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It appears that you want the "rough, introductory answers", so feel fine to ask more detailed questions if anything is unclear/potentially wrong. Also, I only have very general knowledge about enumera …

There could be different ways to give meaning to the phrase "explicit construction". In an algebro-geometric sense, an expicit construction comes from more classical Borel-Weil-Bott theorem of which …

See also the question Unitary representations of $SL(2, \mathbb R)$.

I've heard about this construction on the lecture about higher representation theory: Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K_0$ is the universal enveloping algeb …

You can't assume the module is irreducible, since many aren't! However, if you want to learn something about modules of $sl_2$ it helps to make the following observations: Each module is a sum of i …

It's not entirely trivial, but here's the sketch of the proof if you would like to finish it yourself. Note that people usually denote complex numbers by $\mathbb C$: (1) Every irreducible represe …

Some time ago I was told there's an interesting classical Satake correspondence which I will write as $$[\mathop{\mathrm{disk}} \Rightarrow G] \,\backslash\, [\mathop{\mathrm{disk}^\times} \Rightarr …

I haven't heard this question before, but I would approach it in a following way: you know some basic relations between irreducible representations of a groups — e.g. their # is the # of conjugacy cl …

There is a result about the real representations of a simple Lie group $G$ which is known as Soergel conjecture or Vogan duality. We'll focus on the formulation that for a fixed character of $G$ $\cup …

Let's use the notation of [A=>B] for Hom(A, B). Take a 1-dimensional algebraic torus Gm and higher-dimensional torus T and let's live in the category of commutative algebraic groups over k. Out of f …

This question comes from my notes, heavily edited, thus slightly unusual structure. For Lie groups one can reformulate character theory as saying that C ⊗ K(G\ pt) = C[T/W] = C[ X* ]W where …

There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called Peter-Weyl theorem. Turns out for some reason I automatically think that there is a …

I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group G being SL(2, R), can be completely described and that there is a discre …

There are lots of good points in other answers, so I want to to add one specific thing about why representations are uniquely defined by their characters. Irreducible representations are uniquely det …

Could this be related to Drinfeld associator?