"Then the claim would be immediate by
semisimplicity if one can show that
every irreducible representation of
$G(\mathbb{C})$ (or perhaps of Lie
groups in some more general class than
these) occurs as a subquotient of a
tensor power of a faithful one. How
might one prove the latter? (Can one
prove the latter for compact real
groups in a manner similar to the
proof for finite groups, and then pass
to semisimple complex groups by the
unitary trick?)"
Yes. The analysis is not so pretty, but it is elementary. Let $K$ be a compact Lie group, $V$ a faithful representation, and $W$ any other representation. Just as in the finite group case, $\mathrm{Hom}_K(W,V^{\otimes N}) \cong (W^* \otimes V^{\otimes N})^K$, and the dimension of the latter is $\int_K \overline{\chi_W} \cdot \chi_V^{N}$, where $\chi_V$ and $\chi_W$ are the characters of $V$ and $W$, and the integral is with respect to Haar measure. Let $d_V$ and $d_W$ be the dimensions of $V$ and $W$.
We now come to a technical nuisance. Let $Z$ be those elements of $K$ which are diagonal scalars in their action on $V$; this is a closed subgroup of $S^1$. For $g$ not in $Z$, we have $|\chi_V(g)| < d_V$. We first present the proof in the setting that $Z = \{ e \}$.
Choose a neighborhood $U$ of $\{ e \}$ small enough to be identified with an open disc in $\mathbb{R}^{\dim K}$. On $U$, we have the Taylor expansion $\chi_V(g) = d_V \exp(- Q(g-e) + O(g-e)^3)$, where $Q$ is a positive definite quadratic form; we also have $\chi_W(g) = d_W + O(g-e)$. Manipulating $\int_U \overline{\chi_W} \chi_V^N$ should give you
$$\frac{d_W \pi^{\dim K/2}}{\det Q} \cdot d_V^N \cdot N^{-\dim K/2}(1+O(N^{-1/2}))$$
Meanwhile, there is some $D<d_V$ such that $|\chi_V(g)| < D$ for $g \in K \setminus U$.
So the integral of $\overline{\chi_W} \chi_V^N$ over $K \setminus U$ is $O(D^N)$, which is dominated by the $d_V^N$ term in the $U$ integral.
We deduce that, unless $d_W=0$, we have $\mathrm{Hom}_K(W, V^{\otimes N})$ nonzero for $N$ sufficiently large.
If $Z$ is greater than $\{e \}$, then we can decompose $W$ into $Z$-isotypic pieces. Let $\tau$ be the identity character of the scalar diagonal matrices, and let the action of $Z$ on $W$ be by $\tau^k$. (If $\tau$ is finite, then $k$ is only defined modulo $|Z|$; just fix some choice of $k$). Then we want to consider maps from $W$ to $V_N:=V^{k+Nd_V} (\det \ )^{-N}$. $V_N$ is constructed so that $\overline{\chi_W} \chi_{V_N}$ is identically $d_W$ on $Z$; one then uses the above argument with a neighborhood of $Z$ replacing a neighborhood of the origin.
A very active field of research (and to my understading, may fall into the "fundamental" category) is Domain Decomposition methods (DDM), which can be understood in the geometrical numerical and computational sense. In this last, parallel algorithms are being explored. Although many of these methods are based on Lagrange multipliers, efforts are also made to make a sensible domain decomposition without using them, through indirect methods, like Green's functions, from which some collocation methods can be derived (see "General Theory of Domain Decomposition: Indirect Methods Ismael Herrera, Robert Yates, Martin Diaz", Numerical Methods of Partial Differential Equations, Wiley). In this same paper are mentioned the Steklov-Poincare operators, which I believe is a line of research on its own right. And come to think of it, collocation methods are also a line of research.
Your related question: "[...]fundamentally new types of functional spaces[...]" You don't mention which functional spaces you are aware of, but I can mention Sobolev spaces, which support weak derivatives, and these in turn can be used in problems involving "jumps" or some type of discontinuity (for example, in combustion/explosion problems, see in this instance "physics of shock waves and High-Temperature Hydrodynamic phenomena" by Ya. B. Zeld'dovich and Yu. P. Raizer"). See also Godlewski and Raviart (1996), "Numerical approximation of hyperbolic systems of conservation laws", Applied Mathematical Sciences 118 (Springer, New York). Sobolev spaces have been used also for DDM, for example in the same paper by Herrera, Yates and Diaz.
There is this book "Navier-Stokes Equations and Turbulence"
edited by C. Foias, which devotes some pages to the Banach-Tarski paradox. Maybe that famous paradox (a theorem, in fact) may provide some new avenues of research into some types of PDE's.
Best Answer
This goes back to the beginning of the subject of unitary representations of locally compact noncompact groups. Wigner was looking for all possible generalizations of the Dirac equation to higher spin, and developing the representation theory of the Poincaré group is how he obtained his results (Bargmann did this independently, so they published together). See here: https://www.pnas.org/content/34/5/211