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Take $\mathfrak{g}$ a simple complex Lie algebra and $U(\mathfrak{g})$ its universal enveloping algebra and $Z(\mathfrak{g})$ the center of this last one. From universal property of the universal enveloping algebra we know that the category of $\mathfrak{g}$-modules is equivalent to the category of $U(\mathfrak{g})$-modules. My question are three:

Is it true that, since character theory, two irreducible representation of $\mathfrak{g}$ is determined by action of $Z(\mathfrak{g})$ and not all $U(\mathfrak{g})$ when $\mathfrak{g}$ is finite dimensional?

I read often that the center $Z(\mathfrak{g})$ is important for knowing representation theory of $\mathfrak{g}$, is for the precedent reason?

amWhy
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wood
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1 Answers1

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Your second question is (possibly accidentally) incredibly broad. Yes, the center of the universal enveloping algebra is very important in many aspects of representation theory...

For your first question, I believe Harish-Chandra proved that, for a given character of the center of the universal enveloping algebra, there are only finitely_many irreducible $\mathfrak g,K$-modules with that character. In very small cases, there may be just a single one, but not more generally (though I do not immediately know a counter-example).

At the very least, technical reasons will always make the center of a non-commutative ring be significant, if it's non-trivial (think of Schur's lemma). And the universal enveloping algebra of a Lie algebra is important because it is an associative algebra with the same representation theory as the original Lie algebra.

paul garrett
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  • Thank you, it has been useful. Could I ask for some references about the (maybe) Harish-Chandra theorem which you are talking about? – wood Dec 22 '21 at 01:46
  • Unfortunately I do not have a precise recollection of which paper where the result occurred... – paul garrett Dec 29 '21 at 22:23