I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, \alpha) = \begin{bmatrix} \cos(\alpha) & -\sin(\alpha) & t_1 \\ \sin(\alpha) & \cos(\alpha) & t_2 \\ 0 & 0 & 1 \end{bmatrix} $$ We can define the "quasi-regular" representation of this group on functions $f : \mathbb{R}^2 \rightarrow \mathbb{R}$: $$T(g) f(x) = f(g^{-1} x),$$ where $x=(x_1, x_2, 1)$ is a homogeneous coordinate for a point in the plane. Under suitable restrictions on $f$ (e.g. square integrability), this defines a representation of $M(2)$ in a Hilbert space. My source [1] proceeds to derive the irreducible representations of $M(2)$ on this Hilbert space, and arrives at the complex valued functions: $$ \psi_n^\lambda(r, \theta) = (-i)^n J_n(\lambda r) e^{i n \theta} $$ where $(r, \theta)$ are the polar coordinates of $x$. If I understood correctly, each irrep is spanned by the functions corresponding to a single value for the complex number $\lambda$, and all values of $n \in \mathbb{N}$.
What bothers me about this result is that we started with a function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ (or $\mathbb{C} \rightarrow \mathbb{R}$), and end up with a basis for a space of functions that can return complex values. My questions are thus:
1) Is the notion "real-irreducible" consistent? By real irrep I mean a real subspace of a real space that is invariant under the action of the group, and whose orthogonal complement is also invariant.
2) Has this been studied before? Are there books or papers that state the results for real values representations?
3) Could we simply take $\mathfrak{Re}(\psi_n^\lambda)$ and $\mathfrak{Im}(\psi_n^\lambda)$ as a basis for the function space? Or would it be redundant? Would it be real-irreducible? How does this generalize to other groups, i.e. can I just take the complex results and take the real an imaginary parts to obtain real bases?
Thanks in advance!
[1] N.J. Vilenkin, Special Functions and the Theory of Group Representations, 1968.