Introduction to
Buildings and Combinatorial Representation Theory
American Institute of Mathematics (AIM)
March 26, 2007

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

Last update: 21 August 2014

Weyl characters

Your favourite group G (probably SL3()) corresponds to W={chambers} and P={dots} The irreducible G-modules L(λ) are indexed by λ(P)+ and char(L(λ))= μP Card(B(λ)μ) xμ, where B(λ)μ= {Littelmann paths of typeλand endμ}. If G=G(((t))), K=G([[t]]), andU-= {(10*1)} then G/K is the loop Grassmanian and G=λ(P)+ KtλKandG= μPU- tμK. The MV cycles of type λ and weight μ are the elements of MV(λ)μ= {irreducible components ofKtλKU-tμK}, and char(L(λ))= μCard (MV(λ)μ) xμ.

Hecke algebras

The spherical and affine Hecke algebras are Hsph=C(K\G/K) andH=C(I\G/I), where G = G(((t))) | | K = G([[t]]) Φ G() | | | I = Φ-1(B) B, whereB= {(**0*)}. The Satake map is [X]W= Z(H) Z(H)10= 10H10 =Hsph f f10 Pλ=Z(H) 10Xλ 10=χKtλK "obvious" basis and Pλ are the Hall-Littlewood polynomials. Pλ=μP Cardq(𝒫(λ)μ) xμ, where 𝒫(λ)μ= {Hecke paths of typeλand endμ} {slices ofG/KinKtλKU-tμK} and Cardq(𝒫(λ)μ) =p𝒫(λ)μ (# of𝔽qpoints in slicep). After normalization, Pλ|q-1=0= char(L(λ)).


The group B is a Borel subgroup of G=G() and G/B=flag variety=building. The cell decomposition of G/B is G=wWBwB. Idea: The points of W are regions, or chambers. W= s1,s2| s12=s22=1 ,s1s2s1= s2s1s2 1 s1 s2 s1s2 s2s1 s1s2s1 If w=si1si is a minimal length path to w then BwB={xi1(c1)si1xi(c)siB|c1,,c}, wherexi(c)=1+c Ei,i+1, with Ei,i+1 the matrix with a 1 in the (i,i+1) entry and all other entries 0.
IDEA: The points of G/B are regions, or chambers. x1(1)s1B x1(1)s1x2(2)s2B x1(1)s1x2(5.3)s2B x1(0)s1B x1(7)s1B x1(π/31)s1B x1(-4)s1B

Just as the building of W, the Coxeter complex, has relations s1s2s1=s2 s1s2 the building of G/B also has relations x1(c1)s1 x2(c2)s2 x1(c3)s1= x2(c3)s2 x1(c1c3-c2)s1 x2(c3)s2 An apartment is a subbuilding of G/B that looks like W.

The Borel subgroup of G=G(((t))) is I and G/Iis theaffine flag variety with G=wW IwI,whereW =WP is theaffine Weyl group The affine building G/I has sectors sinceG=vW U-vI.

MV polytopes

Let T={(*00*)} andletVbe aT -module with T-invariant inner product , (such that v,v=0v=0). Let 𝔥=Lie(T)and V={[v]|vV,v0}, where [v]=span{v}. The moment map on V is μ: V 𝔥* [v] μv μv(h)= hv,vv,v. Now let V=L(γ) be a simple G-module (G=G()) with highest weight vector v+. Then B[v+]=[v+] andG[v+] V is the image of G/B in V. The moment map on G/B (associated to γ) is μ: G/B V 𝔥* gB g[v+] μgv+ Joel(Kamnitzer)'s favourite case is G/K with γ=ω0 (the fundamental weight corresponding to the added node on the extended Dynkin diagram) and μ(MV cycle of typeλand weightμ)= (MV polytope of typeλand weightμ)


Let G=G(((t))). ((t))= { at+ a+1t+1+ |,ai } . Points of G/I are gI,whereg= (gij),gij ((t)). The valuation on ((t)) v(at+a+1t+1+) =, is like log v(f1f2)= v(f1)+v(f2) andv(f1+f2) =min(v(f1),v(f2)). Then v(gI) is a tropical point of v(G/I), the tropical flag variety. An amoeba, or tropical subvariety, is the image, under v, of a subvariety of G/I.

Notes and References

These are a typed copy of /Volumes/Data/Users/arun/Work2007/Bites2007/aimtalk3.26.07.pdf the text of a talk at the American Institute of Mathematics in Palo Alto on March 26, 2007.

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