Overview of connections between buildings and representation theory, and open problems

Michael Kapovich

Why are we here? The goal of this workshop is to walk along the perimeter of the following triangle:

Euclidean buildings convexity G

Satake isomorphism

MV-cycles, LS-paths

Combinatorial representation theory G^{∨}

Tevelev: amoebas live naturally in buildings Berenstein-Zelevinsky: polytopal model; Knutson-Tao: honeycombs Gleiser-Postnikov …

Tropical geometry

amoebas

It appears that convexity is a common theme in both “building” and “tropical” approaches to the representation theory.

In this talk I will mostly concentrate on the edge of the triangle connecting buildings and representation theory. Connections between representation theory and tropical geometry will be explored in the talks by Berenstein and Knutson. Connections between tropical geometry and buildings are very recent: It will be discussed in the talk of Tevelev.

Geometric questions about Euclidean buildings: We will think of a Euclidean building as the geometric realization of an abstract simplicial complex. This way we have well-defined concepts of straight lines and lengths of geodesics.

\begin{matrix}• & p & • & q & • & r & λ⃗ & μ⃗ & ν⃗ & A \\ • & p & q & λ⃗∈Δ_{+} \end{matrix}

λ⃗, μ⃗, ν⃗ are vectors in Δ_{+} = P_{+}^{∨} ⊗ ℝ_{+} and represent side-lengths of a triangle in the Euclidean building. Definition of Δ_{+}-valued distance between points in the Euclidean building.

•p • q  •0 v⃗  w(v⃗) = λ = length(v⃗)^{w∈W}  P_{+}^{∨} ⊗ ℝ_{+} = Δ_{+} d⃗(p, q) = λ⃗ (move p⃗q to origin and rotate)

Question. What are the triangle inequalities in this geometry? I.e., what are the restrictions on the side-length vectors of a triangle in the Euclidean building. Note that, a priori, it’s not clear that these restrictions are given by inequalities.

Example. In the case of SL_{2}(ℚ_{p}), the building is a regular tree.

1 •^1

• Δ_+ = ℝ_+

triangle inequalities: the usual triangle inequality It turns out that there is a system of linear homogeneous inequalities with integer coefficients (triangle inequalities) TI on (λ, μ, ν) ∈ Δ×Δ×Δ that are necessary and sufficient conditions for there to be a triangle with side-lengths λ, μ, ν that answer the above question.

Key tool for the proof: Analyzing certain convex functions on Euclidean and spherical buildings.

0  C ⊂ Δ^{3} solution set of TI generalized Belkale-“Klyachko’s” inequalities (A_{n} case) same as Berenstein-Sjamaar inequalities

Question. Irredundancy of the triangle inequalities?

In A_{n} case: Knutson-Tao-Woodward proved irredundancy of TI

In B_{n}, C_{n} cases: C_{B_{n}} = C_{C_{n}} (cones—solution sets—are the same) In particular, C depends only on W, but TI_{B_{n}} ≠ TI_{C_{n}} (inequalities redundant) TI_{B_{n}} are Belkale-Klyachko/Berenstein-Sjamaar inequalities However there is a smaller system of inequalities TI^{BK}—Belkale-Kumar inequalities—smaller, which also describe the cone C.

Conjecture (Belkale–Kumar). TI^{BK} are irredundant.

The inequalities TI come from Schubert calculus over G(ℂ)/P (ℂ), where P (ℂ) are maximal parabolic subgroups. The inequalities TI have the following form:

⟨λ, w_{λ}(ϖ_{i})⟩ + ⟨μ, w_{μ}(ϖ_{i})⟩ + ⟨ν, w_{ν}(ϖ_{i})⟩ ≤ 0.

Here ϖ_{i}, i = 1, …, ℓ are the fundamental weights and w_{λ}, w_{μ}, w_{ν} ∈ W are triples such that

[C_{w_{λ}}] · [C_{w_{μ}}] · [C_{w_{ν}}] = [point],

where C_{w} is the Schubert cycle in the Grassmanian Gr_{ϖ_{i}} = G(ℂ)/P_{ϖ_{i}}(ℂ) corresponding to the coset wW_{ϖ_{i}} ∈ W/W_{ϖ_{i}}. The product is the intersection product in the integer homology ring H_{∗}(Gr_{ϖ_{i}}, ℤ). Belkale-Kumar inequalities have the same form except that they consider a deformed product in H_{∗}(Gr_{ϖ_{i}}, ℤ).

Connection to Hecke Algebra: We now impose certain integrality conditions on the triangles in Euclidean buildings.

1. Consider λ, μ, ν ∈ P_{+}^{∨}.

2. Moreover, consider only triangles in the Euclidean building whose vertices are special

vertices of the building.

A_{n} case: all vertices special

⋯

point is \underline{special} if its stabilizer in W̃ is isomorphic to W

• special  non-special B_{2} = C_{2}

Problem (Restricted triangle/Hecke problem). What are the restrictions on the side-lengths λ, μ, ν ∈ P_{+}^{∨} so that there exists a triangle in the Euclidean building with special vertices and side-lengths λ, μ, ν?

C ⋯  · · · · ·  C_{Hecke} = solution set of Hecke problem

Obvious necessary condition: λ + μ + ν ∈ Q^{∨} ⊂ P^{∨} (congruence condition) Reason for the obvious necessary condition

• • _{1}_{1} ^{1} • ^{1}  \begin{matrix}λ = μ = ν = 1 \\ 1 + 1 + 1 = 3 ∉ 2ℤ = Q^{∨} \end{matrix}

It turns out that (λ, μ, ν) ∈ C_{Hecke} iff

f_{λ} ∗ f_{μ} ∗ f_{ν} = ∑ c_{λμν}(δ) · f_{δ}, (where c_{λμν}(0) ≠ 0 with 0 the trivial character).

Here f_{λ}, f_{μ}, f_{ν} are K-bi-invariant functions on G_{ℚ_{p}} and ∗ denotes the convolution product. Moreover, c_{λμν}(0) = #[triangles (appropriately normalized)]/G.

Problem (Representation Theory problem Rep for G^{∨}). Find necessary and sufficient conditions on λ, μ, ν such that

(V (λ) ⊗ V (μ) ⊗ V (ν))^{G^{∨}} ≠ {0},

where V (γ) is the irreducible representation of G^{∨}(ℂ) with the highest weights γ.

Let C_{Rep} denote the solution set of this problem. It turns out that (Kapovich, Leeb, Millson):

C_{Rep} ⊂ C_{Hecke} ⊂ C

Remark. C_{Rep} and C_{Hecke} are both discrete, but C is continuous.

For the A_{n} root system (SL_{n+1} case), the sets are “the same”:

C_{Hecke}(A_{n}) = C_{Rep}(A_{n}) = C ∩ (P^{∨})^{3} ∩ {λ + μ + ν ∈ Q^{∨}}

Requiring that (λ, μ, ν) belongs to the lattice

L = (P^{∨})^{3} ∩ {λ + μ + ν ∈ Q^{∨}}

is called the integrality condition. The first equality above was proved in the 1960s by Hall and Klein, and later reproved by Kapovich-Millson (buildings), Haines (algebraic geometry). The second equality was proven by Knutson-Tao (through “tropical geometry”), Derksen-Wyman (algebraic geometry), Kapovich and Millson (buildings) and Belkale (algebraic geometry). The equality C_{Rep} = C ∩ (P^{∨})^{3} is equivalent to the saturation conjecture.

For other root systems, have (sometimes) strict inequalities: For all non-simply laced root systems we have:

C_{Rep} ⫋ C_{Hecke} ⫋ C ∩ integrality condition

Remark. While C_{Rep} is a semigroup, C_{Hecke} is not a semigroup in general (B_{2}, G_{2}).

Conjecture (Generalized saturation conjecture). Equalities hold for all simply-laced root systems. Equivalently, saturation holds for all simply-laced root systems.

Example. Below is the picture of how C_{Rep} looks like for B_{2}:

•  •  •  • •  •  • • • •  • • • •  • • • •  C singularλ, μ, ∉ C_{Rep}  (λ, μ, ν) ∈ C_{Rep} B_{2}

For B_{2}, the semigroups C ∩ integrality conditions and C_{Rep} differ only on the faces of the cone C, where two of the three dominant weights are multiples of ϖ_{2}. On these faces, (λ, μ, ν) ∈ C_{Rep} if and only if λ + μ + ν ∈ 2P^{∨}.

Example. Root system G_{2}. Pick a ray ρ = ℝ_{+} · δ in the cone C, where δ is a primitive vector in the lattice

L := {(λ, μ, ν) ∈ P^{∨} : λ + μ + ν ∈ Q^{∨}}.

Then either

ρ ∩ L = C_{Rep} ∩ L

or

ρ ∩ L = {δ} ∪ C_{Rep} ∩ L.

Partial evidence for the generalized saturation conjecture: for A_{n}, D_{4},

C_{Rep} = C_{Hecke} = C ∩ integrality conditions

D_{4} was done by a massive computer computation. Plus computer experiments for other simply-laced root systems.

Conjecture (in the non-simply laced case (Knutson-Tao)). Suppose (λ, μ, ν) ∈ C is so that (λ + μ + ν)(t) = 1 for all t ∈ T such that Z_{G(ℂ)}(t) is semisimple. Here T is the maximal torus of G(ℂ) and Z(t) denotes the centralizer of t. Then

(λ, μ, ν) ∈ C_{Rep}.

Conjecture (Kapovich, Millson). For every root system, the intersection of C_{Rep} with the interior of the cone C is saturated. Moreover, if λ, μ, ν are regular, then

(λ, μ, ν) ∈ C_{Rep} ⟺ (λ, μ, ν) ∈ C ∩ L;

i.e.,

C_{Rep} ∩ (regular) = C ∩ (regular).

Theorem (Kapovich, Leeb, Millson).

k_{R} · C_{Hecke} ⊂ C_{Rep},
k_{R} · C ∩ L ⊂ C_{Hecke},

where k_{R} = lcm(a_{1}, …, a_{ℓ}) when the highest root for G^{∨} has the form θ = ∑_{i=1}^{ℓ} a_{i}α_{i}.