A REMARK ON SCHUBERT CELLS AND DUALITY OF ORBITS ON FLAG MANIFOLDS

SIMON GINDIKIN* AND TOSHIHIKO MATSUKI

Abstract. It is known that the closure of an arbitrary K_{ℂ}-orbit on a flag manifold is expressed as a product of a closed K_{ℂ}-orbit and a Schubert cell ([M2], [Sp]). We already applied this fact to the duality of orbits on flag manifolds ([GM]). We refine here this result and give its new applications to the study of domains arising from the duality.

1. Duality of orbits on flag manifolds

Let G_{ℂ} be a connected complex semisimple Lie group and G_{ℝ} a connected real form of G_{ℂ}. Let K_{ℂ} be the complexification in G_{ℂ} of a maximal compact subgroup K of G_{ℝ}. Let X = G_{ℂ}/P be a flag manifold of G_{ℂ} where P is an arbitrary parabolic subgroup of G_{ℂ}. Then there exists a natural one-to-one correspondence between the set of K_{ℂ}-orbits S and the set of G_{ℝ}-orbits S′ on X given by the condition:

(A) S ↔ S′ ⟺ S ∩ S′ is non-empty and compact

([M3]). In the following, we will identify orbits S with K_{ℂ}-P double cosets and S′ with G_{ℝ}-P cosets.

We defined in [GM] a subset C(S) of G_{ℂ} by

C(S) = {x ∈ G_{ℂ} | xS ∩ S′ is non-empty and compact in X = G_{ℂ}/P }

where S′ is the G_{ℝ}-orbit on X given by (A).

If S is closed, then S′ is open ([M1]) and so the condition

xS ∩ S′ is non-empty and compact in G_{ℂ}/P

implies

xS ⊂ S′.

Hence the set C(S)_{0} is the cycle domain (cycle space) for S′ ([WW]) where C(S)_{0} denote the connected component of C(S) containing the identity.

On the other hand, let S_{0} denote the unique open K_{ℂ}-B double coset in G_{ℂ} where B is a Borel subgroup of G_{ℂ} contained in P. (We will keep this notation for the whole note.) Then S′_{0} is the only one closed G_{ℝ}-B double coset in G_{ℂ} and the condition

xS_{0} ∩ S′_{0} is non-empty and compact in G_{ℂ}/B

implies

xS_{0} ⊃ S′_{0}.

*The first author was partly supported by NSF-grant DMS-0070816. He also thanks the Max-Planck-Institut für Mathematik at Bonn for their hospitality.

Let {S_{j} | j ∈ J} be the set of K_{ℂ}-B double cosets in G_{ℂ} of codimension one and T_{j} = S_{j}^{cl} denote the closure of S_{j}. The sets T_{j} will play an important role in our constructions.

The compliment of S_{0} in G_{ℂ} is written as

⋃_{j∈J} T_{j}

(by Theorem 2 in Section 2). So the set C(S_{0}) is the compliment of the infinite family of complex hypersurfaces

T_{j}^{−1}g (j ∈ J, g ∈ S′_{0})

and hence the connected component C(S_{0})_{0} is Stein.

This domain is sometimes called the “Iwasawa domain” since it is a maximal domain where all Iwasawa decompositions can be holomorphically extended from G_{ℝ}.

In [GM], we defined

C = ⋂ C(S)

where we take the intersection for all K_{ℂ}-orbits on X on all flag manifolds X = G_{ℂ}/P of G_{ℂ} and conjectured

C = D̃_{0}Z

(Conjecture 1.3) where D_{0} = D̃_{0}/K_{ℂ} is the domain introduced by [AG] (which is sometimes denoted as Ω_{AG}) and Z is the center of G_{ℂ}. For connected components, it means

C_{0} = D̃_{0}. (B)

It is proved in Proposition 8.3 of [GM] that

C_{0} = C(S_{0})_{0}.

In other words, C(S)_{0} is minimal when S = S_{0}. We believe that it is one of central facts of this theory since it gives a very strong estimate of all C(S) through C(S_{0}) only. So the conjecture (B) is equivalent to

C(S_{0})_{0} = D̃_{0}.

Recently, the inclusion C(S_{0})_{0} ⊂ D̃_{0} is proved by Barchini ([B]). On the other hand, the opposite inclusion C(S_{0})_{0} ⊃ D̃_{0} is proved for all classical cases ([GM], [KS]) and exceptional Hermitian cases ([GM]). It is announced that it is proved for all spaces in [H]. (We had no chance to see this preprint.)

Remark 1. In [FH], the authors deduce the inclusion C_{0} ⊂ D̃_{0} from their result about C(S) for closed S and Proposition 8.1 in [GM]. As we showed above, this inclusion is already the consequence of Proposition 8.3 in [GM] and [B]. So it does not need the results in [FH].

2. Schubert cells in the category of K_{ℂ}-B double cosets

The principal idea of our considerations in [GM] was that C(S)_{0} will be essentially independent of neither S nor the flag manifold X = G_{ℂ}/P. To justify it, we need to build bridges between C(S) for different S and for it we need to see connections between different K_{ℂ}-orbits. It turns out that Schubert cells are very efficient tool for such considerations as in Section 2 and Section 8 in [GM]. They give a possibility to obtain an important information about general C(S) from a consideration of simplest S. Here we refine connections between K_{ℂ}-orbits and Schubert cells and give more examples of applications.

For a simple root α in the root system with respect to the order defined by B, we can define a parabolic subgroup

P_{α} = B ∪ Bw_{α}B

of G_{ℂ} such that dim_{ℂ} P_{α} = dim_{ℂ} B + 1.

Lemma 1. Let S_{1} be a K_{ℂ}-B double coset. Then we have:

(i) If dim_{ℂ} S_{1}P_{α} = dim_{ℂ} S_{1}, then S_{1}^{cl}P_{α} = S_{1}^{cl}.

(ii) If dim_{ℂ} S_{1}P_{α} = dim_{ℂ} S_{1} + 1, then there exists a K_{ℂ}-B double coset S_{2} such that S_{1}^{cl}P_{α} = S_{2}^{cl}.

Proof. Though this lemma follows easily from [M2] Lemma 3, we will give a proof for the sake of completeness. Write S_{1} = K_{ℂ}gB. Then we have a natural bijection

(g^{−1}K_{ℂ}g ∩ P_{α})\P_{α}/B ≅ K_{ℂ}\K_{ℂ}gP_{α}/B = K_{ℂ}\S_{1}P_{α}/B

by the map x ↦ gx.

(i) If dim_{ℂ} S_{1}P_{α} = dim_{ℂ} S_{1}, then (g^{−1}K_{ℂ}g ∩ P_{α})B/B is Zariski open in P_{α}/B = P^{1}(ℂ) and hence it is dense. So we have

S_{1}^{cl} = (K_{ℂ}gB)^{cl} ⊃ S_{1}P_{α} ⊃ S_{1}

and therefore S_{1}^{cl} = S_{1}^{cl}P_{α}.

(ii) Suppose dim_{ℂ} S_{1}P_{α} = dim_{ℂ} S_{1} + 1. Then there exists a p ∈ P_{α} such that (g^{−1}K_{ℂ}g ∩ P_{α})pB/B is Zariski open in P_{α}/B = P^{1}(ℂ) since the number of K_{ℂ}-B double cosets in G_{ℂ} is finite. If we write S_{2} = K_{ℂ}gpB, then we have

(S_{2})^{cl} ⊃ S_{1}P_{α} ⊃ S_{2}

and therefore S_{2}^{cl} = S_{1}^{cl}P_{α}. q.e.d.

Theorem 1. Let S_{1} be a K_{ℂ}-B double coset in G_{ℂ} and w an element of W. Then we have:

(i) S_{1}^{cl}(BwB)^{cl} = S_{2}^{cl} for some K_{ℂ}-B double coset S_{2}.

(ii) (minimal expression) There exists a w′ ∈ W such that w′ < w (Bruhat order), ℓ(w′) = dim_{ℂ} S_{2} − dim_{ℂ} S_{1} and that

S_{1}^{cl}(Bw′B)^{cl} = S_{2}^{cl}.

Here ℓ(w′) = dim_{ℂ} Bw′B − dim_{ℂ} B is the length of w′.

Proof. (i) This follows from Lemma 1 because every Schubert cell (BwB)^{cl} is written as

(BwB)^{cl} = P_{α_{1}} ⋯ P_{α_{ℓ}}

where w = w_{α_{1}} ⋯ w_{α_{ℓ}} is a minimal expression of w ∈ W.

(ii) By Lemma 1, we can choose a subsequence β_{1}, …, β_{q} (q = dim_{ℂ} S_{2}−dim_{ℂ} S_{1}) of α_{1}, …, α_{ℓ} such that

dim_{ℂ} S_{1}^{cl}P_{β_{1}} ⋯ P_{β_{k}} = dim_{ℂ} S_{1}^{cl}P_{β_{1}} ⋯ P_{β_{k−1}} + 1

for k = 1, …, q and that

S_{2}^{cl} = S_{1}^{cl}(BwB)^{cl} = S_{1}^{cl}P_{α_{1}} ⋯ P_{α_{ℓ}} = S_{1}^{cl}P_{β_{1}} ⋯ P_{β_{q}} = S_{1}^{cl}(Bw′B)^{cl}

with w′ = w_{β_{1}} ⋯ w_{β_{q}}. q.e.d.

Remark 2. S_{1}^{cl}(BwB)^{cl} = S_{2}^{cl} implies S_{1}^{cl} ⊂ S_{2}^{cl}. But S_{1}^{cl} ⊂ S_{2}^{cl} does not always imply S_{1}^{cl}(BwB)^{cl} = S_{2}^{cl} for some w (c.f. [M2]).

Definition 1. For every K_{ℂ}-B double coset S, we can define, by Theorem 1, a subset J(S) of J by

J(S) = {j ∈ J | S^{cl}(BwB)^{cl} = T_{j} for some w ∈ W }.

Lemma 2. Let S be a non-open K_{ℂ}-B double coset. Then there exists a simple root α such that

dim_{ℂ} SP_{α} = dim_{ℂ} S + 1.

Proof. Write G_{ℂ} = (Bw_{0}B)^{cl} = P_{α_{1}} ⋯ P_{α_{m}} with the longest element w_{0} in W. If

dim_{ℂ} SP_{α} = dim_{ℂ} S

for all simple roots α, then we have, by Lemma 1,

G_{ℂ} = S^{cl}G_{ℂ} = S^{cl}P_{α_{1}} ⋯ P_{α_{m}} = S^{cl},

a contradiction. q.e.d.

Theorem 2. If ℓ(w) < codim_{ℂ}S, then

S^{cl}(BwB)^{cl} ⊂ T_{j}

for some j ∈ J(S).

Proof. Since codim_{ℂ}S^{cl}(BwB)^{cl} = d > 0, we can choose simple roots α_{1}, …, α_{d−1} such that

codim_{ℂ}S^{cl}(BwB)^{cl}P_{α_{1}} ⋯ P_{α_{d−1}} = 1

by Lemma 2. Since (BwB)^{cl}P_{α_{1}} ⋯ P_{α_{d−1}} = (Bw′B)^{cl} for some w′ ∈ W, we have

S^{cl}(BwB)^{cl} ⊂ S^{cl}(Bw′B)^{cl} = T_{j}

for some j ∈ J(S). q.e.d.

3. Applications

Definition 2. For every subset J′ in J, we define a domain Ω(J′) in G_{ℂ} by

Ω(J′) = {x ∈ G_{ℂ} | xT_{j} ∩ S′_{0} = ϕ for all j ∈ J′}_{0}.

We can prove the following corollary:

Corollary Let S be a closed K_{ℂ}-P double coset in G_{ℂ}. Write S = S_{1}^{cl} with the dense K_{ℂ}-B double coset S_{1} in S. Then we have

C(S)_{0} = Ω(J(S_{1})).

Remark 3. (i) We can see C(S_{0})_{0} = Ω(J). By the same argument as for C(S_{0})_{0} in Section 1, we can prove Ω(J′) is Stein for every subset J′ in J. So the Steinness of C(S)_{0} ([W]) becomes a corollary of this equivalence C(S)_{0} = Ω(J(S_{1})) (c.f. [HW]).

(ii) It is clear that Ω(J′) ⊃ Ω(J) for every subset J′ in J. So we have

C(S)_{0} ⊃ C(S_{0})_{0}.

But this inclusion was already proved in Proposition 8.3 in [GM]. This is natural because the way of proof of the corollary below is essentially the same as that of Proposition 8.3 in [GM]. So the above corollary may be considered as its refinement.

Proof of Corollary. Let x be an element on the boundary of C(S)_{0}. Then

xS ∩ S′_{2}P ≠ ϕ

for some G_{ℝ}-P double coset S′_{2}P in the boundary of S′. Here we take S_{2} as the dense K_{ℂ}-B double coset contained in S_{2}P. Since S is right P -invariant, we have

xS ∩ S′_{2} ≠ ϕ and dim_{ℂ} S_{2} > dim_{ℂ} S.

Applying Theorem 1 (ii) to the pair (S_{2}^{cl}, G_{ℂ}), we can take a w ∈ W such that ℓ(w) = codim_{ℂ} S_{2} and that

S_{2}(BwB)^{cl} = G_{ℂ}.

So we have S_{2}(BwB)^{cl} ⊃ S_{0} and hence

S′_{2} ⊂ S′_{0}(Bw^{−1}B)^{cl}.

Since xS ∩ S′_{2} ≠ ϕ, we have

xS ∩ S′_{0}(Bw^{−1}B)^{cl} ≠ ϕ.

Hence

xS(BwB)^{cl} ∩ S′_{0} ≠ ϕ

which implies xT_{j} ∩ S′_{0} ≠ ϕ for some j ∈ J(S_{1}) by Theorem 2. Thus x ∉ Ω(J(S_{1})).

Conversely, suppose

xT_{j} ∩ S′_{0} ≠ ϕ

for some T_{j} = S(BwB)^{cl} = S_{1}^{cl}(BwB)^{cl}. Note that j ∈ J(S_{1}) by Definition 1 and that we may assume ℓ(w) = codim_{ℂ} S − 1 = codim_{ℂ} S_{1} − 1 by Theorem 1 (ii). Then we have

xS ∩ S′_{0}(Bw^{−1}B)^{cl} ≠ ϕ

and hence

xS ∩ S′_{3} ≠ ϕ

for some K_{ℂ}-B double coset S_{3} such that S′_{3} ⊂ S′_{0}(Bw^{−1}B)^{cl}. Hence S_{3}(BwB)^{cl} ⊃ S_{0} and therefore dim_{ℂ} S_{3} ≥ dim_{ℂ} G_{ℂ} − ℓ(w) > dim_{ℂ} S. So we have

S′_{3} ∩ S′ = ϕ

because S′ is the union of G_{ℝ}-B double cosets S′_{4} satisfying S_{4} ⊂ S. Hence we have

xS ⊄ S′

and therefore

x ∉ C(S). q.e.d.

Remark 4. (i) The condition ℓ(w) = codim_{ℂ}S − 1 does “not always” imply

codim_{ℂ} S^{cl}(BwB)^{cl} = 1.

Counter examples exist already for G_{ℝ} = SU(2, 1).

(ii) The construction of the domain Ω(J(S_{1})) is essentially equivalent to the construction of “Schubert domain” in [HW]. Unfortunately, their basic definition needs a correction and after this correction their proof of Corollary 3.2 corresponding to our Corollary is not complete. We can see that the proof of Corollary using the results in Section 2 is extremely simple. Let us explain the connection between these two constructions introducing notations in [HW].

Take a Borel subgroup B_{0} of G_{ℂ} so that G_{ℝ}B_{0} is closed in G_{ℂ}. A Borel subgroup B of G_{ℂ} is called an “Iwasawa Borel subgroup” if

B = g_{0}B_{0}g_{0}^{−1} for some g_{0} ∈ G_{ℝ}.

Let Z = G_{ℂ}/Q be a flag manifold. Then we can take Q so that Q ⊃ B_{0}. Every Schubert cell Y in Z for B is written as

Y = (Bg_{0}wQ)^{cl} = (g_{0}B_{0}wQ)^{cl}

with some w ∈ W. Let C_{0} be a closed K_{ℂ}-Q double coset (do not miss with C_{0} of Section 1!). The “incidence divisor” H_{Y} is written as

H_{Y} = {g | gC_{0} ∩ Y ≠ ϕ} = Y C_{0}^{−1} = (g_{0}B_{0}wQ)^{cl}C_{0}^{−1} = (C_{0}(Qw^{−1}B)^{cl}g_{0}^{−1})^{−1}.

In this point, in [HW], it is written: “If codim Y ≤ dim C_{0} + 1, then H_{Y} is a hypersurface in G_{ℂ}.” It is wrong as we remarked in (i).

But if codim H_{Y} = 1, then

H_{Y}^{−1} = C_{0}(Qw^{−1}B_{0})^{cl}g_{0}^{−1} = T_{j}g_{0}^{−1}

for some j ∈ J′ = J(C_{1}) (where C_{1} is the dense K_{ℂ}-B_{0} double coset in C_{0}) and g_{0} ∈ G_{ℝ} by our notation.

So their definition of y(D) should be corrected to

y(D) = {Y = (g_{0}B_{0}wQ)^{cl} | codim H_{Y} = 1}.

Here D is the G_{ℝ}-orbit dual to C_{0}. (The condition Y ⊂ Z \ D follows from codim H_{Y} = 1 because

Y ∩ D = ϕ ⟺ DY^{−1} = D(Qw^{−1}B_{0})^{cl}g_{0}^{−1} ∌ e
⟺ D(Qw^{−1}B_{0})^{cl} ∌ g_{0}
⟺ D(Qw^{−1}B_{0})^{cl} ∩ G_{ℝ}B_{0} = ϕ
⟺ C_{0}(Qw^{−1}B_{0})^{cl} ∩ K_{ℂ}B_{0} = ϕ
⟺ codim C_{0}(Qw^{−1}B_{0})^{cl} ≥ 1.)

The Schubert domain is defined as

Ω_{S}(D) = {G_{ℂ} \ ( ⋃_{Y ∈y(D)} H_{Y} )}_{0} .

This definition is equivalent to ours because

g ∉ ⋃_{Y ∈y(D)} H_{Y} ⟺ g^{−1} ∉ T_{j}g_{0}^{−1} for all j ∈ J′ and g_{0} ∈ G_{ℝ}
⟺ g^{−1}G_{ℝ}B_{0} ∩ T_{j} = ϕ for all j ∈ J′
⟺ G_{ℝ}B_{0} ∩ gT_{j} = ϕ for all j ∈ J′.

(iii) So in their Corollary 3.2 in [HW], (2) codim_{Z}(Y) = q + 1 does not imply (3) H_{Y} is a hypersurface in Ω. Thus the proof of Corollary 3.2 is incomplete.

Remark 5. The problem of the description of the domain of cycles C(S)_{0} for groups G_{ℝ} of Hermitian type is simpler than the general case. Firstly, in this case, D_{0} = D̃_{0}/K_{ℂ} has a very simple description: D_{0} ≅ G_{ℝ}/K × \overline{G_{ℝ}/K} (Proposition 2.2 in [GM]). As usual, the equality C(S)_{0} = D̃_{0} for S (↔ S′) of nonholomorphic type is reduced to two inclusions. The proof of C(S)_{0} ⊂ D̃_{0} in [WZ1] had a mistake which was corrected in [WZ2]. The opposite inclusion was checked in [WZ1] for classical Hermitian groups. In Proposition 2.4 of [GM], we gave a very simple proof of this inclusion for arbitrary groups of Hermitian type which is free of case-by-case considerations: the use of Schubert cells makes this fact almost trivial. The note [WZ2] also contains this fact with a proof referred to [HW] (which is incomplete as is explained in Remark 4) but without an appropriate reference on the preceding proof in [GM]. Moreover it asserts a misleading statement that the preprint [GM] does not contain a direct proof.

References

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Simon Gindikin, Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghysen Road, Piscataway, NJ 08854-8019, U.S.A.

  E-mail address: gindikin@math.rutgers.edu

Toshihiko Matsuki, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-8501, Japan

  E-mail address: matsuki@math.h.kyoto-u.ac.jp