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

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 ${𝔥}_{ℤ}$ be a $ℤ\text{-lattice}$ in ${𝔥}_{0}$ that contains ${Q}^{\vee }=ℤ\text{-span}\phantom{\rule{0.2em}{0ex}}\left\{{h}_{1},\dots ,{h}_{n}\right\}\text{.}$

$Theloop groupis the Tits group G=G0 (ℂ((t))) (6.1)$

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 ${𝔥}_{ℤ}$ 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

$U-be the subgroup ofG generated byx-α(f) forα∈Rre+ andf∈ℂ ((t)), (6.2)$

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

$G = G0 (ℂ((t))) ⊆ ⊆ K = G0 (ℂ[[t]]) ⟶evt=0 G0(ℂ) ⊆ ⊆ ⊆ I = evt=0-1 (B(ℂ)) ⟶evt=0 B(ℂ). (6.3)$

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

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

$xα+jδ(c)= xα(ctj) andtλ∨= hλ∨(t-1), (6.4)$

and, for $c\in {ℂ}^{×},$ define

$nα+jδ(c)= xα+jδ(c) x-α-jδ (-c-1) xα+jδ(c), (6.5) nα+jδ= nα+jδ(1), and h(α+jδ)∨ (c)= nα+jδ(c) nα+jδ-1 (6.6)$

analogous to (3.3).

The group

$W~= { tλ∨w∣ λ∨∈𝔥ℤ,w∈ W0 } withtλ∨ tμ∨= tλ∨+μ∨ andwtλ∨ =twλ∨w, (6.7)$

acts on ${𝔥}_{0}^{*}\oplus ℂ\delta$ by

$v(μ+kδ)=vμ+k δandtλ∨ (μ+kδ)=μ+ ( k- ⟨λ∨,μ⟩ ) δ (6.8)$

for $v\in {W}_{0},$ ${\lambda }^{\vee }\in {𝔥}_{ℤ},$ $\mu \in {𝔥}_{ℤ}^{*}$ and $k\in ℤ\text{.}$ Then ${n}_{\alpha +j\delta }\left(c\right)={t}_{-j{\alpha }^{\vee }}{n}_{\alpha }\left(c\right)={n}_{\alpha }\left(c{t}^{j}\right),$

$nαxβ+kδ (c)nα-1= nαxβ (ctk)nα-1 =xsαβ ( εα,βctk ) =xsα(β+kδ) (εα,βc)$

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

$tλ∨ xβ+kδ(c) tλ∨-1= xβ+kδ ( t-⟨λ∨,β⟩ c ) = x tλ∨ (β+kδ) (c).$

Thus the root subgroups

$χα+jδ= { xα+jδ(c) ∣c∈ℂ } satisfywχα+jβ w-1= χw(α+jδ) (6.9)$

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

$N~=N(ℂ((t))) generated bynα(g) ,hλ∨(g), forg∈ℂ ((t))×, (6.10)$

$\alpha \in {R}_{\text{re}},$ and ${\lambda }^{\vee }\in {𝔥}_{ℤ}\text{.}$ The homomorphism $\stackrel{~}{N}\to {W}_{0}$ from (3.9) lifts to a surjective homomorphism (see [Mac1971, p. 26 and p. 28])

$N~ ⟶ W~ nα+jδ ⟼ t-jα∨sα tλ∨ ⟼ tλ∨ with kernelHgenerated by hλ(d), d∈ℂ[[t]]× .$

Define

$R~reI= ( Rre++ ℤ≥0δ ) ⊔ ( -Rre++ ℤ≥0δ ) and R~reU=- Rre++ℤδ (6.11)$

so that

$χα+jδ⊆I if and only if α+jδ∈R~reI and χα+jδ⊆U- if and only if α+jδ∈R~reU. (6.12)$

Note that ${\stackrel{~}{R}}_{\text{re}}^{I}\bigsqcup \left(-{\stackrel{~}{R}}_{\text{re}}^{I}\right)={\stackrel{~}{R}}_{\text{re}}^{U}\bigsqcup \left(-{\stackrel{~}{R}}_{\text{re}}^{U}\right)={R}_{\text{re}}+ℤ\delta \text{.}$

## Notes and References

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