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.