LS Galleries and MV Cycles

Stephane Gaussent (with Peter Littelmann)

G group  G^{∨}
SL_{3}  PSL_{3}
SO_{5}  Sp_{4}

Fix A, P (the character lattice), P^{∨} (the co-character lattice), (α_{i}) (the simple roots), (α_{i}^{∨}) (the co-roots)

Path model for G^{∨} Littelmann, 1994 p : [0, 1] → P^{∨} ⊗ ℝ

λ ∈ (P^{∨})^{+} model of V (λ) irreducible representation

of highest weight λ

Mirković-Vilonen, 1996 Lusztig-Ginzburg: IH^{∗}(Gr\overline{λ}) ≅ V (λ) IH^{∗}(Gr\overline{λ}) intersection cohomology λ ∈ (P^{∨})^{+}

⨁_{λ∈P^{∨}} # Irred(Z_{μ}^{λ})

Bruhat-Tits building

⊂

⊃

Remarks.

1. Mirković-Vilonen found cycles in IH^{∗}(Gr\overline{λ}) that give a canonical basis for V (λ).

2. The first inclusion is combinatorial; the second inclusion is via geometry (inside the building).

To compare the two models, we need to replace the path model by the gallery model.

Combinatorics Let λ ∈ (P^{∨})^{+} be regular. (λ is a dominant co-weight, regular)

C_{f}  0 λ

The apartment is 𝔸 = P^{∨} ⊗ ℝ.

The affine Weyl group is W^{a} = W ⋉ Q^{∨}, where Q^{∨} is the coroot lattice. An apartment is a Coxeter complex. labelling of it Fix a minimal alcove gallery γ_{λ} from 0 to λ.

A_{2}

• α_{2}^{∨} • α_{1}^{∨}

1 2 0

1

0

2 0 2

0

2 0 1 1 0 0 1 2 0 1 2 1 2 1 0

2 0

[s_{1}s_{2}, s_{0}, 1, s_{1}]

•

typ(γ_{1}) = (0, 2, 1)

λ

γ_{λ}

If γ_{λ} ↔ w_{λ} ∈ W^{a} has minimal decomposition w_{λ} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{r}}, then typ(γ_{λ}) = (i_{1}, i_{2}, …, i_{r}) (this depends on the walls the minimal alcove gallery crosses). Let Γ(γ_{λ}) be the set of all galleries in 𝔸 starting at 0 and having type typ(γ_{λ}). Then we have a bijection

W × {1, s_{i_{1}}} × ⋯ × {1, s_{i_{r}}} ⟶^{≃} Γ(γ_{λ})
[δ_{0}, δ_{1}, …, δ_{r}] ⟼ (0 ⊂ \overline{Δ}_{0} ⊃ Δ′_{1} ⊂ \overline{Δ}_{1} ⊃ ⋯ Δ′_{r} ⊂ \overline{Δ}_{r} ⊃ μ) = δ,

where \overline{Δ}_{i} denotes an alcove (more precisely, \overline{Δ}_{i} = δ_{0}δ_{1} ⋯ δ_{i}(Δ_{f}), where Δ_{f} is the fundamental alcove), Δ′_{j} is a face of codimension 1. Note that type(μ) = type(λ) and π(δ) = μ (end of the gallery).

Root operators Let α be a simple root, f_{α}, e_{α}.

• Δ_{j} = Δ_{j−1} k_{min}  μ δ j_{max} μ − α^{∨} f_{α}(δ)  α^{∨}

Let H_{α,m} with m minimum such that there exists j so that Δ′_{j} is in H_{α,m}. Let j_{max} be the maximum of the indices j such that Δ′_{j} is in H_{α,m}. Let k_{min} be the minimum of the indices k such that Δ′_{k} is in H_{α,m+1}. Then f_{α}(δ) is obtained from δ by reflecting the part of δ between j_{max} and k_{min} and translating the part after k_{min}. Define e_{α} in the same way. Then

π(f_{α}δ) = μ − α^{∨},  π(e_{α}δ) = μ + α^{∨},  f_{α}(e_{α}δ) = e_{α}(f_{α}δ) = δ.

In the following example, λ = α_{1}^{∨} + α_{2}^{∨} and we draw all the galleries in Γ(γ_{λ}).

A_{2}

λ

γ_{λ}

α_{2}^{∨} α_{1}^{∨}
δ = f_{α_{2}}(γ_{λ})

f_{α_{1}}^{2}f_{α_{2}}(γ_{λ})
f_{α_{2}}f_{α_{1}}^{2}f_{α_{2}}(γ_{λ})

folded gallery

γ

f_{α_{1}}f_{α_{2}}(γ_{λ})

Remarks. See the definition of dim(γ) later.

1. dim(γ) = 1 2. dim(δ) = 3

3. dim(f_{α_{1}}f_{α_{2}}(γ_{λ})) = 2 The operators preserve Γ(γ_{λ}), f_{α}, e_{α} : Γ(γ_{λ}) → Γ(γ_{λ}). Let Γ^{+}(γ_{λ}) = {γ ∈ Γ(γ_{λ}) positively folded}. A gallery δ is positively folded if δ_{j} = 1 ⇒ \overline{Δ_{j−1}} = \overline{Δ_{j}} is on the positive side of the wall at j.

A_{2}

λ
α_{2}^{∨} α_{1}^{∨}

not positively folded galleries positively folded galleries

− +
\overline{Δ}_{j−1} = \overline{Δ}_{j} in δ ⇝ δ = [δ_{0}, δ_{1}, …, δ_{j−1}, 1, δ_{j+1}, …, δ_{r}]

The operators preserve the set Γ^{+}(γ_{λ}). Define LS galleries Let δ ∈ Γ(γ_{λ}).

Then dim(δ) = #{H_{α,m} left positively by δ}, where H_{α,m} denotes an affine hyperplane. H_{α,m}

− +

not counted in dim(δ)

H_{α,m}

− +

counted in dim(δ) H_{α,m}

Δ_{j−1} = Δ_{j}
− +

counted in dim(δ) Peculiar rule for the starting alcove: one looks at several walls going through 0 Proposition. If δ ∈ Γ^{+}(γ_{λ}) and π(δ) = μ, then dim(δ) ≤ ρ(λ + μ), where ρ = (∑_{α>0} α)/2 is half the sum of the positive roots.

Definition. A gallery δ ∈ Γ^{+}(γ_{λ}) is LS if dim(δ) = ρ(λ + μ).

The operators preserve Γ_{LS}^{+}(γ_{λ}).

Theorem.

1. character formula:

CharV (λ) = ∑_{ν} ♯Γ_{LS}^{+}(γ_{λ}, ν)e^{ν},

where Γ_{LS}^{+}(γ_{λ}, ν) is the set of all LS galleries ending in ν.

2. The graph built on Γ_{LS}^{+}(γ_{λ}) with the root operators is isomorphic to the crystal graph of V (λ) of G^{∨}.

Remark. If λ is singular, there is still a well-defined alcove containing λ in C_{f} (the fundamental chamber) realizing the minimum distance to Δ_{f} and one can adapt the preceeding to get the same result (with P. Baumann).

\underline{Geometry} Let G ⊃ B ⊃ T (be a borus as Jantzen is saying). Then B ⊃ U ⊃ T, U^{−} ⊂ B^{−}, and B^{−} ∩ B = T. We have ℬ^{sph} (spherical building), ℬ = ℬ(G, K) (affine building), and ℬ^{∞} = ℬ(G(K), B(K)) (spherical building at infinity). Note that ℬ(G, K) = G(K) × 𝔸/ ∼ for some equivalence relation ∼, and

ℬ(G, K) ≅ {parahoric subgroups P ⊃ I = ev_{0}^{−1}(B_{ℂ})}.

• • • • • Affine building of type A_{1}

Affine Grassmanian

Gr = G(K)/G(O) = ∐_{λ∈(P^{∨})^{+}} G(O)t_{λ}G(O)/G(O) = ∐_{μ∈P^{∨}} U^{∓}(K)t_{μ}G(O)/G(O)

Mirković-Vilonen cycles: The intersections U^{−}(K) · t_{μ} ∩ G(O) · t_{λ} are of pure dimension ρ(λ + μ) (Mirković-Vilonen).

An MV cycle Z_{μ}^{λ} is the closure of an irreducible component in U^{−}(K) · t_{μ} ∩ G(O) · t_{λ} in \overline{Gr^{λ}} = \overline{G(O) · t_{λ}}.

Suppose λ is regular (again, one can manage the singular case as well), and let ˆ∑(γ_{λ}) be the set of all galleries in the affine building ℬ starting at 0 of type typ(γ_{λ}). Then

ˆ∑(γ_{λ})  ^{π} \overline{Gr^{λ}} = \overline{G(O) · t_{λ}} G(O) · γ_{λ}  _{≃} G(O) · t_{λ}

Here the upper arrow is a Bott-Samelson resolution of singularities.

Remarks.

1. G(O) · γ_{λ} is the set of all minimal galleries.

2. ˆ∑(γ_{λ}) = G(O) ×_{I} P_{i_{1}} ×_{I} ⋯ ×_{I} P_{i_{r}}/I, where P_{i_{1}} = I{1, s_{i_{1}}}I.

3. π([g_{0}, g_{1}, …, g_{r}]) = g_{0}g_{1} ⋯ g_{r}t_{λ_{fund}}, where λ_{fund} ∈ Δ_{f} such that typ(λ_{fund}) = typ(λ)

Theorem.

U^{−}(K) · t_{μ} ∩ G(O) · t_{λ} ≅ ∐_{\begin{matrix}δ∈Γ^{+}(γ_{λ}), \\ π(δ)=μ \end{matrix}} C(δ) ∩ G(O) · γ_{λ},

where C(δ) is a cell of dimension dim(δ) and therefore, each LS gallery gives an open of a unique MV cycle.