Resolution of nilpotent orbits

for symmetric pairs

Kyo Nishiyama (Kyoto University)

September 5–10, 2004 Conference on Nilpotent Orbits and

Representation Theory 2004 (NORTh 6) at Fuji-Zakura So

1. Equivariant double fibration

G, G′: reductive linear algebraic groups /ℂ W: finite dim representation of G × G′

⟹ \begin{cases}Y := W//G = Spec ℂ[W]^{G} \\ X := W//G′ = Spec ℂ[W]^{G′} \end{cases}

: affine quotients of W,

where ℂ[W] = (ring of regular functions on W ) ℂ[W]^{G} = (G-invariants )

W
↙ ↘
G ↷ X  Y ↶ G′

X/G = (set of G-orbits in X ) Y/G′ = (set of G′-orbits in Y )

Problem 1 What is the relation between X/G and Y/G′?

An answer to Problem 1:

Theorem 2

Under natural Assumptions below, there is an embedding θ : Y/G′ ↪ X/G

• Description of the embedding:

W ⟶^{φ} X: G-equiv. quotient map //G′ W ⟶^{ψ} Y: G′-equiv. quotient map //G

∀O′ ⊂ Y: G′-orbit, ∃O ⊂ X: G-orbit s.t. φ(ψ^{−1}(\overline{O′})) = \overline{O}

⟹ θ : Y/G′ ∋ O′ ↦ O ∈ X/G

θ preserves nilpotency & closure relation Classical example G = GL_{n}, G′ = GL_{m} W = M_{m,n} ⊕ M_{n,m} = ℂ^{n} ⊗ ℂ^{m} ⊕ (ℂ^{n} ⊗ ℂ^{m})^{∗} Assumption: n ≥ 2m

⟹ \begin{cases}Y = W//G = M_{m} \\ X = W//G′ = Det _{m}(M_{n}) \end{cases}

where Det _{m}(M_{n}) = {A ∈ M_{n} | rank A ≤ m}

: determinantal variety of rank m GL_{m} ↷ M_{m} = Y: Ad action GL_{n} ↷ Det _{m}(M_{n}) = X: restrict. of Ad action • Quotient maps

(A, B) ∈ M_{m,n} ⊕ M_{n,m} φ(A, B) = AB ∈ M_{m} = Y ψ(A, B) = ^{t}(BA) ∈ Det _{m}(M_{n}) = X • Correspondence

θ : M_{m}/ Ad GL_{m} → Det _{m}(M_{n})/ Ad GL_{n}

θ(G′ · A) = G · [ \begin{matrix}A & 0 \\ X & 0 \end{matrix} ], X: generic

• Extreme cases: Regular orbits :

rank A = m ⟹ θ(G′ · A) = G · [ \begin{matrix}A & 0 \\ 0 & 0 \end{matrix} ]

Nilpotent orbits : A: nilpotent of type λ = (λ_{1}, …, λ_{l}) ⟹ θ(G′ · A): nilpotent of type λ + (1, …, 1) Young diagramatic notaion

O′ = □ ; □ ; □ ; {□ \atop □} ; {□ \atop □} ; □ ; □ ; □  ⇒  O = □ ; □ ; □ ; ■ ; □ ; □ ; ■ ; □ ; □ ; ■ ; {□ \atop ■} ; {□ \atop ■} ; {□ \atop ■} ; ■ ; ■

Assumptions

Put

𝔑 = ψ−1(ψ(0)): null cone (for the action of G)

Assumption 1 (a) The quotient map ψ : W → Y is flat (⇔ ℂ[W] is flat /ℂ[W]^{G})

(b) ∃ an open dense G-orbit O_{0} ⊂ 𝔑

(c) 𝔑 ≃ W ×_{Y} {0} is reduced

(d) Generically, the fiber of φ : W → X is a single (hence closed) G′-orbit

(e) Put W^{∘} := ∐(closed G′-orbits ) ⟹ ψ^{−1}(y) ∩ W^{∘} ≠ ∅ (∀y ∈ Y)

Under the above assumptions (a) – (e), Theorem 2 holds: ∃ embedding θ : Y/G′ ↪ X/G

∀O′ ⊂ Y: G′-orbit

∃O = θ(O′): G-orbit ⊂ X s.t.

(1) \overline{O} = φ(ψ^{−1}(\overline{O′}))

\overline{O} ≃ (W ×_{Y} \overline{O′})//G′ (as G-variety )

(2) O′_{1} < O′_{2} ⟹ θ(O′_{1}) < θ(O′_{2}) (closure relation)

(3) O′: nilpot. ⟹ O = θ(O′): nilpot.

Examples of G′, G and W

• Tensor product of natural representations

W = G ⊗ G′ condition
O_{n} ⊗ O_{m}  2m < n
O_{n} ⊗ Sp_{2m}  4m < n
O_{n} ⊗ SL_{m}  2m < n
Sp_{2n} ⊗ O_{m}  m ≤ n
Sp_{2n} ⊗ Sp_{2m} 2m ≤ n
Sp_{2n} ⊗ SL_{m}  m ≤ n

• Contruction by GL_{n}, Case (I) G = GL_{n} ↷ V = ℂ^{n} W = (V ⊕ V^{∗}) ⊗ U

U = G′ condition
O_{m}  2m ≤ n
Sp_{2m}  4m ≤ n
SL_{m}  2m ≤ n

• Contruction by GL_{n}, Case (II) G = GL_{n} ↷ V = ℂ^{n} G′ = G′_{+} × G′_{−} ↷ U^{+} ⊕ U^{−} W = V ⊗ U^{+} ⊕ V^{∗} ⊗ U^{−}

U^{+} = G′_{+} U^{−} = G′_{−} condition
O_{p}  O_{q}  p + q ≤ n
Sp_{2p}  Sp_{2q}  2p + 2q ≤ n
O_{p}  Sp_{2q}  p + 2q ≤ n
SL_{p}  SL_{q}  p + q ≤ n

• Opposite cases Can exchange the roles of G and G′

• General linear groups G = GL_{n} ↷ V = ℂ^{n} G′ = GL_{m} ↷ U = ℂ^{m} Assume 2m ≤ n W = V ⊗ U^{∗} ⊕ V^{∗} ⊗ U History

• Case arising from dual pairs: The theorem is primitively noticed by Roger Howe (1970’s)

• G_{ℂ}-nilpotent orbits of reductive dual pairs:

Daszkiewicz-Kraśkiewicz-Przebinda (1996, 1997)

• Nilpotent orbits of symmetric pairs related to dual pair:

N. (2000)

Daszkiewicz-Kraśkiewicz-Przebinda (2002)

Ohta (preprint)

N.-Ochiai-Zhu (to appear in TAMS)

• Beyond nilpotent orbits:

Ohta (private communication, 2003)

N. (preprint)

2. Lifting of nilpotent orbits of symmetric pair

• symplectic group W = ℂ^{N} : Hermitian inner product (,) ⟨u, v⟩ = Im(u, v) : symplectic form /ℝ

⇒ W = W_{ℝ} : symplectic space /ℝ

W_{ℝ} = ℝ^{N} ⊕ √−1ℝ^{N}

= X ⊕ Y : complete polarization symplectic group Sp(W_{ℝ})

⊃ U(ℂ^{N}) : unitary group (max cpt subgrp)

• Dual pair (G_{ℝ}, G′_{ℝ}) (G_{ℝ}, G′_{ℝ}) : subgroups in Sp(W_{ℝ})

mutually commutant to each other in Sp(W_{ℝ})

K_{ℝ} ⊂ G_{ℝ}, K′_{ℝ} ⊂ G′_{ℝ}: max compact subgrps s.t. K_{ℝ} · K′_{ℝ} ⊂ U(W)

K, K′: ℂ-fication of K_{ℝ}, K′_{ℝ}

⟹ K × K′ ↷ W: holomorphic action 𝔤 = (Lie G_{ℝ})_{ℂ}: ℂ-fied Lie algebra 𝔤 = 𝔨 ⊕ 𝔭: Cartan decomposition ℂ-fication of moment maps:

\begin{cases}ψ : W → Y = W//K ⊂ 𝔭′ \\ φ : W → X = W//K′ ⊂ 𝔭 \end{cases}

Assume (G_{ℝ}, G′_{ℝ}) is irreducible and strictly in the stable range (G′_{ℝ}: small) ⇒ K × K′ ↷ W satisfies Assumption 1 Y = 𝔭′ ⇒ ∃embedding θ : 𝔭′/ Ad K′ ↪ 𝔭/ Ad K by Theorem 2

SSR := strictly in the stable range Example: dual pair O_{p,q}(ℝ) × Sp_{2n}(ℝ) SSR ⟺ 2n < min {p, q}

(Rmk. p + q ≤ n is also SSR in the opposite direction)

• ℂ-fied moment maps:

[figure: W = W_{+} ⊕ W_{−}; φ; ψ; 𝔭; 𝔭′_{+} ⊕ 𝔭′_{−} = 𝔭′]
[figure: M_{p,n} ⊕ M_{q,n}; φ; ψ; M_{p,q}; Sym _{n} ⊕ Sym _{n}]

For (A, B) ∈ M_{p,n} ⊕ M_{q,n}

\begin{matrix}φ(A, B) = A^{t}B ∈ M_{p,q} \\ ψ(A, B) = (^{t}AA, ^{t}BB) ∈ Sym _{n} ⊕ Sym _{n} \end{matrix}

• Lifting of rank k orbit K = O_{p}(ℂ) × O_{q}(ℂ), K′ = GL_{n}(ℂ)

O′_{k} := {A ∈ Sym _{n} | rank A = k} ⊂ 𝔭′_{+}

: nilpotent Ad K′-orbit

O_{k} = θ(O′_{k}) ⊂ 𝔭: theta lift of O′_{k} : nilpotent Ad K-orbit

ψ^{−1}(O′_{k}) ∩ W^{∘} = {(A, B) ∈ M_{p,n} ⊕ M_{q,n} | (∗) }

(∗) \begin{cases}rank ^{t}AA = k \\ ^{t}BB = 0 \\ rank A = rank B = n \end{cases}

⟹ O_{k} ≃ (ψ^{−1}(O′_{k}) ∩ W^{∘})/GL_{n} • Geometric structure of the orbit O_{k}

𝔾_{n}(ℂ^{p}): Grassmannian of n-dim subsp in ℂ^{p} T_{p}: tautological bundle on 𝔾_{n}(ℂ^{p})

Z(k; p) := { V ∈ 𝔾_{n}(ℂ^{p}) |

rank V = k (as a quadratic space )}

T (k; p) = T_{p}|_{Z(k;p)} : restriction to Z(k; p)

ψ^{−1}(O′_{k})∩W^{∘} ↪ End (T (k; p))⊠End (T (0; q)^{∗})

(Rmk. T (0; q) is the tautological bundle on isotropic Grassmannian)

Theorem 3 ∃ O_{p} × O_{q}-equiv birational map

Φ : T (k; p) ⊠ T (0; q)^{∗} → \overline{O_{k}}

which gives a rsolution of singularity

• Two extreme cases k = 0 O_{0} = lifiting of the trivial orbit

⟺ [(+ −)^{n}(−+ )^{n}(+ )^{p−2n}(−)^{q−2n}]

T (0; p) ⊠ T (0; q)^{∗} → \overline{O_{0}}: resol of singularity T (0; p): vector bundle over isotropic

Grassmannian

k = n

O_{n} ⟺ [(+ −+ )^{n}(+ )^{p−2n}(−)^{q−n}]

T (n; p): vector bundle over symmetric space O_{p}/O_{n} × O_{p−n} • Spherical nilpotent orbits for symmetric pair O_{0}: spherical nilpotent K-orbit in 𝔭

P_{n}: partitions of length at most n σ_{α}^{(p)} : irred finite dim representation of O_{p}

with ht wt α = (α_{1}, …, α_{n}, 0, …, 0) ∈ P_{n}

ℂ[\overline{O_{0}}] ≃ \begin{matrix}∑ & ⊕ \\ α∈P_{n} \end{matrix} σ_{α}^{(p)} ⊠ σ_{α}^{(q)}

O_{0} (n < min {p/2, q/2}) gives a large part of spherical orbits

Classification of spherical nilpotent K-orbits in 𝔭 — D. R. King (2004)

(cf. N. (2004) for U(p, p))

3. Unimodular action on bilinear forms

SL_{m}(ℂ) ↷ M_{m}(ℂ) = ℂ^{m} ⊗ ℂ^{m} via g · A = gA^{t}g (g ∈ SL_{m}, A ∈ M_{m})

(cf. –Doković-Sekiguchi-Zhao (preprint), Ochiai)

Resolution via the action of GL_{n}

G = GL_{n}, G′ = SL_{m} V = ℂ^{n}: natural representation of G U = ℂ^{m}: natural representation of G′ W = (V ⊕ V^{∗}) ⊗ U = M_{n,m} ⊕ M_{n,m}

2m ≤ n ⟹ W satisfies Assumption 1.

Double fibration by quotient maps:

W

φ↙ ↘ψ
X = W//G′  W//G = Y
\begin{cases}X = 𝔾_{m}^{aff} (V ⊕ V^{∗}) ↶ GL_{n} \\ Y = U ⊗ U = M_{m}(ℂ) ↶ SL_{m} \end{cases}

𝔾_{m}^{aff} (V): affine cone of Grassmannian in ⋀^{m} V with Plücker embedding

∃ embedding

θ : M_{m}/SL_{m} ↪ 𝔾_{m}^{aff} (V ⊕ V^{∗})/GL(V) θ induces (not necessarily injective) map

θ^{∘} : M_{m}/SL_{m} → 𝔾_{m}(V ⊕ V^{∗})/GL(V) θ^{∘} is injective on “nilpotent” orbits K_{ℂ}-orbits on isotropic Grassmannian

V ⊕ V^{∗} carries natural symplectic bilinear form ⟨, ⟩, and symmetric bilinear form (,)

s.t. V ⊕ V^{∗} is a complete polarization

Sp(V ⊕ V^{∗}), O(V ⊕ V^{∗}): isometry groups ⟹ Sp(V ⊕ V^{∗}) ∩ O(V ⊕ V^{∗}) = GL(V)

𝕀𝔾_{m}^{Sp} = 𝕀𝔾_{m}^{Sp}(V ⊕ V^{∗}), 𝕀𝔾_{m}^{O} = 𝕀𝔾_{m}^{O}(V ⊕ V^{∗}) : isotropic Grassmannians

Theorem 4 For O′ ∈ M_{m}/SL_{m}, put O = θ^{∘}(O′) ∈ 𝔾_{m}(V ⊕ V^{∗})/GL(V)

O ∈ 𝕀𝔾_{m}^{Sp}/GL(V) ⟺ O′ ⊂ Sym _{m}
O ∈ 𝕀𝔾_{m}^{O}/GL(V) ⟺ O′ ⊂ Alt_{m}

Rmk. 𝕀𝔾_{m}^{Sp}/GL(V) ≃ P_{m}\Sp(V ⊕ V^{∗})/GL(V) P_{m}: a maximal parabolic subgroup in Sp “K_{ℂ}” = GL(V): ℂ-fied max cpt subgroup (cf. Matsuki: #P \G_{ℂ}/K_{ℂ} < ∞)

Lifting of the trivial orbit

O′ = {0}: the trivial orbit in Y = M_{m} O^{1} = θ(O′): lifted orbit in X = 𝔾_{m}^{aff} (V ⊕ V^{∗})

𝔑 = ψ^{−1}(0) : null cone

= {(A, B) ∈ M_{n,m} ⊕ M_{n,m} | ^{t}BA = 0}

⟹ \overline{O^{1}} = 𝔑//G′

O^{1} is a G-spherical variety with the normal closure in X = 𝔾_{m}^{aff} (V ⊕ V^{∗})

ℙO^{1}: projective image of O^{1} in 𝔾_{m}

Theorem 5 𝕀𝔾_{m}^{Sp} ∩ 𝕀𝔾_{m}^{O} is a normal, GL(V)-spherical variety which contains ℙO^{1} as an open dense orbit P_{m}: partitions of length at most m τ_{α}^{(m)} : irred finite dim representation of GL_{m}

with ht wt α = (α_{1}, …, α_{m}) ∈ P_{m}

α ⊙ β = (α, 0, …, 0, β^{∗}) ∈ ℤ^{n} (α, β ∈ P_{m})
β^{∗} = (−β_{m}, −β_{m−1}, …, −β_{1})

(need the assumption 2m ≤ n)

• Decomposition of the regular function ring ℂ[\overline{O^{1}}] as a representation of GL_{n} = GL(V)

ℂ[\overline{O^{1}}] ≃ ℂ[𝔑]^{G′}  (G′ = SL_{m})

≃ \begin{matrix}∑ & ⊕ \\ k≥0 \end{matrix} \begin{matrix}∑ & ⊕ \\ α−β^{∗}=k1_{m} \end{matrix} τ_{α⊙β}^{(n)} (α, β ∈ P_{m})

⟹ \overline{O^{1}} is G-spherical ⟹ so is O^{1} and ℙO^{1} Rmk. ℂ[\overline{O^{1}}] = (homog coord ring of 𝕀𝔾_{m}^{Sp}∩𝕀𝔾_{m}^{O})

Open Problems

(1) Find a pair (G, G′) of exceptional type and a repr W satisfying Assumption 1

(2) V, U: irred reprs of G, G′ Classify (V, U) for which W = V ⊗ U satisfies Assumption 1.

(3) Give description of θ : Y/G′ ↪ X/G in a combinatorial way.

(4) Find a representation theoretic interpretation of θ

(cf. theta correspondence by Howe; Daszkiewicz-Przebinda (1996), N.-Zhu (preprint))

(5) Find the relation between the singularity of \overline{O′} and that of \overline{θ(O′)}