Algebraic character

Algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of semisimple Lie groups.

Let ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra with a fixed Cartan subalgebra ${\displaystyle {\mathfrak {h}},}$ and let the abelian group ${\displaystyle A=\mathbb {Z} [[{\mathfrak {h}}^{*}]]}$ consist of the (possibly infinite) formal integral linear combinations of ${\displaystyle e^{\mu }}$, where ${\displaystyle \mu \in {\mathfrak {h}}^{*}}$, the (complex) vector space of weights. Suppose that ${\displaystyle V}$ is a locally-finite weight module. Then the algebraic character of ${\displaystyle V}$ is an element of ${\displaystyle A}$ defined by the formula:

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