Kostant theorem on character values at the Coxeter element
introduction
- in $\widehat{G}_{sc}$, the simply connected cover of $\widehat{G}$, there is a unique regular conjugacy class whose image in $\widehat{G}$ has order $h$ (which is the Coxeter conjugacy class).
Coxeter conjugacy class
- The Coxeter elements of the Weyl group lift to the normalizer of a maximal torus and yield a single conjugacy class in the Lie group
Kostant theorem
- thm
Let $G$ be a semi-simple simply connected group over $\C$, and $\pi$ a finite dimensional irreducible representation of $G$. Then the character $\Theta_\pi$ of $\pi$ at the element $c(G)$ takes one of the values $1,0,-1$.
comments by Jim Humphreys
- In the setting of a simply connected compact simple Lie group, the theorem says that any finite dimensional irreducible representation has character value 0,1. or −1 at the (lift of a) Coxeter element.
- Here the group could equally well be a complex Lie group or algebraic group.
Green-Lehrer-Lusztig
- This reminds me of another 1976 theorem, proved in a different area of representation theory.
- collaboration by J.A. Green, G.I. Lehrer, and G. Lusztig, On the degrees of certain group characters, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 1-4.
- They deal with the complex irreducible characters of the finite group G of rational points of a reductive algebraic group over a finite field.
- Under mild conditions the regular unipotent elements form a single conjugacy class, and the theorem states that any irreducible character takes value 0,1, or −1 at such an element.
dual group
We recall that to a reductive algebraic group $G$ over $\C$, there is associated the dual group $\widehat{G}$ which is a reductive algebraic group over $\C$ with root datum which is the dual to that of $G$. Fix a maximal torus $T$ in $G$, and a maximal torus $\widehat{T}$ in $\widehat{G}$, such that there is a canonical isomorphism between the character group of $T$ and the co-character group of $\widehat{T}$, and as a result we have the identifications $$\widehat{T}(\C) = {\rm Hom} [\C^\times, \widehat{T}] \otimes_\Z \C^\times = {\rm Hom}[T,\C^\times]\otimes_\Z \C^\times,$$ all the homomorphisms being algebraic. Thus given a character $\chi: T \rightarrow \C^\times$, it gives rise to a co-character, $$\widehat{\chi}: \C^\times \longrightarrow \widehat{T}(\C)= {\rm Hom}[T,\C^\times]\otimes_\Z \C^\times, $$ given by $z \longrightarrow \chi \otimes z$.
For a semi-simple simply connected algebraic group $G$ over $\C$, let $\rho$ be half the sum of positive roots of a maximal torus $T$ in $G$ (for any fixed choice of positive roots). It is clear from the definition of $\rho$ that the pair $(T, \rho)$ is well-defined up to conjugacy by $G(\C)$; in particular, the restriction of the character $\rho$ to $Z$, the center of $G(\C)$, is a well defined character of $Z$ which is of order $\leq 2$, to be denoted by $\rho_{Z}: Z \rightarrow \Z/2$. By (Pontrajagin) duality, we get a homomorphism $\rho^\vee_{Z}: \Z/2 \rightarrow Z^\vee$ where $Z^\vee$ denotes the character group of $Z$.
Let $\widehat{G}_{sc}$ be the simply connected cover of $\widehat{G}$ whose center can be identified to $Z^\vee$, the character group of $Z$. We shall see later that the image of the nontrivial element in $\Z/2$ under the homomorphism $\rho^\vee_{Z}: \Z/2 \rightarrow Z^\vee$ gives an important element in the center of $\widehat{G}_{sc}$ which determines whether an irreducible selfdual representation of $\widehat{G}_{sc}$ is orthogonal or symplectic.
computational resource
expositions
- Koszul, Jean-Louis. “Travaux de B. Kostant Sur Les Groupes de Lie Semi-Simples.” Accessed April 11, 2014. http://www.numdam.org/numdam-bin/fitem?id=SB_1958-1960__5__329_0.
articles
- Prasad, Dipendra. 2014. “Half the Sum of Positive Roots, the Coxeter Element, and a Theorem of Kostant.” arXiv:1402.5504 [math], February. http://arxiv.org/abs/1402.5504.
- Prasad, Dipendra. 2014. “A Character Relationship on $GL_n$.” arXiv:1402.5505 [math], February. http://arxiv.org/abs/1402.5505.
- Green, J. A., G. I. Lehrer, and G. Lusztig. ‘On the Degrees of Certain Group Characters’. The Quarterly Journal of Mathematics 27, no. 1 (3 January 1976): 1–4. doi:10.1093/qmath/27.1.1.
- Kostant, On Macdonald's η-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), no. 2, 179-212