Let $G$ be a reductive (complex or real) Lie group with a finite number of connected components. Can one construct an algorithm to generate any finite-dimensional representation $\rho$ of $G$ (meaning construct the matrix form $p(g), \text{ }\forall g \in G $) .
More precisely, in the finite-dimensional case, one can construct all the conjugacy classes resulting in a construction of all the representations of the group (such as in GAP). For compact Lie groups, one can compute the character of representations and by orthogonality relationship identify the irreducible ones (even if constructing the character table of the group can be very expensive). In the non compact case one can use the corresponding Lie Algebra and Cartan subalgebra to find the characters of irreducible representations. However it is not trivial to me how one could reconstruct the group irreducible representations from that.
What I would like to construct is an algorithm that takes a reductive (complex or real) Lie group with a finite number of connected components as entries, its corresponding Lie algebra, and the weight of a finite-dimensional irreducible representation and output the matrix form of the corresponding representation. Do you have any reference of related attempts to that or any general idea of this concern?