$W$ is the ;

Weyl group

$\Delta ^{+}$ is the set of the of the root system $\Delta$ ;

positive roots

$\rho$ is the half-sum of the positive roots, often called the Weyl vector;

$\lambda$ is the of the irreducible representation $V$ ;

highest weight

$\varepsilon (w)$ is the determinant of the action of $w$ on the ${\mathfrak {h}}\subset {\mathfrak {g}}$ . This is equal to $(-1)^{\ell (w)}$ , where $\ell (w)$ is the length of the Weyl group element, defined to be the minimal number of reflections with respect to simple roots such that $w$ equals the product of those reflections.

Cartan subalgebra

Λ is a highest weight,

λ is some other weight,

m_Λ(λ) is the multiplicity of the weight λ in the irreducible representation V_Λ

ρ is the Weyl vector

The first sum is over all positive roots α.

Hans Freudenthal's formula is a recursive formula for the weight multiplicities that gives the same answer as the Kostant multiplicity formula, but is sometimes easier to use for calculations as there can be far fewer terms to sum. The formula is based on use of the Casimir element and its derivation is independent of the character formula. It states^[14]

where

W is the complex Weyl group of $H_{\mathbb {C} }$ with respect to $G_{\mathbb {C} }$

$W_{\lambda }$ is the stabilizer of $\lambda$ in W

Harish-Chandra showed that Weyl's character formula admits a generalization to representations of a real, reductive group. Suppose $\pi$ is an irreducible, admissible representation of a real, reductive group G with infinitesimal character $\lambda$ . Let $\Theta _{\pi }$ be the Harish-Chandra character of $\pi$ ; it is given by integration against an analytic function on the regular set. If H is a Cartan subgroup of G and H' is the set of regular elements in H, then

Here

and the rest of the notation is as above.

The coefficients $a_{w}$ are still not well understood. Results on these coefficients may be found in papers of Herb, Adams, Schmid, and Schmid-Vilonen among others.

Character theory

Algebraic character

Demazure character formula

Weyl integration formula

Kirillov character formula

Fulton, William and Harris, Joe (1991). Representation theory: a first course. New York: Springer-Verlag. 0387974954. OCLC 22861245.^[1]

ISBN

Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, 978-3319134666

ISBN

Humphreys, James E. (1972), , Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7.

Introduction to Lie Algebras and Representation Theory

Infinite dimensional Lie algebras, V. G. Kac, 0-521-37215-1

ISBN

Duncan J. Melville (2001) [1994], , Encyclopedia of Mathematics, EMS Press

"Weyl–Kac character formula"

(1925), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I", Mathematische Zeitschrift, 23, Springer Berlin / Heidelberg: 271–309, doi:10.1007/BF01506234, ISSN 0025-5874, S2CID 123145812

Weyl, Hermann

(1926a), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. II", Mathematische Zeitschrift, 24, Springer Berlin / Heidelberg: 328–376, doi:10.1007/BF01216788, ISSN 0025-5874, S2CID 186229448

Weyl, Hermann

(1926b), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. III", Mathematische Zeitschrift, 24, Springer Berlin / Heidelberg: 377–395, doi:10.1007/BF01216789, ISSN 0025-5874, S2CID 186232780