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. 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 g {displaystyle {mathfrak {g}}} be a semisimple Lie algebra with a fixed Cartan subalgebra h , {displaystyle {mathfrak {h}},} and let the abelian group A = Z [ [ h ∗ ] ] {displaystyle A=mathbb {Z} ]} consist of the (possibly infinite) formal integral linear combinations of e μ {displaystyle e^{mu }} , where μ ∈ h ∗ {displaystyle mu in {mathfrak {h}}^{*}} , the (complex) vector space of weights. Suppose that V {displaystyle V} is a locally-finite weight module. Then the algebraic character of V {displaystyle V} is an element of A {displaystyle A} defined by the formula: where the sum is taken over all weight spaces of the module V . {displaystyle V.} The algebraic character of the Verma module M λ {displaystyle M_{lambda }} with the highest weight λ {displaystyle lambda } is given by the formula