UCLA Colloquium

Saturation problems via affine Grassmannians

and triangles in buildings

Thomas Haines University of Maryland

April 24, 2003 On www.arxiv.org:

[KLM] M. Kapovich, B. Leeb, J. Millson, The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, math.RT/0210256.

[H] T. Haines, Structure constants for Hecke and representation rings, math.RT/0304176.

Related Preprint (not yet on server):

M. Kapovich, B. Leeb, J. Millson, Polygons in symmetric spaces and buildings.

Tensor products of representations G - complex linear algebraic group, T a maximal torus. X^{∗}(T) = weights. Theory of highest weight: Any dominant weight λ ↦ V_{λ} an irreducible representation of G, with ”highest weight” λ. Example: GL_{n}(ℂ) ⊃ diag. torus ; dominant weights ↔ non-increasing n-tuples (a_{1}, a_{2}, …, a_{n}) ∈ ℤ^{n}. (1^{r}, 0^{n−r}) ↔ Λ^{r}(ℂ^{n}), and (k, 0^{n−1}) ↔ Sym ^{k}(ℂ^{n}). Question: How to describe

R_{G}^{3} := {(α, β, γ) ∈ X^{∗}(T)_{dom}^{3} | (V_{α} ⊗ V_{β} ⊗ V_{γ})^{G} ≠ 0}
= {⋯ | triv. rep. 𝕀 ∈ V_{α} ⊗ V_{β} ⊗ V_{γ}}
= {⋯ | V_{γ^{∗}} ∈ V_{α} ⊗ V_{β}}.

Here γ^{∗} is the dominant weight indexing the (irred.) dual of V_{γ}.

∃ Many Methods to answer question (or more precisely, compute the multiplicity n_{α,β}^{λ}, the number of times V_{γ} occurs in V_{α} ⊗ V_{β}):

– (GL_{n}) Littlewood-Richardson rule

– Berenstein-Zelevinsky polytopes (→ honycomb model for GL_{n} (Knutson-Tao))

– Littelmann path model

– Kashiwara’s crystal bases/graphs Also: New algorithm due to [KLM], in terms of multiplication in Hecke rings R_{GL_{n}}^{3} is the set of integral points in the following (identical) polyhedral cones in (ℝ^{n})^{3}:

Eigenvalues of a sum The set of triples (α, β, γ) of dominant vectors in ℝ^{n} (entries in non-increasing order) such that there exist Hermitian matrices A, B, C such that the set of eigenvalues of A resp. B resp. C is α, resp. β, resp. γ, and

A + B + C = 0.

Generalized triangle inequalities The triples as above which satisfy a system of inequalities defined using Schubert calculus:

Let Δ ⊂ ℝ^{n} ≅ X^{∗}(T) ⊗ ℝ denote the cone of dominant (real) weights. Fundamental coweight (1^{i}, 0^{n−i}) = λ ∈ X_{∗}(T) = a linear functional on Δ.

λ ↦ parabolic P_{λ} ↦ Schubert variety GL_{n}/P_{λ}.

GTI’s: for each λ and each triple w_{1}, w_{2}, w_{3} ∈ W λ ⊂ X_{∗}(T), impose the inequality

w_{1}(α) + w_{2}(β) + w_{3}(γ) ≤ 0

whenever

[X_{w_{1}}] · [X_{w_{2}}] · [X_{w_{3}}] = [pt] in H_{∗}(G/P) .

GTI’s for general G defined similarly. In that context get

Theorem 1 (Leeb-Millson) (α, β, γ) satisfy GTI’s if and only if there is a triangle in the symmetric space G(ℂ)/K having ”geodesic side lengths” α, β, γ. (Here G is such that G = Ĝ, and K = a maximal cpt. subgroup.)

Rank 1 symm. space = disc, Δ = ℝ_{>0} = usual notion of ”side length”. [Picture].

In general, by Cartan decomposition G(ℂ) = Kexp (Δ)K: directed geodesic \overline{x_{1}x_{2}} ∈ K·\overline{0exp (δ)} , for unique δ ∈ Δ.

General features of R_{G}^{3}

• R_{G}^{3} is a semi-group: (α_{1}, β_{1}, γ_{1}), (α_{2}, β_{2}, γ_{2}) ∈ R_{G}^{3} ⇒ (α_{1} + α_{2}, β_{1} + β_{2}, γ_{1} + γ_{2}) ∈ R_{G}^{3}.

• (Knutson-Tao) If G = GL_{n}, then R_{G}^{3} is saturated: For N > 1,

(Nα, Nβ, Nγ) ∈ R_{G}^{3} ⇒ (α, β, γ) ∈ R_{G}^{3}.

(Used in proof of above description of R_{GL_{n}}^{3} in terms of GTI’s.)

For general G, R_{G}^{3} is usually NOT saturated.

Questions: When exactly does the saturation property hold for other groups? Is there a ”uniform” geometric approach to this?

Saturation Problems for Hecke algebras

New notation: G - linear alg. group over finite field 𝔽_{q}, T max. torus. Old G, T are now the Langlands dual (over ℂ) Ĝ, T̂ of the new G, T. X^{∗}(T) = X_{∗}(T̂), X_{∗}(T) = X^{∗}(T̂).

G(𝔽_{q}((t))) := ”loop group LG of G” ⊃ max’l compact K := G(𝔽_{q}[[t]]). Then have Hecke algebra H = C_{c}(K\G(𝔽_{q}((t)))/K) with convolution ∗.

α ∈ X_{∗}(T)_{dom} provides a basis element for H:

f_{α} := char (Kα(t)K).

[By Cartan decomposition: K\LG/K ↔ X_{∗}(T)_{dom} . (GL_{n}:”elem. divisors”).]

Definition f_{α} ∗ f_{β} ∗ f_{γ} = ∑_{λ} c_{α,β,γ}^{λ}f_{λ}.

Question: Describe

H_{G}^{3} := {(α, β, γ) ∈ X_{∗}(T)_{dom}^{3} | c_{α,β,γ}^{0} ≠ 0}.

Hecke saturation

Theorem 2 (KLM) For each group G, ∃ k = k(G) ∈ ℕ such that for N > 1, (Nα, Nβ, Nγ) ∈ H_{G}^{3} ⇒ (kα, kβ, kγ) ∈ H_{G}^{3}.

Corollary 1 For H_{GL_{n}}^{3} is saturated (k(GL_{n}) = 1).

Idea of proof: The GTI’s also characterize side lengths of triangles in Bruhat-Tits building for LG. The GTI’s are homogeneous, hence solution set in Δ^{3} saturated (in strong sense). But want triangles in building with special points as vertices. This is where k(G) comes in. [Picture]

How the sets are related

New notation: R_{G}^{3} written instead of

R3 ; Ĝ = {(α, β, γ) ∈ (X^{∗}(T̂)_{dom} )^{3} | (V_{α}×V_{β}⊗V_{γ})Ĝ ≠ 0}.

Combinatorial methods give: Theorem 3 (KLM) For general G, R_{G}^{3} ⊂ H_{G}^{3}. Theorem 4 (P. Hall) H_{GL_{n}}^{3} ⊂ R_{GL_{n}}^{3}. Corollary 2 R_{GL_{n}}^{3} is saturated (!).

[KLM] result uses Lusztig’s q-analogue of weightmultiplicity formula (tricky, but valid for every group.) [P. Hall] result uses combinatorics of Hall algebras – special to GL_{n}. In general H_{G}^{3} is strictly bigger than R_{G}^{3} (e.g. G_{2}, SO (5)).

Affine Grassmannians

Let’s define the Affine Grassmannian (for GL_{n}). Let k = 𝔽̄_{q}, F = k((t)), O = k[[t]].

Define

Q(k) = {O−lattices in F^{n} } = G(k((t)))/G(k[[t]]),

an ind-scheme defined over 𝔽_{q}.

Finite-dimensional pieces: inv gives ”distance” between two lattices:

inv(L, L′) := λ ∈ X_{∗}(T)_{dom} , if L = gO^{n}, L′ = g′O^{n}, and g^{−1}g′ ∈ K(diag (t^{λ_{1}}, …, t^{λ_{n}}))K.

Affine Schubert variety: Q̄_{μ} = {L ∈ Q | inv(O^{n}, L) ≤ μ}. (≤ the standard partial order on dominant coweights...) These are fin.-dim’l projective varieties, over 𝔽_{q}.

• K-orbits are the Schubert cells Q_{μ}

• Q = ∐_{μ} Q_{μ}, and the boundary of Q̄_{μ} is the union of the Q_{λ}’s, where λ < μ.

Can do all this for general G (G-bundles on curves, with trivialization outside a fixed point).

Consider P_{K}(Q), the K-equivariant perverse sheaves on Q. (Geometric analogue of Hecke ring).

P_{K}(Q) is a semi-simple, abelian category. Simple objects: the intersection complexes IC(Q̄_{μ}).

(IC complexes compute intersection cohomology...)

There is a convolution ∗ : P_{K}(Q) × P_{K}(Q) → P_{K}(Q), making P_{K}(Q) a Tannakian category.

By Drinfeld, Ginzburg, Mirkovic-Vilonen, Ngô-Polo... , we have the famous geometric version of the usual Satake isomorphism from the Hecke algebra of LG to the rep. ring of Ĝ.

Theorem 5 (Geometric Satake Isomorphism) There is an isomorphism (Rep(Ĝ), ⊗) ≅ (P_{K}(Q ), ∗), such that V_{μ} corresponds to IC(Q̄_{μ}).

Corollary 3 in P_{K}(Q),

IC_{μ_{1}} ∗ ⋯ ∗ IC_{μ_{r}} = _{λ}^{∑} dim (V_{μ_{•}}^{λ}) IC_{λ}.

Geometric reformulation

λ ∈ X_{∗}(T)_{dom} , μ_{•} = (μ_{1}, …, μ_{r}) ∈ X_{∗}(T)_{dom}^{r} . Write |μ_{•}| := ∑_{i} μ_{i}. Rep(μ_{•}, λ): V_{μ_{•}}^{λ} ≠ 0, where V_{μ_{1}} ⊗ ⋯ ⊗ V_{μ_{r}} = ⊕_{λ≤|μ_{•}|}V_{μ_{•}}^{λ} ⊗ V_{λ}. Hecke(μ_{•}, λ): c_{μ_{•}}^{λ} ≠ 0, where f_{μ_{1}} ∗ ⋯ ∗ f_{μ_{r}} = ∑_{λ≤|μ_{•}|} c_{μ_{•}}^{λ}f_{λ}, where f_{μ} = char (Kt_{μ}K). Define the twisted product Q̃_{μ_{•}} to be {L_{•} = (L_{1}, …, L_{r}) ∈ Q^{r} | inv(L_{i−1}, L_{i}) ≤ μ_{i}, ∀i}. Definition

m_{μ_{•}} : Q̃_{μ_{•}} → Q̄_{|μ_{•}|}

given by L_{•} ↦ L_{r}.

This is the geometric analogue of convolution in Hecke algebra. (Used to define convolution in category P_{K}(Q) due to Drinfeld, V. Ginzburg, and studied in geometric Langlands program...)

Key fact used in this definition:

Theorem 6 (Mirkovic-Vilonen, Ngô-Polo) The birational morphism m_{μ_{•}} is locally trivial and semi-small, in the sense of Goresky-MacPherson.

The semi-smallness means that the fibers are not too large: if y ∈ Q_{λ} ⊂ Q̄_{|μ_{•}|}, then

dim (m_{μ_{•}}^{−1}(y) ∩ Q_{μ′}) ≤ \frac{1}{2}(dim (Q_{μ′_{•}}) − dim (Q_{λ}));

where RHS is also ⟨ρ, |μ′_{•}| − λ⟩, where ρ = halfsum of positive roots.

Can now reformulate Rep(μ_{•}, λ) and Hecke(μ_{•}, λ) in terms of fibers of m_{μ_{•}}:

Theorem 7 (H) 1) V_{μ_{•}}^{λ} ≠ 0 iff dim (m_{μ_{•}}^{−1}(y)) = ⟨ρ, |μ_{•}|−λ⟩. Any irreducible component of such max’l dimension meets the open stratum Q_{μ_{•}}.

2) If y is 𝔽_{q}-rational, then c_{μ_{•}}^{λ} ≠ 0 iff m_{μ_{•}}^{−1}(y) ∩ Q_{μ_{•}} ≠ ∅.

Get: geometric proof of Theorem 3. To prove above result, use

Rm_{μ_{•},∗}(IC(Q̃_{μ_{•}})) = IC_{μ_{1}} ∗ ⋯ ∗ IC_{μ_{r}}

and basic properties of IC, to show that V_{μ_{•}}^{λ} has a basis in canonical correspondence with the irreducible components of max’l possible dimension in the fiber m_{μ_{•}}^{−1}(y).

Also get

• proof of Theorem 4: need

Lemma 1 (H) For GL_{n}, if μ_{i} all minuscule, then all fibers of m_{μ_{•}} are equidimensional. [use Spaltenstein-Springer varieties]

• For general G, Weil conjectures give [KLM] formula:

c_{μ_{•}}^{λ}(q) = dim (V_{μ_{•}}^{λ})q^{⟨ρ,|μ_{•}|−λ⟩}+{lower deg terms }.

This gives a new (not very fast) algorithm to compute dim (V_{μ_{•}}^{λ})!

Generalizations?

Above methods work well for arbitrary groups provided λ, μ_{i} are sums of minuscule coweights. I have ”almost proved” the following natural Conjecture 1 Suppose λ, μ_{i} are sums of minuscule coweights. Then for N > 1,

dim (V_{Nμ_{•}}^{Nλ}) ≠ 0 ⇒ dim (V_{μ_{•}}^{λ}) ≠ 0. Groups with minuscule coweights: PGL (n+1) (n); SO (2n + 1) (1); GSp (2n) (1); SO (2n) (3); E_{6} (2); E_{7} (1). THE END.