Kostant theorem on character values at the Coxeter element

수학노트
둘러보기로 가기 검색하러 가기

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).
  • The Coxeter elements of the Weyl group lift to the normalizer of a maximal torus and yield a single conjugacy class (Coxeter conjugacy class) in the Lie group


Kostant theorem

background

  • A well known classical result due to Kostant, based on previous work of Macdonald, expresses certain powers of the Dedekind \(\eta\)-function as power series, summing over the irreducible representations of suitable Lie groups.
  • An important factor in these expressions, \(\epsilon(\bar{\lambda})\), was shown to obtain only the values \(0\), \(1\), and \(-1\) when the group is simply laced.
  • The proof used the representation theory of Lie groups. \(\epsilon(\bar{\lambda})\) was expressed as the trace of the action of a Coxeter element acting on \(V_\bar{\lambda}^T\). Here \(V_\bar{\lambda}\) is an irreducible representation of the Lie group, corresponding to the dominant weight \(\bar{\lambda}\), and \(V_\bar{\lambda}^T\) is the subspace of vectors in \(V_\bar{\lambda}\) invariant under the action of a torus \(T\).

statement

  • Let \(K\) be a compact, simple, simply connected, simply laced Lie group, and let \(D\) be its set of dominant weights.
  • For \(\bar{\lambda}\in D\), let \(V_{\bar{\lambda}}\) be an irreducible \(K\)-module corresponding to \(\bar{\lambda}\); and let

\[ c(\bar{\lambda}) := (\bar{\lambda}+\rho,\bar{\lambda}+\rho) - (\rho,\rho), \] where \(\rho\) is half the sum of all the positive roots.

  • Let \(V_\bar{\lambda}^T\) be the zero-weight subspace of \(V_\bar{\lambda}\), i.e., the subspace of all vectors which are pointwise invariant under a fixed maximal torus \(T\) in \(K\).
  • Let \(W\) be the Weyl group of \(K\), and let

\[ \theta_\bar{\lambda} : W \longrightarrow \rm Aut V_\bar{\lambda}^T \] be the representation of \(W\) on \(V_\bar{\lambda}^T\). Denote \[ \epsilon(\bar{\lambda}) := \rm tr\, \theta_\bar{\lambda}(\tau), \] where \(\tau\) is any Coxeter element in \(W\).

thm (Kostant)

For any simple, simply connected and simply laced compact Lie group \(K\), \[ \phi(x)^{\dim K} = \sum_{\bar{\lambda}\in D} \epsilon(\bar{\lambda}) \cdot \dim V_\bar{\lambda} \cdot x^{c(\bar{\lambda})} \] and also \[ \epsilon(\bar{\lambda})\in\{0,1,-1\}\qquad(\forall\bar{\lambda}\in D). \]


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.
  • 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.

\[ 1\to T\to N(T)\to W\to 1 \]

thm (Kostant)

Let \(G\) be a semi-simple simply connected algebraic 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 \(1,0,-1\).

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.


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.


related items

computational resource


expositions


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.
  • Kostant, Bertram. ‘Powers of the Euler Product and Commutative Subalgebras of a Complex Simple Lie Algebra’. Inventiones Mathematicae 158, no. 1 (2004): 181–226. doi:10.1007/s00222-004-0370-7.
  • Adin, Ron M., and Avital Frumkin. ‘Rim Hook Tableaux and Kostant’s \(\eta\)-Function Coefficients’. arXiv:math/0201003, 31 December 2001. http://arxiv.org/abs/math/0201003.
  • 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

questions

원본 주소 "https://wiki.mathnt.net/index.php?title=Kostant_theorem_on_character_values_at_the_Coxeter_element&oldid=44016"