Introduction toBuildings and Combinatorial Representation TheoryAmerican Institute of Mathematics (AIM)March 26, 2007

Last update: 21 August 2014

Weyl characters

Your favourite group ${G}^{\vee }$ (probably ${SL}_{3}\left(ℂ\right)\text{)}$ corresponds to $W={chambers} and P∨={dots}$ The irreducible ${G}^{\vee }\text{-modules}$ $L\left({\lambda }^{\vee }\right)$ are indexed by ${\lambda }^{\vee }\in {\left({P}^{\vee }\right)}^{+}$ and $char(L(λ∨))= ∑μ∨∈P∨ Card(B(λ∨)μ∨) xμ∨,$ where $B(λ∨)μ∨= {Littelmann paths of type λ∨ and end μ∨}.$ If $G=G(ℂ((t))), K=G(ℂ[[t]]), andU-= {(10⋱*1)}$ then $G/K$ is the loop Grassmanian and $G=⨆λ∨∈(P∨)+ Ktλ∨KandG= ⨆μ∨∈P∨U- tμ∨K.$ The MV cycles of type ${\lambda }^{\vee }$ and weight ${\mu }^{\vee }$ are the elements of $MV(λ∨)μ∨= {irreducible components of Ktλ∨K∩U-tμ∨K‾},$ and $char(L(λ∨))= ∑μ∨Card (MV(λ∨)μ∨) xμ∨.$

Hecke algebras

The spherical and affine Hecke algebras are $H∼sph=C(K\G/K) andH∼=C(I\G/I),$ where $G = G(ℂ((t))) ∪| ∪| K = G(ℂ[[t]]) ⟶Φ G(ℂ) ∪| ∪| ∪| I = Φ-1(B) ⟶ B, whereB= {(**⋱0*)}.$ The Satake map is $ℂ[X]W= Z(H∼) ⟶∼ Z(H∼)10= 10H∼10 =H∼sph f ⟼ f10 Pλ∨ =Z(H∼) ← 10Xλ∨ 10=χKtλ∨K "obvious" basis$ and ${P}_{{\lambda }^{\vee }}$ are the Hall-Littlewood polynomials. $Pλ∨=∑μ∨∈P∨ Cardq(𝒫(λ∨)μ∨) xμ∨,$ where $𝒫(λ∨)μ∨= {Hecke paths of type λ∨ and end μ∨} ⟷{slices of G/K in Ktλ∨K∩U-tμ∨K}$ and $Cardq(𝒫(λ∨)μ∨) =∑p∈𝒫(λ∨)μ∨ (# of 𝔽q points in slice p).$ After normalization, $Pλ∨|q-1=0= char(L(λ∨)).$

Buildings

The group $B$ is a Borel subgroup of $G=G\left(ℂ\right)$ and $G/B=flag variety=building.$ The cell decomposition of $G/B$ is $G=⨆w∈WBwB.$ Idea: The points of $W$ are regions, or chambers. $W= ⟨ s1,s2 | s12=s22=1 ,s1s2s1= s2s1s2 ⟩ 1 {s}_{1} {s}_{2} {s}_{1}{s}_{2} {s}_{2}{s}_{1} {s}_{1}{s}_{2}{s}_{1}$ If $w={s}_{{i}_{1}}\cdots {s}_{{i}_{\ell }}$ is a minimal length path to $w$ then $BwB={xi1(c1)si1⋯xiℓ(cℓ)siℓB | c1,…,cℓ∈ℂ}, wherexi(c)=1+c Ei,i+1,$ with ${E}_{i,i+1}$ the matrix with a $1$ in the $\left(i,i+1\right)$ entry and all other entries $0\text{.}$
IDEA: The points of $G/B$ are regions, or chambers. ${x}_{1}\left(1\right){s}_{1}B {x}_{1}\left(1\right){s}_{1}{x}_{2}\left(\sqrt{2}\right){s}_{2}B {x}_{1}\left(1\right){s}_{1}{x}_{2}\left(5.3\right){s}_{2}B {x}_{1}\left(0\right){s}_{1}B {x}_{1}\left(\sqrt{7}\right){s}_{1}B {x}_{1}\left(\pi /31\right){s}_{1}B {x}_{1}\left(-4\right){s}_{1}B$

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 $W\text{.}$

The Borel subgroup of $G=G\left(ℂ\left(\left(t\right)\right)\right)$ is $I$ and $G/I is the affine flag variety$ with $G=⨆w∈W∼ IwI,whereW∼ =W⋉P∨$ $is the affine Weyl group$ The affine building $G/I$ has sectors $sinceG=⨆v∈W∼ U-vI.$

MV polytopes

Let $T={(*0⋱0*)} andlet V be a T -module$ with $T\text{-invariant}$ inner product $⟨,⟩$ (such that $⟨v,v⟩=0⇔v=0\text{).}$ Let $𝔥=Lie(T)and ℙV={[v] | v∈V,v≠0},$ where $\left[v\right]=\text{span}\left\{v\right\}\text{.}$ The moment map on $ℙV$ is $μ: ℙV ⟶ 𝔥* [v] ⟼ μv μv(h)= ⟨hv,v⟩⟨v,v⟩.$ Now let $V=L\left(\gamma \right)$ be a simple $G\text{-module}$ $\text{(}G=G\left(ℂ\right)\text{)}$ with highest weight vector ${v}^{+}\text{.}$ Then $B[v+]=[v+] andG[v+]⊆ ℙV$ is the image of $G/B$ in $ℙV\text{.}$ The moment map on $G/B$ (associated to $\gamma \text{)}$ is $μ: G/B ⟶ ℙV ⟶ 𝔥* gB ⟼ g[v+] ⟼ μgv+$ Joel(Kamnitzer)'s favourite case is $G/K$ with $\gamma ={\omega }_{0}$ (the fundamental weight corresponding to the added node on the extended Dynkin diagram) and $μ(MV cycle of type λ∨ and weight μ∨)= (MV polytope of type λ∨ and weight μ∨)$ 

Tropicalization

Let $G=G\left(ℂ\left(\left(t\right)\right)\right)\text{.}$ $ℂ((t))= { aℓtℓ+ aℓ+1tℓ+1+⋯ | ℓ∈ℤ,ai∈ℂ } .$ Points of $G/I$ are $gI,whereg= (gij),gij ∈ℂ((t)).$ The valuation on $ℂ\left(\left(t\right)\right)$ $v(aℓtℓ+aℓ+1tℓ+1+⋯) =ℓ,$ is like log $v(f1f2)= v(f1)+v(f2) andv(f1+f2) =min(v(f1),v(f2)).$ Then $v\left(gI\right)$ is a tropical point of $v\left(G/I\right),$ the tropical flag variety. An amoeba, or tropical subvariety, is the image, under $v,$ of a subvariety of $G/I\text{.}$

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.