On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group

Mokler, Claus

Canadian journal of mathematics, Tome 53 (2001) no. 1, pp. 195-211

Résumé

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\mathbb{C}$ . Let $\text{T}{{\text{r}}_{{{\Lambda }_{1}}\,,\ldots ,\,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}}$ be the characters of the fundamental irreducible representations of $G$ , defined as convergent series on a certain part ${{G}^{\text{tr}-\text{alg}}}\,\subseteq \,G$ . Following Steinberg in the classical case and Brüchert in the affine case, we define the Steinberg map $\chi \,:=\,\left( \text{T}{{\text{r}}_{{{\Lambda }_{1}},\ldots ,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}} \right)$ as well as the Steinberg cross section $C$ , together with a natural parametrisation $\omega :{{\mathbb{C}}^{n}}\times {{\left( {{\mathbb{C}}^{\times }} \right)}^{n-l}}\to C$ . We investigate the local behaviour of $\text{ }\!\!\chi\!\!\text{ }$ on $C$ near $\omega \left( \,\left( 0,\ldots 0 \right)\,\times \,\left( 1,\ldots ,1 \right)\, \right)$ , and we show that there exists a neighborhood of $\left( 0,...,0 \right)\,\,\times \,\,\left( 1,...,1 \right)$ , on which $\text{ }\!\!\chi\!\!\text{ }\,\circ \,\omega $ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on $C$ .

Détail

DOI : 10.4153/CJM-2001-008-4

Mots-clés : 22E65, 17B65

Bibliographie

[B] [B] Brüchert, G., Trace class elements and cross-sections in Kac-Moody groups. Canad. J. Math. 50(1998), 972–1006. Google Scholar

[H,Sl] [H,Sl] Helmke, S. and Slodowy, P., Loop Groups, Principal Bundles over Elliptic Curves and Elliptic Singularities. Abstracts of the AnnualMeeting of the Mathematical Society of Japan, Hiroshima, September 1999, Session: ‘Infinite dimensional Analysis’, 67–77. Google Scholar

[K] [K] Kac, V. G., Infinite dimensional Lie algebras,. Cambridge University Press, 1990. Google Scholar

[K,P1] [K,P1] Kac, V. G. and Peterson, D. H., Infinite flag varieties and conjugacy theorems. Proc. Nat. Acad. Sci. USA 80(1983), 1778–1782. Google Scholar

[K,P2] [K,P2] Kac, V. G. and Peterson, D. H., Defining relations on certain infinite-dimensional groups. Proceedings of the Cartan conference, Lyon 1984, Astérisque, 1985, Numéro hors serie, 165–208. Google Scholar

[M] [M] Mokler, C., A formal Chevalley restriction theorem for Kac-Moody groups. Compositio Math., to appear. Google Scholar

[Mo,Pi] [Mo,Pi] Moody, R. V. and Pianzola, A., Lie AlgebrasWith Triangular Decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1995. Google Scholar

[Sl 1] [Sl 1] Slodowy, P., Simple singularities and simple algebraic groups. Springer Lecture Notes in Math. 815, 1980. Google Scholar

[Sl 2] [Sl 2] Slodowy, P., A character approach to Looijenga's invariant theory for generalized root systems. Compositio Math. 55(1985), 3–32. Google Scholar

[Sl 3] [Sl 3] Slodowy, P., An adjoint quotient for certain groups attached to Kac-Moody algebras. In: Infinite-dimensional groups with applications, Math. Sci. Res. Inst. Publ. 4, Springer, 1985, 307–334. Google Scholar

[St] [St] Steinberg, Robert, Regular Elements of Semisimple Algebraic Groups. Inst. Hautes Études Sci. Publ. Math. 25(1965), 49–80. Google Scholar

[Ti] [Ti] Tits, J., Uniqueness and Presentation of Kac-Moody Groups over fields. J. Algebra 105(1987), 542–573. Google Scholar