Hypoellipticity on a product of compact Lie groups

In this thesis we study the behaviour of hypoelliptic operators on compact Lie groups. We prove that irreducible unitary representations of a compact group G are finite-dimensional. The Peter–Weyl theorem states that the matrix coefficients of such representations form a orthogonal basis of L2(G), so that every function in L2(G) has a unique Fourier series representation. The Casimir element of G is a translation-invariant elliptic second-order differential operator, analogous to the usual Laplace operator. The matrix entries of each irreducible representation are eigenfunctions of the Laplace operator, with respect to the same eigenvalue. Such eigenvalues form a sequence that accumulates at infinity and any class of equivalence in the unitary dual uniquely determines an eigenspace of finite dimension. Through the operator symbol, smooth functions and distributions are characterised by the rate of growth of their Fourier coefficients and Sobolev spaces are defined. A differential operator P is globally hypoelliptic on G if for each distribution u such that Pu is smooth, then u is a smooth function. The theory and the results that we present over a compact Lie group can be applied in the case of a product of compact Lie groups. We explain necessary and sufficient conditions for global solvability and global hypoellipticity of a vector field X=X1+aX2 over G=G1×G2 with a constant, and of a vector field on G perturbed by a smooth function. The differential operator P is locally hypoelliptic on G if for every open subset U of G, the restricted operator is globally hypoelliptic on U. It is possible to construct differential operators that are globally hypoelliptic but not locally hypoelliptic, as can be seen in the example on the 2-dimensional torus by Fujiwara and Omori. In the last chapter that example is generalized to the case of a product of compact Lie groups G=G1×G2.