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Orthogonal and symplectic orbits in the affine flag variety of type A

Kam Hung Tong

Abstract

It is a classical result that the set $K\backslash G /B$ is finite, where $G$ is a reductive algebraic group over an algebraically closed field with characteristic not equal to two, $B$ is a Borel subgroup of $G$, and $K = G^θ$ is the fixed point subgroup of an involution of $G$. In this paper, we investigate the affine counterpart of the aforementioned set, where $G$ is the general linear group over formal Laurent series, $B$ is an Iwahori subgroup of $G$, and $K$ is either the orthogonal group or the symplectic group over formal Laurent series. We construct explicit bijections between the double cosets $K \backslash G/B$ and certain twisted affine involutions. This is the first combinatorial description of $K$-orbits in the affine flag variety of type A.

Orthogonal and symplectic orbits in the affine flag variety of type A

Abstract

It is a classical result that the set is finite, where is a reductive algebraic group over an algebraically closed field with characteristic not equal to two, is a Borel subgroup of , and is the fixed point subgroup of an involution of . In this paper, we investigate the affine counterpart of the aforementioned set, where is the general linear group over formal Laurent series, is an Iwahori subgroup of , and is either the orthogonal group or the symplectic group over formal Laurent series. We construct explicit bijections between the double cosets and certain twisted affine involutions. This is the first combinatorial description of -orbits in the affine flag variety of type A.

Paper Structure

This paper contains 15 sections, 18 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction
  2. Classical background
  3. Affine analogs
  4. Orbits of the orthogonal group
  5. Orbits of the special orthogonal group
  6. Orbits of the symplectic group
  7. Preliminaries
  8. Loop groups
  9. Extended affine Weyl group
  10. Discussion of related work
  11. Orbits of orthogonal groups
  12. Orbits of the orthogonal group
  13. Choices of $K$ and $B$
  14. Orbits of the special orthogonal group
  15. Orbits of the symplectic group

Key Result

Theorem 1.3

In the case where $K = \mathsf{O}_n(\mathbb{K}(\space(t)\space))$ and $G = \textsf{GL}_n(\mathbb{K}(\space(t)\space))$, for each double coset $\mathcal{O} \in K\backslash G/B$, there exists a unique $w \in \textsf{eSymAPM}_n$ such that $g^Tg = w$ for some $g \in \mathcal{O}$. Moreover, for each $w

Theorems & Definitions (44)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 34 more