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On convexity and the Iwasawa decomposition of split real and complex Kac-Moody groups

Paul Zellhofer, Ralf Köhl

Abstract

We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free and cofree root data. The result can be seen as a first step towards describing the multiplication map in a Kac-Moody group in terms of Iwasawa coordinates. Our method involves a detailed analysis of the geometry of Weyl group orbits in the Cartan subalgebra of a real Kac-Moody algebra. It provides an alternative proof of Kostant convexity for semisimple Lie groups and also generalizes a linear analogue of Kostant's theorem for Kac-Moody algebras that has been established by Kac and Peterson in 1984.

On convexity and the Iwasawa decomposition of split real and complex Kac-Moody groups

Abstract

We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free and cofree root data. The result can be seen as a first step towards describing the multiplication map in a Kac-Moody group in terms of Iwasawa coordinates. Our method involves a detailed analysis of the geometry of Weyl group orbits in the Cartan subalgebra of a real Kac-Moody algebra. It provides an alternative proof of Kostant convexity for semisimple Lie groups and also generalizes a linear analogue of Kostant's theorem for Kac-Moody algebras that has been established by Kac and Peterson in 1984.
Paper Structure (11 sections, 20 theorems, 96 equations, 1 figure)

This paper contains 11 sections, 20 theorems, 96 equations, 1 figure.

Table of Contents

  1. Kac-Moody algebras
  2. Convex hulls of Weyl group orbits
  3. Some results from convex geometry
  4. Orbits of regular elements
  5. Kac-Moody groups
  6. The convexity theorem
  7. Proof of the main theorem
  8. Step 1: Highest-weight representations
  9. Step 2: Induction
  10. A linear analogue
  11. Concluding remarks

Key Result

Proposition 1.3

Suppose that Assumption assum:GlobalAssumption is in effect, then $d\geq 2n-\operatorname{rank}(\mathbb{A})$ and the following three properties hold:

Figures (1)

  • Figure 1: Two configurations of $P^{(i)}_{\operatorname{ess}}(t)$.

Theorems & Definitions (39)

  • Remark 1.1
  • Proposition 1.3
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • ...and 29 more