Loop groups and the affine flag variety $G/I$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 16 October 2012

Loop groups and the affine flag variety $G/I$

This section gives a short treatment of loop groups following [Ste1967, Ch. 8] and [Mac1971, §2.5 and 2.6]. This theory is currently a subject of intense research as evidenced by the work in [Gar1995], [GKa2004], [Rém2002], [Rou2006], [GRo0703639].

Let ${𝔤}_0$ be a symmetrizable Kac-Moody Lie algebra and let ${𝔥}_{\mathbb{Z}}$ be a $\mathbb{Z}\text{-lattice}$ in ${𝔥}_0$ that contains $Q^{\vee }=\mathbb{Z}\text{-span}\{h_1,\dots ,h_n\}\text{.}$

$$\begin{array}{cc}\text{The}\text{loop group}\text{is the Tits group}G=G_0\left(ℂ\left(\left(t\right)\right)\right)& \text{(6.1)}\end{array}$$

over the field $𝔽=ℂ\left(\left(t\right)\right)\text{.}$ Let $K=G_0\left(ℂ\left[\left[t\right]\right]\right)$ and $G_0\left(ℂ\right)$ be the Tits group of ${𝔤}_0$ and ${𝔥}_{\mathbb{Z}}$ over the rings $ℂ\left[\left[t\right]\right]$ and $ℂ,$ respectively, and let $B\left(ℂ\right)$ be the standard Borel subgroup of $G_0\left(ℂ\right)$ as defined in (4.2). Let

$$\begin{array}{cc}{U}^{-}\text{be the subgroup of}G\text{generated by}{x}_{-\alpha }\left(f\right)\text{for}\alpha \in R_{\text{re}}^{+}\text{and}f\in ℂ\left(\left(t\right)\right),& \text{(6.2)}\end{array}$$

and define the standard Iwahori subgroup $I$ of $G$ by

$$\begin{array}{cc}\begin{array}{ccc}G& =& G_0\left(ℂ\left(\left(t\right)\right)\right)\\ ⊆ & & ⊆ \\ K& =& G_0\left(ℂ\left[\left[t\right]\right]\right)& \stackrel{{\text{ev}}_{t=0}}{⟶}& G_0\left(ℂ\right)\\ ⊆ & & ⊆ & & ⊆ \\ I& =& {\text{ev}}_{t=0}^{-1}\left(B\left(ℂ\right)\right)& \stackrel{{\text{ev}}_{t=0}}{⟶}& B\left(ℂ\right)\text{.}\end{array}& \text{(6.3)}\end{array}$$

The affine flag variety is $G/I\text{.}$

For $\alpha +j\delta \in R_{\text{re}}+\mathbb{Z}\delta$ and $c\in ℂ,$ define

$$\begin{array}{cc}{x}_{\alpha +j\delta }\left(c\right)=x_{\alpha }\left(c{t}^j\right)\text{and}{t}_{{\lambda }^{\vee }}=h_{{\lambda }^{\vee }}\left(t^{-1}\right),& \text{(6.4)}\end{array}$$

and, for $c\in {ℂ}^{\times },$ define

$$\begin{array}{cc}{n}_{\alpha +j\delta }\left(c\right)=x_{\alpha +j\delta }\left(c\right)x_{-\alpha -j\delta }(-c^{-1})x_{\alpha +j\delta }\left(c\right),& \text{(6.5)}\\ n_{\alpha +j\delta }=n_{\alpha +j\delta }\left(1\right),\text{and}{h}_{{(\alpha +j\delta )}^{\vee }}\left(c\right)=n_{\alpha +j\delta }\left(c\right)n_{\alpha +j\delta }^{-1}& \text{(6.6)}\end{array}$$

analogous to (3.3).

The group

$$\begin{array}{cc}\tilde{W}=\{t_{{\lambda }^{\vee }}w\mid {\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},w\in W_0\}\text{with}{t}_{{\lambda }^{\vee }}{t}_{{\mu }^{\vee }}=t_{{\lambda }^{\vee }+{\mu }^{\vee }}\text{and}w{t}_{{\lambda }^{\vee }}=t_{w{\lambda }^{\vee }}w,& \text{(6.7)}\end{array}$$

acts on ${𝔥}_0^{*}\oplus ℂ\delta$ by

$$\begin{array}{cc}v(\mu +k\delta )=v\mu +k\delta \text{and}{t}_{{\lambda }^{\vee }}(\mu +k\delta )=\mu +(k-⟨{\lambda }^{\vee },\mu ⟩)\delta & \text{(6.8)}\end{array}$$

for $v\in W_0,$ ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},$ $\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and $k\in \mathbb{Z}\text{.}$ Then $n_{\alpha +j\delta }\left(c\right)=t_{-j{\alpha }^{\vee }}{n}_{\alpha }\left(c\right)=n_{\alpha }\left(c{t}^j\right),$

$$n_{\alpha }{x}_{\beta +k\delta }\left(c\right)n_{\alpha }^{-1}=n_{\alpha }{x}_{\beta }\left(c{t}^k\right)n_{\alpha }^{-1}=x_{s_{\alpha }\beta }\left({\epsilon }_{\alpha ,\beta }c{t}^k\right)=x_{s_{\alpha }(\beta +k\delta )}\left({\epsilon }_{\alpha ,\beta }c\right)$$

for $\alpha \in R_{\text{re}},$ and, for ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},$

$$t_{{\lambda }^{\vee }}{x}_{\beta +k\delta }\left(c\right)t_{{\lambda }^{\vee }}^{-1}=x_{\beta +k\delta }\left(t^{-⟨{\lambda }^{\vee },\beta ⟩}c\right)=x_{t_{{\lambda }^{\vee }}(\beta +k\delta )}\left(c\right)\text{.}$$

Thus the root subgroups

$$\begin{array}{cc}{\chi }_{\alpha +j\delta }=\{x_{\alpha +j\delta }\left(c\right)\mid c\in ℂ\}\text{satisfy}w{\chi }_{\alpha +j\beta }{w}^{-1}={\chi }_{w(\alpha +j\delta )}& \text{(6.9)}\end{array}$$

for $w\in \tilde{W}$ and $\alpha +j\delta \in R_{\text{re}}+\mathbb{Z}\delta \text{.}$ These relations are a reflection of the symmetry of the group $G$ under the group defined in (3.8):

$$\begin{array}{cc}\tilde{N}=N\left(ℂ\left(\left(t\right)\right)\right)\text{generated by}{n}_{\alpha }\left(g\right),h_{{\lambda }^{\vee }}\left(g\right),\text{for}g\in ℂ{\left(\left(t\right)\right)}^{\times },& \text{(6.10)}\end{array}$$

$\alpha \in R_{\text{re}},$ and ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}$ The homomorphism $\tilde{N}\to W_0$ from (3.9) lifts to a surjective homomorphism (see [Mac1971, p. 26 and p. 28])

$$\begin{array}{cc}\begin{array}{ccc}\tilde{N}& ⟶& \tilde{W}\\ n_{\alpha +j\delta }& ⟼& t_{-j{\alpha }^{\vee }}{s}_{\alpha }\\ t_{{\lambda }^{\vee }}& ⟼& t_{\lambda }^{\vee }\end{array}& \text{with kernel}H\text{generated by}{h}_{\lambda }\left(d\right),d\in ℂ{\left[\left[t\right]\right]}^{\times }\text{.}\end{array}$$

Define

$$\begin{array}{cc}{\tilde{R}}_{\text{re}}^I=(R_{\text{re}}^{+}+{\mathbb{Z}}_{\ge 0}\delta )\bigsqcup (-R_{\text{re}}^{+}+{\mathbb{Z}}_{\ge 0}\delta )\text{and}{\tilde{R}}_{\text{re}}^U=-R_{\text{re}}^{+}+\mathbb{Z}\delta & \text{(6.11)}\end{array}$$

so that

$$\begin{array}{cc}\begin{array}{cccc}{\chi }_{\alpha +j\delta }\subseteq I& \text{if and only if}& \alpha +j\delta \in {\tilde{R}}_{\text{re}}^I& \text{and}\\ {\chi }_{\alpha +j\delta }\subseteq U^{-}& \text{if and only if}& \alpha +j\delta \in {\tilde{R}}_{\text{re}}^U\text{.}\end{array}& \text{(6.12)}\end{array}$$

Note that ${\tilde{R}}_{\text{re}}^I\bigsqcup (-{\tilde{R}}_{\text{re}}^I)={\tilde{R}}_{\text{re}}^U\bigsqcup (-{\tilde{R}}_{\text{re}}^U)=R_{\text{re}}+\mathbb{Z}\delta \text{.}$

Notes and References

This is section 6 from a paper entitled Combinatorics in affine flag varieties by James Parkinson, Arun Ram and Cristoph Schwer.

page history