Complexity of nilpotent

orbits and the Kostant-Sekiguchi

correspondence Math.RT/0303096

Donald R. King Mathematics Department Northeastern University

Boston, MA 02115

NORTH 6, 2004, Fuji-Zakura So, Japan

Outline of Talk

1. Notation, basic facts and definitions

2. Statement of Main Theorem

3. K-structure of nilpotent orbits (Schmid and Vilonen)

4. Proof of Main Theorem

5. Commutativity of a related Poisson algebra 6. An Application to the G-saturation of the nilradical of a parabolic subalgebra

1. Notation

𝔤 real simple Lie algebra, Cartan involution θ. Cartan decomposition: 𝔤 = 𝔨 ⊕ 𝔭. 𝔤_{C}, 𝔨_{C} and 𝔭_{C}. σ is conjugation of 𝔤_{C} w.r.t. 𝔤. G_{C} is the adjoint group of 𝔤_{C}. G, K, K_{C} conn. subgps corr. to 𝔤, 𝔨, 𝔨_{C}. Ω a nilpotent G-orbit in 𝔤. O a nilpotent K_{C}-orbit in 𝔭_{C}.

Kostant-Sekiguchi correspondence Definition. (orbit “cores”) C(Ω) := {E ∈ Ω|∃ sl(2)-triple {H, E, F } ⊂ 𝔤, θ(E) = −F }.

C(O) := {e ∈ O|∃ normal sl(2)-triple {x, e, f} ⊂ 𝔤_{C}, σ(e) = f}.

C(Ω) (resp., C(O)) is a single K orbit.

The correspondence:

Ω ⊃ C(Ω) ∋ E ↦ K_{C} · \frac{H − i(E + F)}{2} := O_{Ω}

(Ω, O_{Ω} ) is said to be a KS-pair.

Properties of Kostant-Sekiguchi pairs Let (Ω, O) be a KS-pair.

(1) dim_{R} Ω = dim_{R} O.

(2) Ω and O lie in the same G_{C} orbit, O_{C}.

(3) O is a Lagrangian submanifold of O_{C}.

(4) (Vergne) ∃ K-equivariant real analytic isomorphism, V : Ω → O. (5)(Barbasch and Sepanski)

Ω_{1} ⊂ \overline{Ω_{2}} ⟺ O_{1} ⊂ \overline{O}_{2} How does the correspondence affect other invariants of Ω and O that are significant in representation theory?

The Big Picture

V

Ω ⟶ O
symplectic manifold  quasi-affine variety
Hamiltonian K-space  algebraic K_{C}-action

Invariants ↙ ↘

K-rank  K_{C}-rank
K-corank  K_{C}-complexity

Notions of rank

rank of K-action on Ω , r_{K}(Ω):=rank K - rank K^{E′} , where E′ ∈ Ω and the orbit K · E′ is generic.

R[O] denotes the ring of regular functions on O. If λ ∈ K̂, R[O]_{λ} is the λ-isotypic component. m_{O}(λ) is the multiplicity of λ in R[O].

Γ(O) := {λ ∈ K̂|R[O]_{λ} ≠ (0)} is a finitely generated semigroup in the set of dominant weights of K_{C}.

The rank of the K_{C} action on O is the rank of Γ(O). It is denoted r_{K_{C}}(O).

The corank of the K action on Ω

Let w_{Ω} be the canonical symp. form on Ω and Φ : Ω → 𝔨^{∗} denote the moment map.

Lemma. Let E′ ∈ Ω. Set W = T_{E′}(K · E′) and W^{⊥} equal to the orthogonal complement (with respect to w_{Ω} |_{E′}) of W inside T_{E′}(Ω). Then T_{E′}(Ω) is a sum of symplectic vector spaces:

\frac{W}{W ∩ W^{⊥}} ⨁ ((W ∩ W^{⊥}) ⊕ (W ∩ W^{⊥})^{∗}) ⨁ \frac{W^{⊥}}{W ∩ W^{⊥}}.

where,

(1) 𝔨^{Φ(E′)} /𝔨^{E′} ≃ W ∩ W^{⊥} (as 𝔨^{E′} modules) and

(2) 𝔨/𝔨^{Φ(E′)} ≃ \frac{W}{W ∩W^{⊥}} (as 𝔨^{E′} modules).

Proposition. (Vinberg) ∃ open dense subset U ⊂ Ω such that for all E′ ∈ U, the non-negative integer

dim \frac{T_{E′}(K · E′)^{⊥}}{T_{E′}(K · E′) ∩ T_{E′}(K · E′)^{⊥}},

has a constant value. We denote this value by 2ϵ̃_{K}(Ω). The integer 2ϵ̃_{K}(Ω) is said to be the corank of the action of K on Ω.

Remark. If ϵ̃_{K}(Ω) = 0, Ω is said to be multiplicity free as a Hamiltonian K-space (that is, T_{E′}(K · E′) is coisotropic in T_{E′}(Ω) for a dense open subset of elements of Ω).

Multiplicity-free Hamiltonian K-spaces

Let X be a Hamiltonian K-space; C^{∞}(X)^{K} be the Poisson algebra of K-invariant smooth functions on X; and 𝔄(X)_{ra}^{K} be the Poisson algebra of K-invariant real analytic functions on X.

Guillemin and Sternberg have shown:

X is mult. free ⟺ C^{∞}(X)^{K} is comm.

ϵ̃_{K}(Ω) “should” measure the failure of C^{∞}(Ω)^{K} to be commutative. We can show that ϵ̃_{K}(Ω) is a “rough” measure of the failure of 𝔄(Ω)_{ra}^{K} to be commutative.

Definition. Let d = d_{Ω} be the maximum dimension of a K-orbit in Ω and d_{Φ} denote the maximum dimension of a K-orbit in Φ(Ω).

Ω_{d} = {E′ ∈ Ω|dim K · E′ = d }
Ω_{Φ} = {E′ ∈ Ω|dim K · Φ(E′) = d_{Φ} }
= {E′ ∈ Ω|dim 𝔨^{Φ(E′)} is minimum }.

Remark. The subset U in the proposition on slide 9 can be chosen so that U ⊂ Ω_{d} ∩ Ω_{Φ} .

The complexity of the K_{C} action on O

Definition. If B_{𝔨} ⊂ K_{C} is a Borel subgroup, then the c_{K_{C}}(O), the complexity of the K_{C} action, is the codimension of a generic B_{𝔨}-orbit in O. It is also the transcendence degree over C of the field of B_{𝔨}-invariant rational functions on O.

Proposition. (Panyushev) c = c_{K_{C}}(O) is the smallest non-negative integer such that for all λ ∈ Γ(O):

m_{O}(nλ) grows no faster than n^{c} as n → ∞

Spherical nilpotent orbits

Definition. If c_{K_{C}}(O) = 0, O is said to spherical (for K_{C}).

(McGovern and Panyushev) Classification of spherical orbits for 𝔤 simple and complex.

(King, 2003) Classification of spherical orbits for 𝔤 simple and real.

2. Main Theorem

Theorem. Let (Ω, O) be a Kostant-Sekiguchi pair, then

(a) r_{K_{C}}(O) = r_{K}(Ω);

(b) c_{K_{C}}(O) = ϵ̃_{K}(Ω). Corollary. (King, TAMS (2002)) Ω is multiplicity free ⟺ O is spherical. Example. When 𝔤 is simple, the minimal nonzero Ω is multiplicity free, and O_{Ω} is spherical.

Motivation

1. The previous corollary.

2. I. V. Mykytyuk’s result:

ϵ̃_{G}(T^{∗}(G/K)) = c_{G_{C}}(G_{C}/K_{C}),

where K and G are compact groups, K ⊂ G , and K_{C} and G_{C} are there complexifications.

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability, preprint, 2001.

Ingredients in the proof of main theorem

1. The Vergne real analytic isomorphism

2. Results of Schmid and Vilonen on the K structure of O

3. Results of Panyushev on rank and complexity of K_{C} actions

4. Results of A. T. Huckleberry and T. Wurzbacher on K actions on symplectic manifolds

3. The K structure of O

Let E ∈ C(Ω), e ∈ C(O), {H, E, F } ↔ {x, e, f}. Set 𝔰 := RH + RE + RF.

Proposition. (Schmid-Vilonen in “Geometry of nilpotent orbits”)

If V_{O}(𝔰) = [𝔨_{C}, e]⧸[𝔨, e], then ∃ K-invariant real analytic isomorphism:

O ≃ K ×_{K^{𝔰}} V_{O}(𝔰) ≃ T_{C(O)}^{∗}(O).

In addition, there is an isomorphism of K^{𝔰} modules over R:

V_{O}(𝔰) ≃ 𝔨^{x}/𝔨^{𝔰} ⊕ Z.

4. Proof of Main Theorem

The spaces 𝔨^{x}/𝔨^{𝔰} and Z appear in results of Panyushev on the rank and complexity of O.

Thus one can compute n_{O} = n_{O, K}, the real codimension of the largest K-orbit in O, in terms of the rank and complexity, i.e., on the one hand,

n_{O} = r_{K_{C}}(O) + 2c_{K_{C}}(O).  (1)

Proof. Combine results of Schmid-Vilonen with those of Panyushev.

Another formula for n_{O}

Clearly, n_{O} = n_{Ω} . Recall the vector space dec. of T_{E′}(Ω) in terms of W = T_{E′}(K · E′):

\frac{W}{W ∩ W^{⊥}} ⨁ ((W ∩ W^{⊥}) ⊕ (W ∩ W^{⊥})^{∗}) ⨁ \frac{W^{⊥}}{W ∩ W^{⊥}}

where,

(1) 𝔨^{Φ(E′)} /𝔨^{E′} ≃ W ∩ W^{⊥} (as 𝔨^{E′} modules) and

(2) 𝔨/𝔨^{Φ(E′)} ≃ \frac{W}{W ∩W^{⊥}} (as 𝔨^{E′} modules). dim Ω = dim T_{E′}(Ω)

= 2 dim (𝔨^{Φ(E′)} /𝔨^{E′}) + dim (𝔨/𝔨^{Φ(E′)} ) + dim \frac{W^{⊥}}{W ∩ W^{⊥}}

= dim 𝔨^{Φ(E′)} − dim 𝔨^{E′} + dim 𝔨 − dim 𝔨^{E′} + dim \frac{W^{⊥}}{W ∩W^{⊥}}. One shows that for a dense open subset Ũ ⊂ Ω, E′ ∈ Ũ implies: dim 𝔨^{Φ(E′)} − dim 𝔨^{E′} = rank 𝔨^{Φ(E′)} − rank 𝔨^{E′} = r_{K}(Ω) and

dim \frac{W^{⊥}}{W ∩ W^{⊥}} = 2ϵ̃_{K}(Ω).

The preceding facts depend on work of Huckleberry and Wurzbacher.

One concludes that

n_{O} = r_{K}(Ω) + 2ϵ̃_{K}(Ω).  (2)

Once it is shown that r_{K}(Ω) = r_{K_{C}}(O), it follows from (1) and (2) that ϵ̃_{K}(Ω) = c_{K_{C}}(O).

5. Commutativity of 𝔄(Ω)_{ra}^{K}

Definition. Suppose M is a smooth manifold and p ∈ M. f_{1}, …, f_{n} ∈ C^{∞}(M) are functionally independent at p if df_{1}, …, df_{n} are linearly independent at p.

Let ℨ = Z (𝔄(Ω)_{ra}^{K}) denote the center of the Poisson algebra 𝔄(Ω)_{ra}^{K}.

Definition. Let α_{K}(Ω) be the largest non negative integer α such that ∃ dense open set U^{♭} ⊂ Ω and f_{1}, …, f_{α} ∈ 𝔄(Ω)_{ra}^{K} such that:

(1) f_{1}, …, f_{α} are f.i. at each p in U^{♭} and

(2) no nontrivial linear combination of the f_{i} belongs to ℨ.

Remark. α = α_{K}(Ω) is well defined since the intersection of any two dense open sets is open and dense. Clearly α ≤ dim Ω.

α_{K}(Ω) is a measure of the size of 𝔄(Ω)_{ra}^{K} relative to its center. In fact, α_{K}(Ω) = 0 ⟺ Ω is multiplicity free .

α_{K}(Ω) is related to ϵ̃_{K}(Ω) because:

Proposition. Let n = n_{{Ω,K}} be the codimension of a K-orbit of maximal dimension in Ω. ∃ open dense subset U^{♯} ⊂ Ω and f_{1}, …, f_{n} in 𝔄(Ω)_{ra}^{K} which are f.i. on U^{♯}.

More on the Poisson algebra C^{∞}(Ω)^{K}

Definition. f ∈ C^{∞}(Ω) is collective if f = g ∘ Φ for some g ∈ C^{∞}(𝔨^{∗}). 𝔅 denotes the space of collective functions.

For f ∈ C^{∞}(Ω), X_{f} denotes the Hamiltonian vector field associated to f by w_{Ω} . That is, for all smooth vector fields Y on Ω:

df(Y) = w_{Ω} (Y, X_{f}).

C^{∞}(Ω) is a Poisson algebra under the Poisson bracket {·, ·} defined as follows. For f, g ∈ C^{∞}(Ω),

{f, g} = w_{Ω} (X_{f}, X_{g}).

Since C^{∞}(Ω)^{K} is the centralizer of 𝔅 in C^{∞}(Ω), the proposition on slide 23 implies:

Lemma. For E′ in a certain dense open subset of Ω and W = T_{E′}(K · E′) :

(a) W is spanned by: {X_{f}(E′)|f ∈ 𝔅}.

(b) W^{⊥} is spanned by: {X_{f}(E′)|f ∈ 𝔄_{ra}^{K}}.

(c) If M_{E′} denotes a complement of W ∩ W^{⊥} in W^{⊥}, then M_{E′} is spanned by:

{X_{f}(E′)|f ∈ 𝔄_{ra}^{k}, f ∉ ℨ}.

Unfortunately, we only have:

α_{K}(Ω) ≥ 2ϵ̃_{K}(Ω).

However, Theorem. If s = 2ϵ̃_{K}(Ω), ∃ open dense subset U^{∗} of Ω, s.t. for some f_{1}, …, f_{s} ∈ 𝔄_{ra}^{K}:

(1) f_{1}, …, f_{s} are f.i. on U^{∗};

(2) no nontrivial linear combination of the f_{i} belongs to ℨ; and

(3) ∀p ∈ U^{∗}, f_{1}, …, f_{s} are f.i. of 𝔅 at p.

s is the largest integer for which ∃ s functions f_{1}, …, f_{s} ∈ 𝔄_{ra}^{K} and an open dense subset U satisfying (1), (2) and (3).

Finally, we obtain:

Corollary. Let (Ω, O) be a KS-pair, then 2c_{K_{C}}(O) is the largest integer n s.t. ∃ f_{1}, …, f_{n} ∈ 𝔄_{ra}^{K} satisfying (1), (2) and (3) of the previous theorem (on slide 26).

Thus, in a weak sense, the integer c_{K_{C}}(O) measures the failure of the Poisson algebra of K-invariant real analytic functions on Ω to be commutative.

6. An application to the G-saturation of the nilradical of a par. sub. of 𝔤

Proposition. Let (Ω, O) be a KS-pair. Suppose that P = MAN ⊂ G (with 𝔭 = 𝔪 ⊕ 𝔞 ⊕ 𝔫) s.t. Ω is an open subset of G · 𝔫. Then

(a) c_{K_{C}}(O) = c_{K_{C}}(K_{C}/M_{C}) and

(b) r_{K_{C}}(O) = r_{K_{C}}(K_{C}/M_{C}).

Remark. The integer c_{K_{C}}(K_{C}/M_{C}) can be calculated by a result of Heckman.

Example. 𝔤 = sl(n, R). For each Ω there is a 𝔭 such that Ω is open in G · 𝔫. Therefore, the complexity and rank of the corresponding K_{C}-orbit O can be computed.

Some related open questions

1. For general O, determine r_{K_{C}}(O) and c_{K_{C}}(O).

2. Find a set of generators for Γ(O). (For 𝔤 simple and complex, these generators have been found by McGovern.) The orbit method suggests that

Γ(O) ⊂ √−1Φ(Ω) ∩ positive Weyl chamber

3. Determine Γ(O), Γ(\overline{O}), and Γ(\overline{O}^{n}).