Flag Partial Differential Equations and Representations of Lie Algebras

Flag partial differential equations naturally appear in the problem of decomposing the polynomial algebra (symmetric tensor) over an irreducible module of a Lie algebra into the direct sum of its irreducible submodules. Many important linear partial differential equations in physics and geometry are also of flag type. In this paper, we use the grading technique in algebra to develop the methods of solving such equations. In particular, we find new special functions by which we are able to explicitly give the solutions of the initial value problems of a large family of constant-coefficient linear partial differential equations in terms of their coefficients. As applications to representations of Lie algebras, we find certain explicit irreducible polynomial representations of the Lie algebras $sl(n,\mathbb {F}),so(n,\mathbb {F})$ and the simple Lie algebra of type G ₂.