Loop groups and the affine flag variety G/I

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 𝔤0 be a symmetrizable Kac-Moody Lie algebra and let 𝔥 be a -lattice in 𝔥0 that contains Q=-span {h1,,hn}.

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

over the field 𝔽=((t)). Let K=G0([[t]]) and G0() be the Tits group of 𝔤0 and 𝔥 over the rings [[t]] and , respectively, and let B() be the standard Borel subgroup of G0() as defined in (4.2). Let

U-be the subgroup ofG generated byx-α(f) forαRre+ andf ((t)), (6.2)

and define the standard Iwahori subgroup I of G by

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

The affine flag variety is G/I.

For α+jδRre+δ and c, define

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

and, for 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~= { tλw λ𝔥,w W0 } withtλ tμ= tλ+μ andwtλ =twλw, (6.7)

acts on 𝔥0*δ by

v(μ+kδ)=vμ+k δandtλ (μ+kδ)=μ+ ( k- λ,μ ) δ (6.8)

for vW0, λ𝔥, μ𝔥* and k. Then nα+jδ(c)= t-jα nα(c)=nα (ctj),

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

for αRre, and, for λ𝔥,

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 } satisfywχα+jβ w-1= χw(α+jδ) (6.9)

for wW~ and α+jδ Rre+δ. These relations are a reflection of the symmetry of the group G under the group defined in (3.8):

N~=N(((t))) generated bynα(g) ,hλ(g), forg ((t))×, (6.10)

αRre, and λ𝔥. The homomorphism N~W0 from (3.9) lifts to a surjective homomorphism (see [Mac1971, p. 26 and p. 28])

N~ W~ nα+jδ t-jαsα tλ tλ with kernelHgenerated by hλ(d), d[[t]]× .


R~reI= ( Rre++ 0δ ) ( -Rre++ 0δ ) and R~reU=- Rre++δ (6.11)

so that

χα+jδI if and only if α+jδR~reI and χα+jδU- if and only if α+jδR~reU. (6.12)

Note that R~reI (-R~reI)= R~reU (-R~reU)= Rre+δ.

Notes and References

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

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