How to identify characters Suppose I have a compact connected Lie group $G$, and a smooth class function $f : G \to \mathbb{C}$. Suppose that $\| f \|_{L^2(G)} = 1$ and $\Delta f = - \kappa_\pi f$ where $\Delta$ is the Laplace-Beltrami operator on $G$, $\kappa_\pi$ is the Casimir of the irreducible representation $\pi$. I would like to conclude that $f = \chi_\pi$ (the character of the irreducible representation $\pi$). However there is a problem: it may be that there are inequivalent irreducible representations $\pi \neq \pi^\prime$ where $\kappa_\pi = \kappa_{\pi^\prime}$. Then any linear combination $f = a  \chi_\pi + b \chi_{\pi^\prime}$ for which $|a|^2 + |b|^2 = 1$ would satisfy the above requirements but $f$ would not be a character. One could also consider more general linear combinations. In certain cases like $U(1)$ and $SU(2)$ I believe the eigenvalues of $\Delta$ are non-degenerate and so this issue doesn't arise, however I don't think this is true in general.
Is there any analytical property that can be used to identify characters? (By "analytical" I mean something you could apply purely at the level of functions.) One thing you could do is evaluate at the identity. If $f(\mathrm{Id})$ isn't a non-negative integer then it can't be a character. However this probably isn't sufficient.
 A: I wonder if this might be of any help, but characters may be characterized as follows.
A necessary and sufficient set of conditions for a function $\phi:G\to\mathbb C$ to be a nonegative multiple of a character is:

*

*$\phi$ is a continuous class function,


*$\phi$ is positive semi-definite,


*If $\psi$ is any continuous class function such that
$$0\leq \psi \leq \phi,$$
in the sense that $\psi$ and $\phi-\psi$ are positive semi-definite, then there exists a nonnegative constant $\lambda$ such that $\psi=\lambda \phi$.

EDIT.  Below I will try to outline a justification of the assertion made above.
If $G$ is a compact Lie group (actually,  for the time being, all we  need is that  $G$ is a locally compact topological group) then $C_c(G)$,  that
is,  the space of all compactly supported,  continuous, complex valued functions on $G$, is a complex $^\ast$-algebra with
convolution for multiplication, namely
$$
  (f\star g)(t) = \int_Gf(s)g(s^{-1}t)\, ds
  $$
(the integral being taken with respect to Haar measure)
and involution given by
$$
  f^\ast(t) = \overline{f(t^{-1})}.
  $$
This algebra has a canonical norm, and the completion, denoted $C^\ast(G)$, is called the group C$^\ast$-algebra of $G$.
For every (strongly continuous) representation $\pi $ of $G$ on a Hilbert space $H$, one can define the so called integrated
form of $\pi $, which is  a representation $\Pi$ of $C^\ast(G)$ on the same Hilbert space $H$.  The description of
$\Pi$ is pretty explicit on $C_c(G)$:  for every $f$ in this dense subalgebra   one has
$$
  \Pi(f) = \int_G f(t)\pi(t)\, dt,
  $$
where the integral is taken with respect to the weak operator topology.  Once $\Pi$ is defined on $C_c(G)$,  as above,
it may be extended to  $C^\ast(G)$ by continuity.
A key result of Harmonic Analysis states that every representation of $C^\ast(G)$ arises as the integrated form of some
representation of $G$ and in fact  the correspondence $\pi\to\Pi$ is a bijection between the set of
representations of $G$ and those of $C^\ast(G)$.
Regarding characters, states, and all that, if $\phi:G\to \mathbb C$ is a positive semi-definite function, one may also  define an
integrated form
$$
  \Phi:C_c(G) \to \mathbb C,
  $$
by the formula
$$
  \Phi(f) = \int_G f(t)\phi(t)\, dt.
  \tag {$\dagger$}
  $$
One may prove that $\Phi$ extends to $C^\ast(G)$ by continuity, providing a positive linear functional,  namely one that
satisfies
$\Phi(a^\ast a)\geq 0$,
for all $a$ in $C^\ast(G)$.
As before, ($\dagger$) provides a  bijective correspondence between positive semi-definite functions on $G$ and positive linear functionals on
$C^\ast(G)$.
The correspondence $\phi\to\Phi$  may be shown to map class functions to traces (a trace on $C^\ast(G)$ is a linear
functional $\Phi$ satisfying $\Phi(ab)=\Phi(ba)$).
When $G$ is compact, the Peter-Weyl Theorem essentially says that
$$
  C^\ast(G)=\bigoplus_{\pi\in\hat G}M(\pi),
  $$
(a $c_0$-direct sum) where $\hat G$ is the set of equivalence classes of irreducible representations of $G$, and each
$M(\pi)$ is the algebra of $n\times n$ complex matrices, $n$ being the dimension of $\pi$.
If $\phi$ is the character of $G$ associated to a given irreducible representation $\pi$, that is $\phi(g)=\text{tr}(\pi (g))$,
then one may show that the corresponding positive linear functional $\Phi$ is given by
$$
  \Phi\big ((a_\rho )_{\rho \in \hat G}\big ) = \text{tr}(a_\pi ).
  $$
Addressing the characterization proposed in the first part of this answer,
assume  that $\phi$ and $\Phi$ are as above, and that
$\psi$ is a positive semi-definite continuous class function on $G$, such that $\psi\leq\phi$.  Letting $\Psi$ be the
integrated form of $\psi$, we then have that $\Psi$ is a positive trace on $C^\ast(G)$ with $\Psi\leq\Phi$.
As $\Phi$ vanishes on every $M(\rho)$, with $\rho\neq\pi$, the same must be true for $\Psi$, and hence $\Psi$ factors through
a trace on $M(\pi)$.  However,  as we all know, matrix algebras admit a unique trace up to scalar multiples, so  it follows
that
$\Psi$ is a multiple of $\Phi$ which in turn implies  that   $\psi$ is a multiple of $\phi$, as desired.
Conversely, assuming that $\phi$ satisfies conditions (1--3) above, its  integrated form $\Phi$ is a trace,  so its
restriction  to each component
$M(\pi)$ must be a nonnegative scalar multiple,  say $\lambda_\pi$, of the standard trace on $M(\pi)$.  Therefore
$$
  \Phi\big ((a_\pi )_{\pi \in \hat G}\big ) = \sum_{\pi\in\hat G}\lambda_\pi\text{tr}(a_\pi ).
  $$
Picking any $\pi$ such that $\lambda_\pi\neq0$, observe that the character $\phi_\pi$
associated to $\pi$ satisfies
$\lambda_\pi\phi_\pi\leq\phi$ as a consequence of the fact that  their integrated forms obey that inequality.  The conclusion then
follows from point (3).
