Distinguished orbits in regular varieties

KATO Syu^{∗} September 7,2004

∗E-mail: syuchan@ms.u-tokyo.ac.jp

The goal of this talk is to generalize the notion of θ-stable maximal torus from symmetric spaces to much wider class of homogeneous spaces.

Generalized θ-stable maximal torus would help us to

1) do harmonic analysis on the coset space;

2) classify some class of homogeneous spaces.

Generalization:

Symmetric spaces: Spherical G/H + θ ∈ AutG s.t. θ^{2} = id, H = G^{θ}

⇓

Regular varieties: Spherical G/H s.t. H = N_{G}^{♯}(H)

N_{G}^{♯}(H) := {g ∈ N_{G}(H); g fixes 𝔅_{H}(1)} ⊂ N_{G}(H)

Remark: Aut_{G}G/H = N_{G}(H)/H Definition of “θ-stable” torus:

Symmetric spaces: maximal torus T s.t. θ(T) = T

⇓

Regular varieties: difficult since there is NO θ!!!

Char’zation of regular varieries (after Luna) The dense open G-orbit of a G-variety X s.t.

• Smooth projective with finite number of G-orbits;

• Closed G-orbit is unique;

• X\(G/H) is a union of G-stable ncd.

Examples of regular varieties:

1) ∀ adjoint algebraic symmetric spaces;

2) (SL_{p+1} (ℂ) × SL_{p}(ℂ))/ SL_{p}(ℂ);

3) SL_{n}(ℂ)/N_{G}(GL_{p}(ℂ) × SL_{p+q−1} (ℂ)) for n = 2p + q;

4) G = Sp(4)^{∗}(⊃ Sp(2) × Sp(2)) ⊃ Sp(2) × O(2) = H.

∗measured by size

Main ingredients:

• W -action on 𝔅_{H} (due to Knop)

• Gen. max’lly split torus (due to Knop)

• Boundary behaviour of orbits (due to Brion)

• Gen. notion of real roots Richardson-Springer corr.^{†} (Kostant etc...) T: θ-stable max’l torus s.t. dim T^{θ}: min.

{θ-stable max’l torus } /Ad(H)
⇔T \{g ∈ G; gθ(g^{−1}) ∈ N_{G}(T)}/H
⇔𝔅_{H}(= {B-orbits in G/H })

gT g^{−1} ↔ g mod (T × H) ↔ Bg[H] is the corr. In part., W acts on 𝔅_{H}.

Remark on R-S corr. The element g is taken as the “Cayley transform.” by the following: We have ((wg)T (wg)^{−1})^{−θ} ⊂ T^{−θ} for w ∈ W. ∀θ-stable torus ↦ fixed part of an involution in W and vice versa. Every involution of a Coxeter group is a product of pairwise commutative reflections (Deodhar). ⇒ g ↦ a seq. of orth. real roots. ↦ prod. of simple Cayley trans.

Knop’s W -action on 𝔅_{H}^{†}

For Y ∈ 𝔅_{H}, we define rkY = dim Y/[B, B]. ∃W -action ⋆ on 𝔅_{H} s.t. rkY = rk(w ⋆ Y) for all w ∈ W.

∃ analogue of max’lly split θ-stable max’l torus T s.t.

1) Bẇ[H] = w ⋆ (B[H]) for all N_{G}(T) ∋ ẇ ↦ w ∈ W.

2) N_{H}(T)/T is max’l & N_{H}(T)[H] ⊂ T [H].

Notations (for LST)

• P := {g ∈ G; gB[H] = B[H]};

• P = LU : Levi dec. s.t. T ⊂ L;

• W_{L} := N_{L}(T)/T, P^{−} ⇔ P : opposite;

• x_{1} : unique P^{−}-fixed point in X.

Local Structure Theorem^{†} (LST due to BLV) ∃Z ⊂ X: L-stable loc. closed sub. s.t.

• X_{0} := P ×^{L} Z ↪ X : affine open emb.;

• x_{1} ∈ Z ≅ 𝔸^{r} as L/[L, L]-varieties;

• Gx_{1} ≅ G/P^{−}.

LST = boundary behaviour of dense open B-orbit. For ∀Y ∈ 𝔅_{H}, we define

W (Y) := \{w ∈ W; dim X−dim Y = ℓ(w), \overline{Bẇ^{−1}Y} = X\},

where ℓ : W → ℤ is the length func. w.r.t. B. |W (Y)| ≥ 2 in general. Define X_{0}^{w} := ẇX_{0}, x_{w} := ẇx_{1}, X_{−}^{w} := X_{0}^{w} ∩ G/H, and 𝕆_{w} := Bx_{w} ⊂ G/P^{−} for each w ∈ W.

Define W̃_{H} := {w ∈ W; (w⋆B[H]) = B[H]}; Define U_{w} := B ∩ ẇUẇ^{−1} and U^{w} := B^{−} ∩ ẇUẇ^{−1} for each w ∈ W. For each Y ∈ 𝔅_{H}, w ∈ W (Y), and v ∈ W̃_{H}, there exists a T U_{w}-equivariant fibration

Y ∩X_{0}^{w} = U_{w}×(Y ∩T U^{w}ẇv̇[H]) ⟶ w𝕆_{1} ⟶ 𝕆_{w}.

f_{(v,w)}^{Y} := Y ∩ T U^{w}ẇv̇[H] (fiber over x_{w}) . Projection (Y ∩ T U^{w}ẇv̇[H]) ⟶ T ẇv̇[H] is Galois .

Elementary modifications Let α ∈ R: simple & Y ∈ 𝔅_{H} be s.t.

dim P_{α}Y > dim Y.

Pair (α, Y) is called: Type T if P_{α}Y has 3 B-orbits. Type N if P_{α}Y has 2 orbits &

(P_{α}Y)^{𝔾_{m}^{α}} ⊂ Y.

Type U if P_{α}Y ≠ Y & not Type T or N. We can also distinguish them via the stabilizer group of P_{α} at a point of Y.

Weak order graph A labeled graph corr. to elm. relations among 𝔅_{H}. Vertex := 𝔅_{H}; Labels := Simple roots of (G, B) Edges := Y, Y′ ∈ 𝔅_{H} are connected by a edge labeled by α ∈ R if Y′ ⊂ P_{α}Y. The edge is a simple (resp. double) edge if (α, Y) is Type UT (resp. Type N). Define an order ⪯ on 𝔅_{H} gen. by Y ⪯_{α} Y′ ⇔ 1) dim Y′ = dim Y + 1 2) Y′ and Y are connected by an edge of the weak order graph with label α.

Reflections of W̃_{H} T_{0} : identity comp. of the stabilizer of T at [H]. Let s ∈ W̃_{H}(⊂ W) be a reflection s.t. T_{0}^{s} ≠ T_{0}. (i.e. s = s_{α} s.t. α ∈ R & 𝔾_{m}^{α} ⊂ T_{0})

⇓ “Reading” from weak order graph

ṡ_{α}[H] ⊂ U_{α}[H].

(Gen.) real roots α ∈ R is called psuedo-real root (R_{ps}) if 𝔾_{m}^{α} ⊂ T_{0}; real root (R_{re}) if s_{α} ∈ W̃_{H}; spherical real root (R_{re}^{H})

if α ∈ R_{re} & ṡ_{α} ∈ H;

perpendicular real roots (R_{re}^{⊥})

if α ∈ R_{re} & ṡ_{α} ∉ H;

quasi-real root (R_{qu}) if α ∈ ℚR_{re} ∩ R.

R_{ps}^{∗} ⊃ R_{qu} ⊃ R_{re} = R_{re}^{H} ⊕ R_{re}^{⊥}

∗I have no example that R_{ps} ≠ R_{qu}

Structural result: γ, δ ∈ R are called str. orthogonal if γ ⊥ δ & γ ± δ ∉ R. Lemma

• α ∈ R_{re}^{⊥} is str. orth. to ∀β ∈ R_{re}\{±α};

• W̃_{H} preserves both R_{re}^{H} and R_{re}^{⊥};

• α ∈ R_{ps}\R_{qu} is str. orth. to ∀β ∈ R_{qu}.

Main Result-I

1) For each Y ∈ 𝔅_{H}, (w, v) ∈ W (Y) × W̃_{H}, Y ∩ \overline{f{Y \atop (v,w)}} contains a unique min’l dim. T -orbit o_{v}^{Y} ⊂ Y;

2) o_{v}^{Y} is independent of the choice of w ∈ W (Y);

3) The set 𝔗_{H} consisting of all o_{v}^{Y} is stable under the action of N_{G}(T);

4) ∃ W -equivariant surjection 𝔱 : 𝔗_{H} → → 𝔅_{H}; Main Result-II

5) ∃ direct factor W_{H}^{⊥} ⊂ W̃_{H} s.t. W_{H}^{⊥} ≅ (ℤ/2ℤ)^{⊕n} and each 𝔱^{−1}(Y) admits a transitive W_{H}^{⊥}-action;

We call elements of 𝔗_{H} distinguished orbits. For each distinguished T g[H], gT g^{−1} is an analogue of (a H-conj. class of) θ-stable maximal torus.

Remarks

1) H-conj. class of “θ-stable torus” corr. to a B-orbit is NOT unique in general (unique in symmetric case by Richardson-Springer).

2) 𝔱 is an isomorphism in symmetric case.

3) W_{H}^{⊥} is generated by reflections corr. to R_{re}^{⊥}.

4) (T ∩H) fixes w^{−1}o for each o ∈ 𝔗_{H} and w ∈ W (𝔱(o)).

Explicit relation between f and o For each Y ∈ 𝔅_{H}, we fix w ∈ W (Y). T_{Y} := identity comp. of stab. of T on o_{v}^{Y} (v ∈ W̃_{H}). R_{Y}^{w} := {α ∈ R; α(wT_{0}w^{−1}) = 1&𝔾_{m}^{α} ⊂ T_{Y} }. G_{Y}^{w} ⊃ T : the group corr. to R_{Y}^{w}. Proposition We have

f_{(v,w)}^{Y} ⊂ G_{Y}^{w}o_{v}^{Y}

for every v ∈ W̃_{H}. If G : simply-laced ⇒ f_{(v,w)}^{Y} is the open T -orbit in (G_{Y}^{w} ∩ B)o_{v}^{Y}.

How to prove Main Theorem-I Lemma Let Y ∈ 𝔅_{H}. Let w ∈ W (Y). Let α be s.t. (α, Y) is of Type U, T, or N. Then, it is of Type T or N ⇔ ws_{α}w^{−1} ∈ W̃_{G/H} and α(T_{Y}) = 1 holds. In conjunction of this lemma, we prove

1) o_{v}^{Y} ⊂ \overline{f{Y \atop (v,w)}}, 2) o_{v}^{Y} ⊂ G_{Y}^{w}ẇv̇[H], and 3) ṡf_{(v,w)}^{Y} = f_{(v,s_{α}w)}^{s_{α}⋆Y} if (α, Y) is of Type U.

How to prove four assertions-Method Induction on the weak order. Difficult part is ⇒ in Lemma and 3). Let Y ∈ 𝔅_{H} and w ∈ W (Y) be s.t. (α, Y) is of Type UTN. Let Y_{+} ⊂ P_{α}Y be open B-orbit. Then

f_{(v,w)}^{Y} ⊂ U_{α}f_{(v,s_{α}w)}^{Y_{+}}

holds. (Here we used the assumption that G/H is regular var.)

How to prove four assertions-Observation Use the component group of (T ∩ H), which determines

T_{0} _{α∈R_{re}^{H}}^{∏} 𝔾_{m}^{α} ⊂ T.

uniquely.

f_{(v,s_{α}w)}^{Y_{+}} ⊂ (G_{Y_{+}}^{s_{α}w} ∩ B^{−})ṡ_{α}wv[H]

⇓

ẇ^{−1}f_{(v,s_{α}w)}^{Y_{+}} admits an action of (T ∩H)/T_{0}.

How to prove four assertions-Idea Prove explicit relation between f and o

f_{(v,s_{α}w)}^{Y_{+}} ⊂ (G_{Y_{+}}^{s_{α}w} ∩ B)o_{v}^{Y_{+}}

and compare the min’l dim’l T -orbits and its stabilizer to characterize the position of f_{(v,w)}^{Y} via Observation. Ex. If ⟨α, β⟩ = 1 for some α ∈ R_{re}^{H}, then α^{∨}(−1) ⊂ (T ∩ H)/T_{0} acts on (U_{α}\1)T [H] freely!

How to prove Main Theorem-II Proposition For each Y ∈ 𝔅_{H}, we define R_{Y} := {α ∈ R; 𝔾_{m}^{α} ⊂ T_{Y} }. Let G_{Y} ⊃ T be the group corr. to R_{Y}. Then, o_{v}^{Y} is the T_{Y} -fixed point set of

G_{Y} o_{v}^{Y} ∩ Y = G_{Y} f_{(v,w)}^{Y} ∩ Y = G_{Y} ẇv̇[H] ∩ Y.

Moreover,

X ∩ \overline{G_{Y} f_{(v,w)}^{Y}} = W_{G_{Y}} p_{wv}

holds. Here W_{G_{Y}} is the Weyl group of G_{Y}.

How to prove Main Theorem-III Final Step We prove that 𝔗_{H} is N(T)-invariant by 1) Choose nice w ∈ W (Y) s.t. ℓ(s_{α}w) < ℓ(w).

2) Using induction on ⪯ according to its type. By virtue of 1), our construction of N(T)- action is compatible with the construction of Knop’s W -action.

Further application We can “generalize” Vogan’s c-invariant of symmetric Harish-Chandra modules to general (𝔤, H)-modules by using the boundary behaviour along the boundary.