Mirkovic-Vilonen cycles and polytopes

Joel Kamnitzer (with J. Anderson)

Remark. All weights mentioned here are dominant.

For μ ∈ P^{∨}, let t_{μ} ∈ Gr = G(K)/G(O). Write Gr^{λ} = G(O)t_{λ}. If L_{1}, L_{2} ∈ Gr, then ⟨L_{1}, L_{2}⟩ = μ ∈ P_{+}^{∨}. If there is a g ∈ G(K) such that gL_{1} = 1 = t_{0} and gL_{2} = t_{μ}, then the distance between L_{1} and L_{2} is μ. Let

Gr_{λμν} = {(L_{1}, L_{2}, L_{3}) : L_{1} = 1, ⟨L_{1}, L_{2}⟩ = λ, ⟨L_{2}, L_{3}⟩ = μ, ⟨L_{3}, L_{1}⟩ = ν}

= all triangles with side lengths λ, μ, ν and special vertices.

• L_{1} • L_{2}  • L_{3} λ μ ν

Theorem (Geometric Satake^{1}). The number of components of Gr_{λμν} of dimension ⟨ρ, λ + μ + ν⟩ is dim(V_{λ} ⊗ V_{μ} ⊗ V_{ν})^{G^{∨}}.

Then

C_{Rep} = {(λ, μ, ν) : Gr_{λμν} has a component of dimension ⟨ρ, λ + μ + ν⟩},
C_{Hecke} = {(λ, μ, ν) : Gr_{λμν} ≠ ∅}.

Now G = GL_{n} = G^{∨}. \underline{Hives}

• •  • •  •  • •  •  •  • 1 2  4 Δ_{n} = {(i, j, k) : i + j + k = n}

Fill with intgers; i.e., put a number at each vertex so that for all rhombi

A C D B

C + D ≥ A + B over ℤ.

1See Haines’ paper

A hive with boundary values λ, μ, ν:

• •  • •  • •  •  •  • λ_{1} λ_{2} μ_{1}  ν_{1}  ⟷^{bijection}  honeycombs_{λ μ  ν}

Theorem (Knutson-Tao, Woodward, J, Berenstein-Zelevinsky). The number of hives with boundary values λ, μ, ν is dim(V_{λ} ⊗ V_{μ} ⊗ V_{ν})^{GL^{n}}.

If (i, j, k) is such that i + j + k = n, consider

∧^{i} ℂ^{n} ⊗ ∧^{j} ℂ^{n} ⊗ ∧^{k} ℂ^{n} ⊃ det .

Let ϵ_{ijk} be a basis vector for det ⊂ ∧^{i} ⊗ ∧^{j} ⊗ ∧^{k} (ϵ_{ijk} is well-defined up to scalar). Write Gr = G(O)\G(K). For (i, j, k) such that i + j + k = n, define H_{ijk} : Gr_{λμν} → ℤ by

H_{ijk}([g_{i}], [g_{j}], [g_{k}]) = val(g_{1} ⊗ g_{2} ⊗ g_{3} · ϵ_{ijk}) ∈ ∧^{i} K^{n} ⊗ ∧^{j} K^{n} ⊗ ∧^{k} K^{n},

where val : K → ℤ is given by t^{i} + ⋯ ↦ i. If {e_{1}, …, e_{n}} is a basis for V and v ∈ V ⊗ K with v = ∑ a_{i}e_{i}, where a_{i} ∈ K, then val(v) = min_{i} val(a_{i}).

Remarks.

1. If g ∈ GL_{V} (O), then val(v) = val(gv) for all v ∈ V, so the function H_{ijk} is independent of choice of representative.

2. V ⊗ K comes with a filtration V ⊗ K ⊃ V ⊗ t^{i}O.

3. If V is a representation of G, then V ⊗ K is a representation of G(K).

H : Gr_{λμν} → {fillings of Δ_{n}} It’s easy to show that the boundary values are always λ, μ, ν.

H_{n,0,0}(L_{1}, L_{2}, L_{3}) •  H_{n,n−1,0}(L_{1}, L_{2}, L_{3}) •  • •  •

L_{1} L_{2}  L_{3} λ  ν μ  ⇝  L_{1} L_{2}  L_{3} only depends on L_{1} and L_{2}

Theorem.

1. The function H is constant on an open part of each component of Gr_{λμν}.

2. For each component, this constant value is a hive.

•

This gives a bijection between the top-dimensional components of Gr_{λμν} and the hives with boundary λ, μ, ν.

\underline{General G}: Fix λ, μ, ν ∈ P_{+}^{∨}. Note that ν^{∨} = −w_{0}ν. Let S^{μ} = t_{μ} · U(K) and T^{ν} = t_{ν} · U_{−}(K). Write kpf for the Kostant partition function. Recall that

Gr_{λμν} = {(L_{1}, L_{2}, L_{3}) : L_{1} = 1, ⟨L_{1}, L_{2}⟩ = λ, ⟨L_{2}, L_{3}⟩ = μ, ⟨L_{3}, L_{1}⟩ = ν}.

Then by using L_{3} ⇝ t_{ν^{∨}},

# (top-dimensional) components of Gr_{λμν}

= # (top-dimensional) components of {(1, L_{2}, t_{ν^{∨}}) : L_{2} ∈ Gr^{λ}, L_{2}^{−1}t_{ν^{∨}} ∈ Gr^{μ}}.

Since L_{2}^{−1}t_{ν^{∨}} ∈ Gr^{μ} if and only if L_{2} ∈ t_{ν^{∨}}Gr^{−μ},

stable weight multiplicities kpf(λ + μ − ν^{∨}) = # (top-dimensional) components of T^{λ+μ−ν^{∨}} ∩ S^{0} = # (top-dimensional) components of T^{λ} ∩ t_{ν^{∨}}S^{−μ}  ≥ ⊃  weight multiplicities dim V_{λ}(ν^{∨} − μ) = # (top-dimensional) components of Gr^{λ} ∩ S^{ν^{∨}−μ} = # (top-dimensional) components of Gr^{λ} ∩ t_{ν^{∨}}S^{−μ}  ≥ ⊃  tensor product multiplicities dim(V_{λ} ⊗ V_{μ} ⊗ V_{ν})^{G^{∨}} = # (top-dimensional) components of_{Gr_{λμν}} = # (top-dimensional) components of Gr^{λ} ∩ t_{ν^{∨}}Gr^{−μ}

Remarks.

1. The containments are in terms of the top-dimensional components.

2. The second containment is obtained by taking the closure: it’s not true that Gr^{−λ} ⊂ S^{−λ}, but \overline{Gr^{−λ}} ⊂ \overline{S^{−λ}}. To see that Gr^{−λ} ⊂ S^{−λ}, but \overline{Gr^{−λ}} ⊂ \overline{S^{−λ}}, consider ℂ^{×} ⊂^{ρ} T ⊂ G. For s ∈ ℂ^{×} and L ∈ Gr, s · L = t_{ν} for some ν ∈ P^{∨} and

lim_{\begin{matrix}s→0 \\ s∈ℂ^{×} \end{matrix}} s · L = retraction of L.

Then

S^{ν} = {L : lim_{s→0} s · L = t_{ν}}

and

\overline{S^{ν}} = \{L : lim_{s→0} s · L = t_{μ} for some μ ≥ ν\} .

On the other hand, if L ∈ \overline{Gr^{−λ}}, then s · L = t_{μ} and

lim_{s→0} s · L ∈ \overline{Gr^{−λ}}.

Thus, μ ≥ −λ.

It follows that if we want to describe the top-dimensional components, we can just describe the top-dimensional components in T^{λ} ∩ t_{ν^{∨}}S^{−μ} and then specialize. Note that V_{λ}(μ) = kpf(λ − μ) when λ − μ ≪ λ, μ. If S ⊂ Gr is a T -invariant closed subset, then the polytope of S, denoted P (S), is

P (S) = convex hull of {μ : t_{μ} ∈ S} ⊆ t_{ℝ} := ℝ ⊗ P^{∨}

= image of the moment map

and

P (\overline{S^{μ}}) = \{x ∈ t_{ℝ} : x ≥ μ\},

where x ≥ μ means that x − μ is a positive linear combination of positive coroots.

Example. A_{2}

• μ \{ν : ν ≥ μ\}  • ν P (\overline{T^{ν}}) = \{x ∈ t_{ℝ} : x ≤ ν\}

• w_{0}λ  • •  • • λ • P (\overline{Gr^{λ}}) = convex hull of W · λ (Weyl polytope or permutahedron)

ν^{∨} − μ • T^{λ} ∩ t_{ν^{∨}}S^{−μ} • λ
\begin{matrix}• \\ ν^{∨} − μ \end{matrix} •  • •  • • λ • Gr^{λ} ∩ t^{ν^{∨}}S^{−μ}

• λ
• ν^{∨} − μ Gr^{λ} ∩ t_{ν^{∨}}Gr^{−μ}

translate of −μ permutahedron Theorem.

\{ \begin{matrix}top-dimensional \\ components of \\ T^{λ} ∩ S^{μ} \end{matrix} \} ⟶ \{ \begin{matrix}polytopes with \\ highest vertex λ, \\ lowest vertex μ \end{matrix} \}
X ⟼ P (X)

This map is injective. Moreover, X ⊂ \overline{Gr^{λ}} if and only if P (X) ⊂ P (\overline{Gr^{λ}}). Remark. The P (X) in the last theorem are MV polytopes. Let

S^{μ} = U(K) · t^{μ},  T^{μ} = U_{−}(K) · t^{μ},  S_{w}^{μ} = (wUw^{−1})(K) · t^{μ}.
\begin{aligned}& • \\ T^{μ} = S_{w_{0}}^{μ} \end{aligned}  _{S_{s_{1}s_{2}}^{μ}  •}  • S^{μ} = S_{1}^{μ}

Then

P (S_{w}^{μ}) = {x ∈ t_{ℝ} : w^{−1}(x − μ) ≥ 0}.

If \underline{μ} : W → P^{∨}, then

A (\underline{μ}) = ⋂_{w∈W} S_{w}^{\underline{μ}(w)}.

Lemma. If A (\underline{μ}) ≠ ∅, then

P (A (\underline{μ})) = \{x ∈ t_{ℝ} : w^{−1}(x − μ) ≥ 0 for all w\} = ⋂_{w∈W} cones.

Remark. The set \{x ∈ t_{ℝ} : w^{−1}(x − μ) ≥ 0 for all w\} is the convex hull of \underline{μ}(w).

• μ(1)  • μ(w_{0}) • μ(s_{1}s_{2}) P (A (\underline{μ}))

For some good choices of \underline{μ}, \overline{A (\underline{μ})} is a top-dimensional component of S_{1}^{μ(1)} ∩ S_{w_{0}}^{μ(w_{0})}, and all the top-dimensional components arise in this way.

A pseudo Weyl-polytope/generalized permutahedron is

P (M.) = {x ∈ t_{ℝ} : ⟨x, γ⟩ ≤ M_{γ} for all γ ∈ Γ}

for some collection {M_{γ}}_{γ∈Γ}, where the M_{γ} ∈ ℤ. The walls of the permutahedron are perpendicular to Γ = ∪W · w_{i}.

w_{2}  w_{1} s_{2} · w_{2}

Then

P (\overline{Gr^{λ}}) = convex hull of W · λ = \{x ∈ t_{ℝ} : ⟨x, w · w_{i}⟩ ≤ ⟨λ, w_{i}⟩\}.

\begin{aligned}& • \\ 1 \end{aligned}

If P (M.) is a permutahedron, then \underline{μ} : W → P^{∨} (the vertices). For each γ ∈ Γ,

D_{γ} : Gr ⟶ ℤ
[g] ⟼ val(g · v_{γ}) = min_{δ} val Δ_{γδ}(g),

where the expression to the right of the equal sign is in Berenstein’s notation. If γ = w · ω_{i}, then v_{γ} ∈ V_{ω_{i}} is of weight γ. If G = SL_{n}, γ = (0, 0, 1, 0, …, 0), and ω_{1} = (1, 0, …, 0) (note that the corresponding representation is the standard representation), then D_{γ}(g) is the minimum valuation of the entries in the third column of g.

Lemma. If \underline{μ} are the vertices of a generalized permutahedron P (M.), then

A (\underline{μ}) = \{L ∈ Gr : D_{γ}(L) = M_{γ}\} =: A(M.).

and P (\overline{A (\underline{μ})}) = P (M.) if it’s non-empty.

Remark. Showing that A (\underline{μ}) = A(M.) is easy.

We use the fact that ⟨\underline{μ}(w), w · ω_{i}⟩ = M_{w·ω_{i}}.

Let G = SL_{3}, and let g ∈ SL_{3}(K). Then “most of the time” (since we are talking in terms of representations)

D_{1}(g) = val Δ_{1,1}(g),  D_{2}(g) = val Δ_{1,2}(g),  Δ_{12}(g) = val Δ_{13,12}(g),

and we have the Plücker relations

Δ_{1,1}Δ_{12,23} + Δ_{3,3}Δ_{12,12} = Δ_{2,2}Δ_{12,13}.

Remark. The Pl ucker relations have been generalized to other root systems by Berenstein-Fomin-Zelevinsky.

We get

val(ab) = val(a) + val(b), val(a + b) = min{val(a), val(b)},

where the equality in the expression involving val(a + b) holds most of the time. A collection {M_{γ}}_{γ∈Γ} is called a BZ datum if they satisfy the tropical Plücker relations, examples of which are

min{D_{1}(L) + D_{23}(L),  } =  ,  min{M_{1} + M_{23}, M_{3} + M_{12}} = M_{2} + M_{13}.

Theorem. If {M_{γ}} is a BZ datum, then A(M.) is an MV cycle, and these are all the MV cycles. Hence, all the MV polytopes have the form P (M.).