Schubert varieties as variations of Hodge structure

Abstract

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS ‘span’ the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to the Schubert VHS form a basis of the invariant characteristic cohomology associated with the infinitesimal period relation (a.k.a. Griffiths’ transversality).

Appendices

Appendix A: Examples

This appendix contains examples of compact duals $\check{D}$ and their maximal Schubert variations of Hodge structure for simple Hodge groups. With the exceptions of Examples 5.3 and 5.6, we omit discussion of Hodge representations and the corresponding Hodge numbers, as our focus is primarily on Hodge domains, rather than Mumford–Tate domains. The reader interested in additional examples of Hodge representations and Mumford–Tate domains is encouraged to consult [20].

Remark

(Computing $W^\varphi$) For each of the examples that follow, we must compute the set $W^\varphi$ of Sect. 3.4. In practice, this is most easily done with neither the definition (3.12) nor one of the equivalent formulations of Remark 3.6, but as follows. Let

$$\begin{aligned} \varrho ^\varphi = \sum _{i \in I} \omega _i \, \in \, {\Lambda _{\mathrm {wt}}}. \end{aligned}$$

Define a map $W \rightarrow {\Lambda _{\mathrm {wt}}}$ by $w \mapsto w^{-1}(\varrho ^\varphi )$. When restricted to $W^\varphi \subset W$, this map defines a bijection between the $W$-orbit of $\varrho ^\varphi$ and $W^\varphi$. See [2, §4.3] or [8, §3.2.16], for a proof and examples. (In [2], our $W^\varphi$ and $\varrho ^\varphi$ are, respectively denoted $W^\mathbf {p}$ and $\rho ^\mathbf {p}$; in [8], by $W^{\mathfrak {p}}$ and $\delta ^{\mathfrak {p}}$.)

Remark

(Hermitian symmetric Schubert VHS) In many of the examples the set $\Delta (w)$ associated with a maximal Schubert VHS is of the form

$$\begin{aligned} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_w)=0 \} \end{aligned}$$

for a second grading element $\mathtt {T}_w = \mathtt {T}_{j_1} + \cdots + \mathtt {T}_{j_t}$. In these cases, the corresponding Schubert variety $X_w$ is a homogeneously embedded Hermitian symmetric space. This space may be identified with ‘Dynkin diagram hieroglyphics’ as follows. Begin with the Dynkin diagram of ${\mathfrak {g}}_\mathbb {C}$. Remove all nodes (and the adjacent edges) associated with an index $j_s$ of $\mathtt {T}_w$. We now have a collection of connected Dynkin (sub-)diagrams. Mark the nodes associated with an index $i \in I$ of $\mathtt {T}_\varphi$. Delete any connected component that contains no marked nodes. What remains is the marked Dynkin diagram of $X_w$ as a homogeneously embedded Hermitian symmetric space.

The Dynkin diagrams associated with the irreducible compact Hermitian symmetric spaces are given in Table 1. In each diagram, the number of nodes is the rank $r$ of the algebra ${\mathfrak {g}}_\mathbb {C}$. As an illustration, consider Example 5.3 where ${\mathfrak {g}}_\mathbb {C}= \mathfrak {sp}_{10}\mathbb {C}$ and $\mathtt {T}_\varphi = \mathtt {T}_2 + \mathtt {T}_5$. We begin with the Dynkin diagram

of ${\mathfrak {g}}_\mathbb {C}$. Suppose $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3) = 0\}$. We have $\mathtt {T}_w = \mathtt {T}_3$; so we delete the third node in the diagram to obtain

We then mark the second and fifth nodes, which correspond to $\mathtt {T}_\varphi$, to obtain

Whence, $X_w = {\mathbb P}^2 \times {\mathrm {LG}}(2,\mathbb {C}^4) = {\mathbb P}^2 \times {\mathcal Q}^2$.

Table 1 Dynkin diagrams for irreducible compact Hermitian symmetric spaces

1.1 Unitary Hodge groups $G = A_r$

Throughout we fix a complete flag $\mathbb {C}^\bullet = \{ 0 \subset \mathbb {C}^1 \subset \mathbb {C}^2 \subset \cdots \subset \mathbb {C}^{r+1} \}$.

Example 5.1

Consider ${\mathfrak {g}}_\mathbb {C}= \mathfrak {sl}_6\mathbb {C}$ and $\mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_4$ (equivalently, $I = \{2,4\}$). Then $\check{D}$ may be identified with the partial flag variety

$$\begin{aligned} \mathrm {Flag}_{(2,4)}\mathbb {C}^6 = \{ (E^2,E^4) \in \mathrm {Gr}(2,6) \times \mathrm {Gr}(4,6) \, | \, E^2 \subset E^4 \}. \end{aligned}$$

The underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathfrak {su}(2,4)$. To see this, note that the noncompact simple roots are $\Sigma ({\mathfrak {g}}_\mathrm {odd}) = \{ \sigma _2 ,\, \sigma _4 \}$. The Weyl group element $w = (2312)$ maps

$$\begin{aligned} w\sigma _1 = \sigma _3 ,\ w\sigma _2 = -(\sigma _1+\sigma _2+\sigma _3) ,\ w\sigma _3 = \sigma _1 ,\ w\sigma _4 = \sigma _2+\sigma _3+\sigma _4 ,\ w\sigma _5 = \sigma _5. \end{aligned}$$

That is, $w\Sigma$ is a simple system with a single noncompact root, $w\sigma _2$. It now follows from the Vogan diagram classification [26, VI] that ${\mathfrak {g}}_{\mathbb R}= \mathfrak {su}(2,4)$. The basis $\{\mathtt {T}^w_j\}$ dual to $w\Sigma$ is $\mathtt {T}^w_1 = \mathtt {T}_3-\mathtt {T}_2$, $\mathtt {T}^w_2 = \mathtt {T}_4-\mathtt {T}_2$, $\mathtt {T}^w_3 = \mathtt {T}_1-\mathtt {T}_2+\mathtt {T}_4$, $\mathtt {T}^w_4 = \mathtt {T}_4$ and $\mathtt {T}^w_5 = \mathtt {T}_5$; and

$$\begin{aligned} \mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_4 = 2\,\mathtt {T}^w_4 - \mathtt {T}^w_2 . \end{aligned}$$

We have

$$\begin{aligned} {\mathfrak {g}}_0^\mathrm {ss}\, \simeq \mathfrak {sl}_2\mathbb {C}\,\times \,\mathfrak {sl}_2\mathbb {C}\,\times \,\mathfrak {sl}_2\mathbb {C}, \end{aligned}$$

and $\Delta ({\mathfrak {g}}_1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4)=1 \}$. The maximal Schubert integrals are indexed by $W^\varphi _{{\mathcal J},{\mathrm {max}}} = \{ (2312) ,\, (4521) ,\, (4534)\}$, cf. Section 3.5; the corresponding root sets are

$$\begin{aligned} \Delta (4534)&= \{ \sigma _j + \cdots + \sigma _k \, | \, 3 \le j \le 4 \le k \}\ = \, \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2) = 0 \} ,\\ \Delta (2312)&= \{ \sigma _j + \cdots + \sigma _k \, | \, j \le 2 \le k \le 3 \} \ = \, \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_4)=0 \} ,\\ \Delta (4521)&= \{ \sigma _2 ,\, \sigma _1+\sigma _2 \}\,\cup \,\{ \sigma _4 ,\, \sigma _4+\sigma _5 \} \ = \, \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} . \end{aligned}$$

The three maximal Schubert VHS, each of dimension four, are

$$\begin{aligned} X_{(4534)}&= \{ E^\bullet \, | \, E^2 = \mathbb {C}^2 \} \ = \, \mathrm {Gr}( 2 , \mathbb {C}^6/\mathbb {C}^2) \, \simeq \, \mathrm {Gr}(2,\mathbb {C}^4) ,\\ X_{(2312)}&= \{ E^\bullet \, | \, E^4 = \mathbb {C}^4 \} \ = \, \mathrm {Gr}( 2 , 4),\\ X_{(4521)}&= \{ E^\bullet \, | \, E^2 \subset \mathbb {C}^3 \subset E^4 \} \ = \, {\mathbb P}(\mathbb {C}^3)^* \times {\mathbb P}(\mathbb {C}^6/\mathbb {C}^3) \, \simeq \, {\mathbb P}^2 \times {\mathbb P}^2. \end{aligned}$$

Example 5.2

Consider ${\mathfrak {g}}_\mathbb {C}= \mathfrak {sl}_9\mathbb {C}$ and $\mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_4+\mathtt {T}_7$ (equivalently, $I = \{2,4,7\}$). If $w = (456345)$, then $w\Sigma$ is a simple system with a single noncompact root, $w \sigma _5$. By the Vogan diagram classification [26, VI] of real, semisimple Lie algebras, the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is ${\mathfrak {g}}_{\mathbb R}= \mathfrak {su}(5,4)$.

The compact dual $\check{D}$ may be identified with the partial flag variety

$$\begin{aligned} \mathrm {Flag}_{(2,4,7)}\mathbb {C}^9 = \{ (E^2,E^4,E^7) \in \mathrm {Gr}(2,9) \times \mathrm {Gr}(4,9) \times \mathrm {Gr}(7,9) \, | \, E^2 \subset E^4\subset E^7 \} . \end{aligned}$$

The maximal Schubert VHS are

The second column describes the maximal integral $X_w$ as a homogeneously embedded Hermitian symmetric space; the third column gives the additional condition necessary to distinguish $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4+\mathtt {T}_7)=1 \}$.

1.2 Symplectic Hodge groups $G = C_r$

Fix a nondegenerate, skew-symmetric bilinear form ${\varvec{\varsigma }}$ on $\mathbb {C}^{2r}$. Throughout,

$$\begin{aligned} {\mathrm {LG}}(d,2r) = \{ E \in \mathrm {Gr}(d,{2r}) \, | \, {\varvec{\varsigma }}_{|E} = 0\} \end{aligned}$$

will denote the Lagrangian Grassmannian of $\varsigma$-isotropic $d$-planes $E \subset \mathbb {C}^{2r}$.

Example 5.3

This is a counter-example to [29, Theorem 1.1.1(a)].

Let ${\mathfrak {g}}_\mathbb {C}= \mathfrak {sp}_{10}\mathbb {C}$ and $\mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_5$. The underlying real form of Proposition 2.14 is ${\mathfrak {g}}_{\mathbb R}= \mathfrak {sp}(5,{\mathbb R})$. To see this, let $w = (543545) \in W$. Then

$$\begin{aligned} w\sigma _1 = \sigma _1 ,\ w \sigma _2 = \sigma _2+\cdots +\sigma _5 ,\ w\sigma _3 = \sigma _4 ,\ w\sigma _4 = \sigma _3 ,\ w\sigma _5 = -2(\sigma _3+\sigma _4)-\sigma _5. \end{aligned}$$

In particular, $w\Sigma$ is a simple system with $w\sigma _5(\mathtt {T}_\varphi ) = -1$ odd, and all other $w\sigma _j(\mathtt {T}_\varphi )$ even; that is, $w\sigma _5$ is the unique noncompact root of $w \Sigma$. The claim now follows from the Vogan diagram classification of real Lie algebras [26, VI].

Let $U_{\omega _1} = \mathbb {C}^{10}$ be the irreducible ${\mathfrak {g}}_\mathbb {C}$-module of highest weight $\omega _1$. Then $U_{\omega _1}$ is self-dual. Moreover, $\mathtt {H}_\mathrm {cpt} = 2(\mathtt {T}_1+\mathtt {T}_3+\mathtt {T}_4)$ and $\omega _1 = \sigma _1+\cdots +\sigma _4+\tfrac{1}{2}\sigma _5$. So $\omega _1(\mathtt {H}_\mathrm {cpt}) = 6$ is even. It follows from Remark 2.16 that $U_{\omega _1}$ is real. Set $V_\mathbb {C}= U_{\omega _1}$. The Hodge decomposition (2.13) is $V_\mathbb {C}= V_{3/2} \oplus V_{1/2} \oplus V_{-1/2} \oplus V_{-3/2}$. As ${\mathfrak {g}}_0^\mathrm {ss}= \mathfrak {sl}_2\mathbb {C}\oplus \mathfrak {sl}_3\mathbb {C}$-modules,

$$\begin{aligned} V_{3/2} \, \simeq \, \mathbb {C}^2 \otimes \mathbb {C},\quad V_{1/2} \, \simeq \, \mathbb {C}\otimes {\bigwedge }^2\mathbb {C}^3 \end{aligned}$$

and $V_{-p} \simeq V_p^*$. In particular, the Hodge numbers are $\mathbf {h}= (2,3,3,2)$. The Hodge domain $D = \mathrm {Sp}(5,{\mathbb R})/{H_\varphi }$ is the period domain for these Hodge numbers. Note that, this example satisfies the hypotheses of [29, Theorem 1.1.1(a)].

The maximal Schubert varieties are

The second column describes the maximal integral $X_w$ as a homogeneously embedded Hermitian symmetric space; the third column gives the additional condition necessary to distinguish $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_5)=1 \}$. From (3.16) (or a direct dimension count), we see that the maximal Schubert VHS are of dimensions five and six; the two of dimension six are indexed by $w_1 = (545345)$ and $w_2 = (234123)$. Corollary 3.13 assures us that every IVHS is of dimension at most six. By Lemma 3.2(b), both $w_1 X_{w_1}$ and $w_2 X_{w_2}$ are integral varieties. Moreover, from (3.17), we see that $o$ is a smooth point of both $w_1 X_{w_1}$ and $w_2 X_{w_2}$. Therefore, there exist two distinct integrals of maximal dimension through $o \in D$, contradicting [29, Theorem 1.1.1(a)]. (Indeed, this example is also a counter-example to [29, Theorems 5.1.1 & 5.3.1(a)].)

As noted in Theorem 3.12, the line $\hat{{\mathfrak {n}}}_w \subset {\bigwedge }^{|w|}{\mathfrak {g}}_{-1}$ is a $G_0$-highest weight line of weight $-\varrho _w$. For the two Schubert varieties above, $-\varrho _{w_1} = -4\omega _2 + 4\omega _5$ and $-\varrho _{w_2} = 5\omega _2-2\omega _5$. In particular, both $\mathbf {I}_{w_1}$ and $\mathbf {I}_{w_2}$ are trivial ${\mathfrak {g}}_0^\mathrm {ss}$-modules; equivalently, $\mathbf {I}_{w_1} = \hat{{\mathfrak {n}}}_{w_1}$ and $\mathbf {I}_{w_2} = \hat{{\mathfrak {n}}}_{w_2}$. Therefore, $\mathbf {I}_6 = \hat{{\mathfrak {n}}}_{w_1} \oplus \hat{{\mathfrak {n}}}_{w_2}$.

Example 5.4

If ${\mathfrak {g}}_\mathbb {C}= \mathfrak {sp}_{10}\mathbb {C}$ and $\mathtt {T}_\varphi = \mathtt {T}_2$, then $\check{D} \simeq {\mathrm {LG}}(2,\mathbb {C}^{10})$. The unique maximal element of $W^\varphi _{\mathcal J}$ is $(234123)$, for which $\Delta (234123) = \{ \alpha \in \Delta ^+ \, | \, \alpha (\mathtt {T}_2)=1 ,\ \alpha (\mathtt {T}_5) = 0 \}$. The unique maximal Schubert VHS is $X_{(234123)} = \mathrm {Gr}(2,\mathbb {C}^5)$.

1.3 Spin Hodge groups $G = B_r$ and $D_r$

Fix a nondegenerate, symmetric bilinear form ${\mathbf {q}}$ on $\mathbb {C}^m$, for $m=2r,2r+1$. Throughout,

$$\begin{aligned} {\mathrm {Gr}}_{\mathbf {q}}(d,m) = \{ E \in \mathrm {Gr}(d,m) \, | \, {\mathbf {q}}_{|E} = 0\} \end{aligned}$$

will denote the ${\mathbf {q}}$-isotropic $d$-planes $E \subset \mathbb {C}^{m}$. In particular, ${\mathrm {Gr}}_{\mathbf {q}}(1,m)$ is the smooth quadric hypersurface

$$\begin{aligned} {\mathcal Q}^{m-2} \, \subset \, {\mathbb P}^{m-1} . \end{aligned}$$

We also fix a complete isotropic flag $\mathbb {C}^\bullet = \{ 0 \subset \mathbb {C}^1 \subset \cdots \subset \mathbb {C}^m\}$; that is,

$$\begin{aligned} {\mathbf {q}}(\mathbb {C}^a , \mathbb {C}^{m-a}) = 0. \end{aligned}$$

Example 5.5

Consider ${\mathfrak {g}}_\mathbb {C}= \mathfrak {so}_{11}\mathbb {C}= \mathfrak {b}_5$ and $\mathtt {T}_\varphi = \mathtt {T}_3$ (equivalently, $I = \{3\}$). Then $\check{D}\simeq {\mathrm {Gr}}_{\mathbf {q}}(3,{11})$. By the Vogan diagram classification [26, VI] of real, semisimple Lie algebras, the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathfrak {so}(6,5)$. We have $\Delta ({\mathfrak {g}}_1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_3)=1 \}$. The maximal Schubert VHS are given by:

• (a) $w = (34543)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \}$ and

  $$\begin{aligned} X_w = \{ E \in {\mathrm {Gr}}_{\mathbf {q}}(3,{11}) \, | \, {\mathbb {C}}^2 \subset E \subset \mathbb {C}^9 \} \ \simeq \, {{\mathcal Q}}^5 \subset {{\mathbb P}}^6. \end{aligned}$$

• (b) $w = (345421)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4) \le 1 \}$ and

  $$\begin{aligned} X_w&= \{ E \in \mathrm {Gr}_{\mathbf {q}}(3,{11}) \, | \, \mathrm {dim}\,(E\cap \mathbb {C}^3) \ge 2 ,\ E \subset \mathbb {C}^8 \} ,\\ \mathrm {Sing}\,X_w&= \{{\mathbb {C}}^3\} ,\quad \hbox {(the point }o\in \check{D}). \end{aligned}$$

• (c) $w = (3452312)$ with $\Delta (w) = \{ \alpha \in \Delta ({{\mathfrak {g}}}_1) \, | \, \alpha ({\mathtt {T}}_2+{\mathtt {T}}_5) \le 1 \}$   and

  $$\begin{aligned} X_w&= \{ E \in {\mathrm {Gr}}_{\mathbf {q}}(3,{11}) \, | \, \mathrm {dim}\,(E\cap {\mathbb {C}}^4) \ge 2 ,\ E \subset {\mathbb {C}}^7\} ,\\ \mathrm {Sing}\,X_w&= \{ E \in {\mathrm {Gr}}_{\mathbf {q}}(3,{\mathbb {C}}^{11}) \, | \, E \subset {\mathbb {C}}^4\} \ \simeq \, {{\mathbb P}}^3 . \end{aligned}$$

The singular loci in (b) and (c) indicate that these maximal VHS are not homogeneously embedded Hermitian symmetric space (as the latter are necessarily smooth).

Example 5.6

The following is a counter-example to [29, Theorem 1.1.1(b)].

Let ${\mathfrak {g}}_\mathbb {C}= \mathfrak {so}_{13}=\mathfrak {b}_6$ and $\mathtt {T}_\varphi = \mathtt {T}_3+\mathtt {T}_5$. The compact dual $\check{D}$ is the partial flag $\mathrm {Flag}_{3,5}^{\mathbf {q}}(\mathbb {C}^{13})$ of ${\mathbf {q}}$-isotropic planes. The underlying real form of Proposition 2.14 is ${\mathfrak {g}}_{\mathbb R}= \mathfrak {so}(4,9)$. To see this, let $w = (342312) \in W$. Then

$$\begin{aligned} \begin{array}{lll} w\sigma _1 = \sigma _4,\ &{} w\sigma _2 = -(\sigma _1 + \cdots + \sigma _4),\ &{} w\sigma _3 = \sigma _1,\\ w\sigma _4 = \sigma _2,\ &{} w\sigma _5 = \sigma _3 + \sigma _4 + \sigma _5,\ &{} w\sigma _6 = \sigma _6. \end{array} \end{aligned}$$

Note that $w\sigma _2(\mathtt {T}_\varphi ) = -1$, and all other $w\sigma _j(\mathtt {T}_\varphi )$ are even. That is, $w\sigma _2$ is the unique noncompact root of the simple system $w\Sigma$. It follows from the Vogan diagram classification [26, VI] that ${\mathfrak {g}}_{\mathbb R}= \mathfrak {so}(4,9)$.

Let $U_{\omega _1} = \mathbb {C}^{13}$ be the irreducible ${\mathfrak {g}}_\mathbb {C}$-module of highest weight $\omega _1 = \sigma _1+\cdots +\sigma _6$. Then $U_{\omega _1}$ is self-dual. Additionally, $\mathtt {H}_\mathrm {cpt} = 2(\mathtt {T}_1+\mathtt {T}_2+\mathtt {T}_4+\mathtt {T}_6)$ and $\omega _1$. So $\omega _1(\mathtt {H}_\mathrm {cpt}) = 8$ is even, and Remark 2.16 implies that $U_{\omega _1}$ is real. Set $V_\mathbb {C}= U_{\omega _1}$. The decomposition of (2.13) is $V_2\oplus V_1 \oplus V_0 \oplus V_{-1} \oplus V_{-2}$. As ${\mathfrak {g}}_0^\mathrm {ss}= \mathfrak {sl}_3\mathbb {C}\oplus \mathfrak {sl}_2\mathbb {C}\oplus \mathfrak {sl}_2\mathbb {C}$-modules,

$$\begin{aligned} V_2 \, \simeq \, ({\bigwedge }^2\mathbb {C}^3)\otimes \mathbb {C}\otimes \mathbb {C},\quad V_1 \, \simeq \, \mathbb {C}\otimes \mathbb {C}^2\otimes \mathbb {C},\quad V_0 \, \simeq \, \mathbb {C}\otimes \mathbb {C}\otimes \mathrm {Sym}^2\mathbb {C}, \end{aligned}$$

and $V_{-p} \simeq V_p^*$. In particular, the Hodge numbers are $\mathbf {h}= (3,2,3,2,3)$. The Hodge domain $D = \mathrm {SO}(4,9)/H_\varphi$ is the period domain for these Hodge numbers.

The maximal Schubert varieties are

• (a) $w = (564)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0,\ \alpha (\mathtt {T}_4+\mathtt {T}_6)\le 1 \}$ and

  $$\begin{aligned} X_w = \{ (E^3\subset E^5) \in \check{D} \, | \, E^3 = \mathbb {C}^3 ,\ \mathrm {dim}(E^5\cap \mathbb {C}^5) \ge 4 ,\ E^5 \subset \mathbb {C}^8 \}. \end{aligned}$$

  Note that $\mathrm {Sing}\,X_w = \{ \mathbb {C}^3 \subset \mathbb {C}^5 \} = o \in \check{D}$.

• (b) $w = (565321)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_4)=0 \}$, and $X_w = {\mathbb P}^3\times {\mathcal Q}^3$ is a homogeneously embedded Hermitian symmetric space.

• (c) $w = (342312)$ with $\Delta (w) = \{\alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_5)=0 \}$, and $X_w = \mathrm {Gr}(3,{\mathbb {C}}^5)$ is a homogeneously embedded Hermitian symmetric space.

From (3.16) (or a direct dimension count), we see that the maximal Schubert VHS are of dimensions three and six; the two of dimension six are indexed by $w_1 = (565321)$ and $w_2 = (342312)$. Corollary 3.13 assures us that every IVHS is of dimension at most six. From [29, §4.3], we have $q_1^{even} = h^{3,1}\,h^{4,0} = 2\cdot 3 = 6$; thus, the hypotheses of [29, Theorem 1.1.1(b)] are satisfied. By Lemma 3.2(b), both $w_1 X_{w_1}$ and $w_2 X_{w_2}$ are integral varieties. Moreover, from (3.17), we see that $o$ is a smooth point of both $w_1 X_{w_1}$ and $w_2 X_{w_2}$. Therefore, there exist two distinct integrals of maximal dimension through $o \in D$, contradicting [29, Theorem 1.1.1(b)]. (Indeed, this example is also a counter-example to [29, Theorems 5.1.2(a) & 5.3.1(b)].)

As noted in Theorem 3.12, the line $\hat{{\mathfrak {n}}}_w \subset {\bigwedge }^{|w|}{\mathfrak {g}}_{-1}$ is a $G_0$-highest weight line of weight $-\varrho _w$. For the two Schubert varieties above, $-\varrho _{w_1} = -4\omega _3+6\omega _4-3\omega _5$ and $-\varrho _{w_2} = -5\omega _3+3\omega _5$. In particular, as ${\mathfrak {g}}_0^\mathrm {ss}$-modules, $\mathbf {I}_{w_1}\simeq \mathbb {C}\otimes (\mathrm {Sym}^6\mathbb {C}^2)\otimes \mathbb {C}$ and $\mathbf {I}_{w_2}$ is trivial. In particular, the $G_0$-orbit of ${\mathfrak {n}}_{w_1} \in {\mathbb P}\mathbf {I}_{w_1}$ is naturally identified with the Veronese embedding $v_6({\mathbb P}^1)$, while ${\mathbb P}\mathbf {I}_{w_2} = \{{\mathfrak {n}}_{w_2}\}$. The orbit $v_6({\mathbb P}^1)$ indexes the set $\{ g w_1 X_{w_1} \, | \, g \in G_0 \}$; each variety $g w_1 X_{w_1} = \overline{(g N_{w_1} g^{-1})\cdot o}$ is a distinct VHS, of maximal dimension, containing $o \in D$ as a smooth point.

Example 5.7

Consider ${\mathfrak {g}}_\mathbb {C}= \mathfrak {so}_{10} = \mathfrak {d}_5$ and $\mathtt {T}_\varphi = \mathtt {T}_2$. The compact dual $\check{D} = D_5/P_2$ may be identified with the ${\mathbf {q}}$-isotropic 2-planes $\mathrm {Gr}_{\mathbf {q}}(2,\mathbb {C}^{10}) = \{ E \in \mathrm {Gr}(2,\mathbb {C}^{10}) \, | \, {\mathbf {q}}_{|E}=0\}$. We have $\Delta ({\mathfrak {g}}_1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_2)=1 \}$. The maximal Schubert VHS are given by:

• (a) $w = (235432)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1)=0 \}$ and

  $$\begin{aligned} X_w = \{ E \in \mathrm {Gr}_{\mathbf {q}}(2,{10}) \, | \, \mathbb {C}^1 \subset E \subset \mathbb {C}^9 \} \ \simeq \, {\mathcal Q}^6 . \end{aligned}$$

• (b) $w = (235123)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_5)=0 \}$ and

  $$\begin{aligned} X_w = \{ E \in \mathrm {Gr}_{\mathbf {q}}(2,{10}) \, | \, E \subset \mathbb {C}^5 \} \ \simeq \, \mathrm {Gr}(2,\mathbb {C}^5) . \end{aligned}$$

• (c) $w = (234123)$ with $\Delta (w) = \{ \alpha \in \Delta ({{\mathfrak {g}}}_1) \, | \, \alpha (\mathtt {T}_4)=0 \}$ and

  $$\begin{aligned} X_w = \{ E \in {\mathrm {Gr}}_{\mathbf {q}}(2,{10}) \, | \, E \subset {\tilde{\mathbb {C}}}^5 \} \ \simeq \, \mathrm {Gr}(2,{\mathbb {C}}^5) . \end{aligned}$$

  Here, if $\{e_1,\ldots ,e_{10}\}$ is a basis of $\mathbb {C}^{10}$ adapted to the filtration ${\mathbb {C}}^\bullet$, then $\tilde{\mathbb {C}}^5 = \mathrm {span}_\mathbb {C}\{ e_1,\ldots ,e_4,e_6\}$.

• (d) $w = (235431)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_3)\le 1 \}$ and

  $$\begin{aligned} X_w&= \{ E \in \mathrm {Gr}_{\mathbf {q}}(2,{10}) \, | \, \mathrm {dim}\,(E\cap {\mathbb {C}}^2) \ge 1 ,\ E \subset {\mathbb {C}}^{8} \} ,\\ \mathrm {Sing}\, X_w&= \{ {\mathbb {C}}^2 \} = o \, \in \, \check{D}. \end{aligned}$$

• (e) $w = (235412)$ with $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_4+\mathtt {T}_5) \le 1 \}$ and

  $$\begin{aligned} X_w&= \{ E \in {\mathrm {Gr}}_{\mathbf {q}}(2,{10}) \, | \, \mathrm {dim}\,(E\cap {\mathbb {C}}^3) \ge 1 , \ E \subset {\mathbb {C}}^7 \} ,\\ \mathrm {Sing}\,X_w&= \{ E \subset {\mathbb {C}}^3 \} \, \simeq \, {{\mathbb P}}^2. \end{aligned}$$

1.4 The exceptional Hodge group $G = E_6$

Example 5.8

In the case ${\mathfrak {g}}_\mathbb {C}= \mathfrak {e}_6$ and $\mathtt {T}_\varphi = \mathtt {T}_2$, we have $\mathrm {dim}_\mathbb {C}D = 21$. There are six maximal Schubert VHS, and they are each of dimension ten. The maximal Schubert VHS are given by

The second column gives the additional condition necessary to define $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{\alpha \in \Delta \, | \, \alpha (\mathtt {T}_2)=1\}$; the third column describes the integrals that are homogeneously embedded Hermitian symmetric spaces. Above, ${\mathcal S}_5$ denotes the Spinor variety

$$\begin{aligned} {\mathcal S}_5 \, \mathop {=}\limits ^\mathrm{{dfn}}\, D_5/P_5 = {Spin}_{10}(\mathbb {C}){/}P_5 . \end{aligned}$$

Example 5.9

In the case ${\mathfrak {g}}_\mathbb {C}= \mathfrak {e}_6$ and $\mathtt {T}_\varphi = \mathtt {T}_3$, we have $\mathrm {dim}_\mathbb {C}D = 25$. The maximal Schubert VHS are given by

The second column gives the additional condition necessary to define $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{\alpha \in \Delta \, | \, \alpha (\mathtt {T}_3)=1\}$; the third column describes the integrals that are homogeneously embedded Hermitian symmetric spaces.

Example 5.10

In the case ${\mathfrak {g}}_\mathbb {C}= \mathfrak {e}_6$ and $\mathtt {T}_\varphi = \mathtt {T}_4$, we have $\mathrm {dim}_\mathbb {C}D = 29$. The maximal Schubert VHS are given by

The second column gives the additional condition necessary to define $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{\alpha \in \Delta \, | \, \alpha (\mathtt {T}_4)=1\}$; the third column describes the integrals that are homogeneously embedded Hermitian symmetric spaces.

Example 5.11

In the case ${\mathfrak {g}}_\mathbb {C}= \mathfrak {e}_6$ and $\mathtt {T}_\varphi = \mathtt {T}_5$, we have $\mathrm {dim}_\mathbb {C}D = 25$. The maximal Schubert VHS are given by

The second column gives the additional condition necessary to define $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{\alpha \in \Delta \, | \, \alpha (\mathtt {T}_5)=1\}$; the third column describes the integrals that are homogeneously embedded Hermitian symmetric spaces.

Example 5.12

Consider the exceptional ${\mathfrak {g}}_\mathbb {C}= \mathfrak {e}_6$ and $\mathtt {T}_\varphi = \mathtt {T}_2 + \mathtt {T}_5$, we have $\mathrm {dim}_\mathbb {C}D = 25$. The underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 satisfies ${\mathfrak {k}}_{\mathbb R}= \mathfrak {so}(10) \oplus {\mathbb R}$. To see this, note that the noncompact simple roots are $\Sigma ({\mathfrak {g}}_\mathrm {odd}) = \{ \sigma _2 ,\, \sigma _5 \}$. The Weyl group element $w = (2431)$ maps

$$\begin{aligned} w\sigma _1&= -(\sigma _1+\cdots +\sigma _4) ,\quad w\sigma _2 = \sigma _4 ,\quad w\sigma _3 = \sigma _1 ,\\ w\sigma _4&= \sigma _3 , \quad w\sigma _5 = \sigma _2+\sigma _4+\sigma _5 ,\quad w\sigma _6 = \sigma _6. \end{aligned}$$

That is, $w\Sigma$ is a simple system with a single noncompact root, $w\sigma _1$. The claim now follows from the Vogan diagram classification [26, VI]. The basis $\{\mathtt {T}^w_j\}$ dual to $\omega \Sigma$ is $\mathtt {T}^w_1 = \mathtt {T}_5-\mathtt {T}_2$, $\mathtt {T}^w_2 = \mathtt {T}_4-\mathtt {T}_2$, $\mathtt {T}^w_3 = \mathtt {T}_1-\mathtt {T}_2+\mathtt {T}_5$, $\mathtt {T}^w_4 = \mathtt {T}_3-\mathtt {T}_2+\mathtt {T}_5$, $\mathtt {T}^w_5 = \mathtt {T}_5$ and $\mathtt {T}^w_6 = \mathtt {T}_6$; and

$$\begin{aligned} \mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_5 = 2\,\mathtt {T}^w_5 - \mathtt {T}^w_1 . \end{aligned}$$

The maximal Schubert VHS are given by

The second column gives the additional condition necessary to define $\Delta (w)$ as a subset of $\Delta ({\mathfrak {g}}_1) = \{\alpha \in \Delta \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_5)=1\}$; the third column describes the integrals that are homogeneously embedded Hermitian symmetric spaces.

1.5 The exceptional Hodge group $G = F_4$

In the case that ${\mathfrak {g}}$ is the exceptional Lie algebra $\mathfrak {f}_4$, there are 15 grading elements $\mathtt {T}_\varphi$ of the form (3.10). Each is considered in the examples below; these examples, along with Proposition 3.4, yield

Corollary 5.13

Variations of Hodge structure in $\check{D} = \mathrm {F}_4(\mathbb {C})/P_\varphi$ are of dimension at most 7.

For the exceptional $\mathfrak {f}_4$, then there are two possibilities for the underlying real form ${\mathfrak {g}}_{\mathbb R}= {\mathfrak {k}}_{\mathbb R}\oplus {\mathfrak {q}}_{\mathbb R}$ of Proposition 2.14, cf. [26, p. 416]. The first, denoted $\mathrm {F}\,\mathrm {I}$, has maximal compact subalgebra ${\mathfrak {k}}_{\mathbb R}= \mathfrak {sp}(3)\oplus \mathfrak {su}(2)$. The simple roots may be selected so that the unique noncompact simple root is $\sigma _1$. The second, denoted $\mathrm {F}\,\mathrm {II}$, has maximal compact subalgebra ${\mathfrak {k}}_{\mathbb R}= \mathfrak {so}(9)$. The simple roots may be selected so that the unique noncompact simple root is $\sigma _4$.⁶

Example 5.14

Consider the case $\mathtt {T}_\varphi = \mathtt {T}_1$. We have $\mathrm {dim}_\mathbb {C}D = 15$. The underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$. The two maximal Schubert VHS are given by

$$\begin{aligned} w = (1234232)&\hbox {and}&\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)\le 1 \} \,;\\ w = (1234231)&\hbox {and}&\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4)\le 2 \} . \end{aligned}$$

Example 5.15

Consider the case $\mathtt {T}_\varphi = \mathtt {T}_2$. We have $\mathrm {dim}_\mathbb {C}D = 20$. If $w = (2342321)$, then $w\sigma _1$ is the single noncompact root of $w\Sigma$. Therefore, the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$. The two maximal Schubert VHS are given by

$$\begin{aligned} w = (2341)&\hbox {and}&\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_3)\le 1 \} \,;\\ w = (234232)&\hbox {and}&\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1)=0 \} . \end{aligned}$$

The maximal $X_{(234232)}$ is a homogeneously embedded ${\mathrm {LG}}(3,\mathbb {C}^6)$.

Example 5.16

Consider the case $\mathtt {T}_\varphi = \mathtt {T}_3$. We have $\mathrm {dim}_\mathbb {C}D = 20$. If $w = (34)$, then $w\sigma _4$ is the single noncompact root of $w\Sigma$. Therefore, the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {II}$. The unique maximal Schubert VHS is a homogeneously embedded ${\mathbb P}^2$ given by $w = (34)$ and $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0\}$.

Example 5.17

Consider the case $\mathtt {T}_\varphi = \mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 15$. The underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {II}$. The unique maximal Schubert VHS is a homogeneously embedded ${\mathbb P}^2$ given by $w = (43)$ and $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0\}$.

Example 5.18

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_2$. We have $\mathrm {dim}_\mathbb {C}D = 21$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (1) \in W$. Then $w \sigma _1 = -\sigma _1$, $w\sigma _2 = \sigma _1+\sigma _2$ and $w \sigma _j = \sigma _j$, $j=3,4$. So $w\sigma _1$ is the unique noncompact root of $w \Sigma$. The Vogan diagram classification [26, p. 416] yields ${\mathfrak {g}}_{\mathbb R}= \mathrm {F}\,\mathrm {I}$. The two maximal Schubert VHS are homogeneously embedded Hermitian symmetric spaces given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (1) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} ,\ &{} X_w = {\mathbb P}^1 \,;\\ w = (234232) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1)=0 \} , &{} X_w = {\mathrm {LG}}(3,\mathbb {C}^6) . \end{array} \end{aligned}$$

Example 5.19

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_3$. We have $\mathrm {dim}_\mathbb {C}D = 22$. In this case, the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$. To see this, let $w = (3412321) \in W$. Then $w \sigma _1 = -\sigma _1-2\sigma _2-4\sigma _3-2\sigma _4$, $w\sigma _2 = \sigma _2+2\sigma _3$, $w \sigma _3 = \sigma _4$ and $w \sigma _4 = \sigma _1+\sigma _2+\sigma _3$. So $w\sigma _1$ is the unique noncompact root of $w \Sigma$. The Vogan diagram classification [26, p. 416] yields ${\mathfrak {g}}_{\mathbb R}= \mathrm {F}\,\mathrm {I}$. The two maximal Schubert VHS are homogeneously embedded Hermitian symmetric spaces given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (12) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^2 \,;\\ w = (341) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} , &{} X_w = {\mathbb P}^1\times {\mathbb P}^2 . \end{array} \end{aligned}$$

Example 5.20

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 20$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (12{,}321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The three maximal Schubert VHS are homogeneously embedded Hermitian symmetric spaces given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (412) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^2\times {\mathbb P}^1 \,;\\ w = (431) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} ,\ &{} X_w = {\mathbb P}^1\times {\mathbb P}^2 \,;\\ w = (12321) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_4)=0 \} , &{} X_w = {\mathcal Q}^5 \, \subset \, {\mathbb P}^6 . \end{array} \end{aligned}$$

Example 5.21

Let $\mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_3$. We have $\mathrm {dim}_\mathbb {C}D = 22$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (21) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The two maximal Schubert VHS are homogeneously embedded Hermitian symmetric spaces given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (21) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^2 \,;\\ w = (34) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} , &{} X_w = {\mathbb P}^2 . \end{array} \end{aligned}$$

Example 5.22

Let $\mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 22$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (4{,}321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The four maximal Schubert VHS, three of them homogeneously embedded Hermitian symmetric spaces, are given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (43) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} ,\ &{} X_w = {\mathbb P}^2 \,;\\ w = (231) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_3) \le 1 ,\ \alpha (\mathtt {T}_4)=0 \} ,\ \\ w = (421) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^2\times {\mathbb P}^1 \,;\\ w = (232) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_4)=0 \} , &{} X_w = {\mathcal Q}^3 \, \subset \ {\mathbb P}^4 . \end{array} \end{aligned}$$

Example 5.23

Let $\mathtt {T}_\varphi = \mathtt {T}_3+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 21$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {II}$, let $w = (4) \in W$. We leave it to the reader to confirm that $w\sigma _4$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The two maximal Schubert VHS, both homogeneously embedded ${\mathbb P}^1$s, are given by

$$\begin{aligned} \begin{array}{l@{\quad }l} w = (3) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4)=0 \} \,;\\ w = (4) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} . \end{array} \end{aligned}$$

Example 5.24

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_2+\mathtt {T}_3$. We have $\mathrm {dim}_\mathbb {C}D = 23$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (342321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The two maximal Schubert VHS are given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (2) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^1 \,;\\ w = (341) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} ,\ &{} X_w = {\mathbb P}^1 \times {\mathbb P}^2. \end{array} \end{aligned}$$

Example 5.25

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_2+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 23$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (2321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The three maximal Schubert VHS are given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (42) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^1 \times {\mathbb P}^1 \,;\\ w = (431) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2)=0 \} ,\ &{} X_w = {\mathbb P}^1 \times {\mathbb P}^2\,;\\ w = (232) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_4)=0 \} ,\ &{} X_w = {\mathcal Q}^3 \, \subset \, {\mathbb P}^4. \end{array} \end{aligned}$$

Example 5.26

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_3+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 23$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (412321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The two maximal Schubert VHS are given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (31) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4)=0 \} ,\ &{} X_w = {\mathbb P}^1 \times {\mathbb P}^1\,;\\ w = (412) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^2\times {\mathbb P}^1. \end{array} \end{aligned}$$

Example 5.27

Let $\mathtt {T}_\varphi = \mathtt {T}_2+\mathtt {T}_3+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 23$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The two maximal Schubert VHS are given by

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} w = (3) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4)=0 \} ,\ &{} X_w = {\mathbb P}^1\,;\\ w = (421) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_3)=0 \} ,\ &{} X_w = {\mathbb P}^2\times {\mathbb P}^1. \end{array} \end{aligned}$$

Example 5.28

Let $\mathtt {T}_\varphi = \mathtt {T}_1+\mathtt {T}_2+\mathtt {T}_3+\mathtt {T}_4$. We have $\mathrm {dim}_\mathbb {C}D = 24$. To see that the underlying real form ${\mathfrak {g}}_{\mathbb R}$ of Proposition 2.14 is $\mathrm {F}\,\mathrm {I}$, let $w = (42321) \in W$. We leave it to the reader to confirm that $w\sigma _1$ is the unique noncompact root of $w\Sigma$; the claim then follows from the Vogan diagram classification [26, p. 416]. The three maximal Schubert VHS, all homogeneously embedded ${\mathbb P}^1\times {\mathbb P}^1$s, are given by

$$\begin{aligned} \begin{array}{l@{\quad }l} w = (31) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_4)=0 \} \,;\\ w = (41) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_2+\mathtt {T}_3)=0 \} \,;\\ w =(42) ,\ &{} \Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_1+\mathtt {T}_3)=0 \} . \end{array} \end{aligned}$$

1.6 The exceptional Hodge group $G = G_2$

In the case that ${\mathfrak {g}}$ is the exceptional Lie algebra ${\mathfrak {g}}_2$, there are 3 grading elements $\mathtt {T}_\varphi$ of the form (3.10). Each is considered in the examples below; these examples, along with Proposition 3.4, yield

Corollary 5.29

Variations of Hodge structure in $\check{D} = \mathrm {G}_2(\mathbb {C})/P_\varphi$ are of dimension at most 2.

For the exceptional Lie algebra ${\mathfrak {g}}_2$, Vogan diagram classification [26, p. 416] implies that the underlying real form ${\mathfrak {g}}_{\mathbb R}= {\mathfrak {k}}_{\mathbb R}\oplus {\mathfrak {q}}_{\mathbb R}$ of Proposition 2.14 is the split real form with maximal compact subalgebra ${\mathfrak {k}}_{\mathbb R}= \mathfrak {su}(2) \oplus \mathfrak {su}(2)$.

Example 5.30

Suppose ${\mathfrak {g}}_\mathbb {C}= {\mathfrak {g}}_2$ and $\mathtt {T}_\varphi = \mathtt {T}_1$. We have $\mathrm {dim}_\mathbb {C}D = 5$. The unique maximal integral $X_{(1)}$ is a homogeneously embedded ${\mathbb P}^1$ given by $\Delta (1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_1)=1,\ \alpha (\mathtt {T}_2)=0 \}$.

Example 5.31

Suppose ${\mathfrak {g}}_\mathbb {C}= {\mathfrak {g}}_2$ and $\mathtt {T}_\varphi = \mathtt {T}_2$. We have $\mathrm {dim}_\mathbb {C}D = 5$. There is a unique maximal Schubert VHS, $X_{(21)}$, with $\Delta (21) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_2) = 1 ,\ \alpha (\mathtt {T}_1)\le 1\}$.

Example 5.32

Suppose ${\mathfrak {g}}_\mathbb {C}= {\mathfrak {g}}_2$ and $\mathtt {T}_\varphi = \mathtt {T}_1 + \mathtt {T}_2$. We have $\mathrm {dim}_\mathbb {C}D = 5$. There are two maximal Schubert VHS, $X_{(1)}$ and $X_{(2)}$, both are homogeneously embedded ${\mathbb P}^1$’s given by $\Delta (1) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_1)=1,\ \alpha (\mathtt {T}_2)=0 \}$ and $\Delta (2) = \{ \alpha \in \Delta \, | \, \alpha (\mathtt {T}_2)=1,\ \alpha (\mathtt {T}_1)=0 \}$.

Appendix B: Hodge representations of Calabi–Yau type

Definition

A Hodge structure $\varphi$ of weight $n$ on $V$ is of Calabi–Yau type if $\mathrm {dim}_\mathbb {C}V^{m/2} = 1$ in the decomposition (2.13).⁷ A Hodge representation, given by $\rho : G \rightarrow \mathrm {Aut}(V,Q)$ and $\varphi : S^1 \rightarrow G_{\mathbb R}$, is of Calabi–Yau type if the Hodge structure $\rho \circ \varphi$ on $V$ is of Calabi–Yau type.

Proposition 6.1

Let $G$ be a Hodge group. Let $\rho : G \rightarrow \mathrm {Aut}(V,Q)$ and $\varphi : S^1 \rightarrow G_{\mathbb R}$ define a Hodge representation. Without loss of generality, the grading element $\mathtt {T}_\varphi = \sum n_i \mathtt {T}_i$ associated with the circle $\varphi$ by (2.12) is of the form (2.22).⁸ Assume $V_{\mathbb R}$ is irreducible, and let $U$ be the associated irreducible, complex $G$-representation of highest weight $\mu = \mu ^i\omega _i$ (cf. Remark 2.16). Then, the Hodge representation is of Calabi–Yau type if and only if $\{ i \, | \, n_i=0\} \subset \{ i \, | \, \mu ^i =0\}$, and one of the following holds:

• (a) The representation $U$ is real. Equivalently, $\mu = \mu ^*$, and $\sum _{n_i \in 2\mathbb Z} \mu (\mathtt {T}_i)$ is an integer.

• (b) The inequality $\mu (\mathtt {T}_\varphi ) \not = \mu ^*(\mathtt {T}_\varphi )$ holds. (In this case, $U$ is necessarily complex.)

Remark

Assume that $D = G_{\mathbb R}/H_\varphi$ is Hermitian symmetric. In [17, Lemma 2.27], Friedman–Laza identify $\{ i \, | \, n_i=0\} \subset \{ i \, | \, \mu ^i =0\}$ as a necessary condition for the Hodge structure $\rho \circ \varphi$ to be of Calabi–Yau type, and show that $U$ is either real or complex.

Example 6.2

Let ${\mathfrak {g}}$ be one of the exceptional Lie algebras $\mathfrak {e}_8$, $\mathfrak {f}_4$ or ${\mathfrak {g}}_2$. Fix a lattice ${\Lambda _{\mathrm {rt}}}\subset \Lambda \subset {\Lambda _{\mathrm {wt}}}$ and Lie group $G = G_\Lambda$ with Lie algebra ${\mathfrak {g}}$ (cf. Sect. 2.3). By Remark 2.15, all representations $U$ of ${\mathfrak {g}}_\mathbb {C}$ are self-dual. Therefore, by Remark 2.16, all irreducible representations $U$ of ${\mathfrak {g}}_\mathbb {C}$ are either real or quaternionic. Moreover, $\omega _i(\mathtt {T}_j) \in \mathbb Z$ for all $i,j$, cf. [26, Appendix C.2]. That is, for any grading element $\mathtt {T}\in \mathrm {Hom}({\Lambda _{\mathrm {rt}}},\mathbb Z)$, we have $\mathtt {T}\in \mathrm {Hom}(\Lambda ,\mathbb Z)$. This has two consequences:

$\circ$ :

Given any irreducible $G_\mathbb {C}$–representation $U$ of highest weight $\mu \in \Lambda$, the integer $\mu (\mathtt {H}_\mathrm {cpt})$ is even, and it follows from Remark 2.16 that $U$ is real. Thus, $U = V_\mathbb {C}$, for some real vector space $V$.

$\circ$ :

Let $\varphi : S^1 \rightarrow T = \mathfrak {t}/\Lambda ^*$ be the homomorphism of ${\mathbb R}$-algebraic groups determined by $\mathtt {T}$, cf. §2.3. Suppose that (2.22) holds. Then, Theorem 2.17 implies that $(V,\varphi )$ admits the structure of a Hodge representation. Whence, Proposition 6.1 implies that the Hodge representation $(V,\varphi )$ is of Calabi–Yau type if and only if $\mu ^i = 0$ for all $i$ such that $n_i=0$.

Set

$$\begin{aligned} I^\mu = \{ i \, | \, \mu ^i > 0 \} = \{ i \, | \, (\sigma _i,\mu ) \not = 0 \} \quad \hbox {and}\quad I^\varphi = \{ i \, | \, n_i > 0 \} . \end{aligned}$$

Note that

$$\begin{aligned} \{ i \, | \, n_i=0\} \,\subset \, \{ i \, | \, \mu ^i =0\} \quad \hbox {if and only if}\quad I^\mu \subset I^\varphi . \end{aligned}$$

(6.1)

Let $\mathtt {T}_\mu$ be the grading element (2.10) associated with $I^\mu$. Let

$$\begin{aligned} U = \oplus \,U_k^\mu \quad \hbox { and }\quad U = \oplus \,U_\ell \end{aligned}$$

respectively, be the $\mathtt {T}_\mu$ and $\mathtt {T}_\varphi$ graded decompositions (2.4). Set

$$\begin{aligned} {\mathsf {m}}= \mu (\mathtt {T}_\mu )\quad \hbox { and } \quad \mathsf {p}= \mu (\mathtt {T}_\varphi ). \end{aligned}$$

Then, the highest weight line of $U$ is contained in $U^\mu _{\mathsf {m}}\,\cap \, U_\mathsf {p}$.

Lemma 6.3

The highest weight line of $U$ is $U^\mu _{\mathsf {m}}$. Thus, $U^\mu _{\mathsf {m}}\subset U_\mathsf {p}$. Equality holds if and only if $I^\mu \subset I^\varphi$.

Proof of lemma

The lemma is a consequence of two facts. First, note that every weight $\lambda$ of $U$ is of the form $\lambda = \mu -(\sigma _{j_1} + \cdots + \sigma _{j_s})$ for some sequence $\{\sigma _{j_\ell }\}$ of simple roots with the property that each $\mu -(\sigma _{j_1} + \cdots + \sigma _{j_t})$ is also a weight for $1 \le t \le s$. (This is an elementary result from representation theory; see, for example, [26].)

Second, let ${\mathfrak {p}}_\mu$ be the parabolic subalgebra (2.7) determined by $\mathtt {T}_\mu$. Then ${\mathfrak {p}}_\mu$ is the stabilizer of the highest weight line, cf. [8, Proposition 3.2.5 or Theorem 3.2.8]. Consequently, given a simple root $\sigma _i \in \Sigma$, $\lambda =\mu -\sigma _i$ is a weight of $U$ if and only if ${\mathfrak {g}}_{-\sigma _i} \not \subset {\mathfrak {p}}_\mu$; equivalently, $i \in I^\mu$.

These two facts imply that any weight $\lambda \not = \mu$ of $U$ is necessarily of the form $\lambda = \mu - \sigma _i - (\sigma _{j_2} + \cdots + \sigma _{j_s})$ for some $i \in I^\mu$. Therefore, $\lambda (\mathtt {T}_\mu ) = \mu (\mathtt {T}_\mu ) - 1 - (\sigma _{j_2}+\cdots \sigma _{j_s})(\mathtt {T}_\mu ) \le {\mathsf {m}}- 1$. Therefore, $U^\mu _{\mathsf {m}}$ is necessarily the highest weight line.

The discussion preceding the statement of the lemma yields $U^\mu _{\mathsf {m}}\subset U_\mathsf {p}$. Moreover, $\lambda (\mathtt {T}_\varphi ) = \mathsf {p}- (\sigma _i+\sigma _{j_2}+\cdots +\sigma _{j_s})(\mathtt {T}_\varphi )$. So, $U^\mu _{\mathsf {m}}= U_\mathsf {p}$ holds if and only if $I^\mu \subset I^\varphi$. $\square$

Set $-{\mathsf {q}}= -\mu ^*(\mathtt {T}_\varphi )$. By Remark 2.15, the lowest weight line of $U$ is contained in $U_{-{\mathsf {q}}}$. In particular, the $\mathtt {T}_\varphi$-graded decompositions (2.4) of $U$ and $U^*$ are

$$\begin{aligned} U&= U_{\mathsf {p}} \oplus U_{\mathsf {p}-1} \oplus \cdots \oplus U_{1-{\mathsf {q}}} \oplus U_{-{\mathsf {q}}} ,\\ U^*&= U^*_{{\mathsf {q}}} \oplus U^*_{{\mathsf {q}}-1} \oplus \cdots \oplus U^*_{1-\mathsf {p}} \oplus U^*_{-\mathsf {p}} . \end{aligned}$$

The proof of Proposition 6.1 breaks into three cases.

Proof of Proposition B.1 when U is real

First, as noted in Remark 2.16, $U$ is real if and only if $\mu = \mu ^*$ and $\mu (\mathtt {H}_\mathrm {cpt})$ is even. Since $\mathtt {H}_\mathrm {cpt} = 2 \sum _{n_i \in 2\mathbb Z} \mathtt {T}_i$, the latter is equivalent to $\sum _{n_i \in 2\mathbb Z} \mu (\mathtt {T}_i) \in \mathbb Z$.

Given $U$ real, we have $V_\mathbb {C}=U$. So $V_{m/2} = U_\mathsf {p}$. By definition, the Hodge representation is of Calabi–Yau type if and only if $\mathrm {dim}_\mathbb {C}U_\mathsf {p}=1$. By Lemma 6.3, $\mathrm {dim}_\mathbb {C}U_\mathsf {p}= 1$ if and only if $I^\mu \subset I^\varphi$. Thus, $(V,\rho \circ \varphi )$ is a Hodge structure of Calabi–Yau type if and only if $I^\mu \subset I^\varphi$. The proposition now follows from (6.1). $\square$

Proof of Proposition B.1 when U is quaternionic

As noted in Remark 2.16, $U$ is quaternionic if and only if $V_\mathbb {C}= U \oplus U^*$ and $U \simeq U^*$. Then $\mathsf {p}= {\mathsf {q}}$ so that $V_{m/2} = U_\mathsf {p}\oplus U_\mathsf {p}$ is of dimension at least two. Therefore, $(V,\rho \circ \varphi )$ is not a Hodge structure of Calabi–Yau type. $\square$

Proof of Proposition B.1 when U is complex

As noted in Remark 2.16, $U$ is complex if and only if $V_\mathbb {C}= U \oplus U^*$ and $U \not \simeq U^*$. Swapping $U$ and $U^*$ if necessary, we may assume that $\mathsf {p}\ge {\mathsf {q}}$. Then, $V_{m/2} = U_\mathsf {p}\oplus U^*_\mathsf {p}$, and $U^*_\mathsf {p}=0$ if and only if $\mathsf {p}> {\mathsf {q}}$. Therefore, if $\mathsf {p}={\mathsf {q}}$, then $\mathrm {dim}_\mathbb {C}V_{m/2} >1$ and $(V,\rho \circ \varphi )$ is not of Calabi–Yau type. If $\mathsf {p}>{\mathsf {q}}$, then Lemma 6.3 implies that $(V,\rho \circ \varphi )$ gives a Hodge structure of Calabi–Yau type if and only if $I^\mu \subset I^\varphi$. The proposition now follows from (6.1). $\square$

This completes the proof of Proposition 6.1.

Appendix C: Proof of Proposition 3.11

1.1 Preliminaries

Let $I \subset \{ 1,\ldots ,r\}$ be the index set associated with $\mathtt {T}_\varphi$, cf. (3.10) so that

$$\begin{aligned} \mathtt {T}_\varphi = \sum _{i \in I} \mathtt {T}_i . \end{aligned}$$

Let $X_w$ be a maximal Schubert VHS. By Corollary 8.3, this is equivalent to: $\Delta (w)$ is maximal, with respect to containment, among the $\{ \Delta (w') \, | \, w' \in W^\varphi _{\mathcal J}\}$. By Lemma 3.3, the IVHS ${\mathfrak {n}}_w$, defined by (3.15), is abelian; equivalently,

$$\begin{aligned} \hbox {if }\alpha ,\beta \in \Delta (w),\hbox { then }\alpha +\beta \hbox { is not a root.} \end{aligned}$$

(7.1)

By (3.11), we may decompose $\Delta (w)$ as the disjoint union $\sqcup _{i\in I} \Delta _i(w)$ where

$$\begin{aligned} \Delta _i(w) = \{ \alpha \in \Delta (w) \, | \, \alpha (\mathtt {T}_i) = 1 \}. \end{aligned}$$

In our descriptions of the set $\Delta _i(w)$ below, it is essential to keep in mind that $\alpha (\mathtt {T}_j) = 0$ for all $\alpha \in \Delta _i(w)$ and $i\not = j \in I$. (This is due to the fact that the positive roots are of the form $m^a\sigma _a$ with $0 \le m^a \in \mathbb Z$, cf. [26].) Write

$$\begin{aligned} I = \{ i_1 < i_2 < \cdots < i_t \} . \end{aligned}$$

It will be convenient to set

$$\begin{aligned} i_0 \,= \, 0 \quad \hbox {and}\quad i_{t+1} \,=\, r+1 . \end{aligned}$$

1.2 The case $G_\mathbb {C}= \mathrm {SL}_{r+1}\mathbb {C}$

We will use throughout the proof the standard representation theoretic result that the positive roots of $\mathfrak {sl}_{r+1}\mathbb {C}$ are of the form

$$\begin{aligned} \Delta ^+ = \{ \sigma _j + \cdots + \sigma _k \, | \, 1 \le j \le k \le r \} , \end{aligned}$$

(7.2)

cf. [26]. In particular,

$$\begin{aligned} \Delta _{i_s}(w) \, \subset \, \{ \sigma _j + \cdots + \sigma _k \, | \, i_{s-1} \,<\, j \,\le \, i_s \,\le \, k \,<\, i_{s+1} \}. \end{aligned}$$

(7.3)

Step 1:

Suppose that $\Delta _i(w)$ is empty for some $i\in I$. Then $\Delta (w) \subset \Delta (\mathfrak {a}_{i-1} \times \mathfrak {a}_{r-i}) = \Delta (\mathfrak {a}_{i-1}) \cup \Delta (\mathfrak {a}_{r-i})$, where $\mathfrak {a}_{i-1} \times \mathfrak {a}_{r-i} = \mathfrak {sl}_i\mathbb {C}\times \mathfrak {sl}_{r+1-i}\mathbb {C}$ is the semisimple subalgebra of ${\mathfrak {g}}_\mathbb {C}$ generated by the simple roots $\Sigma \backslash \{ \sigma _i\}$. Therefore, $X_w = X_{w^1} \times X_{w^2} \subset (\mathrm {SL}_i\mathbb {C}/P_{I^1}) \times (\mathrm {SL}_{r+1-i}/P_{I^2})$, where $\Delta (w^1) = \Delta (w) \cap \Delta (\mathfrak {a}_{i-1})$ and $\Delta (w^2) = \Delta (w) \cap \Delta (\mathfrak {a}_{r-i})$, and $I^1 = \{ j \in I \, | \, j < i \}$ and $I^2 = \{ j \in I \, | \, i < j \}$. The Schubert variety $X_w$ is Hermitian symmetric if and only if $X_{w^1}$ and $X_{w^2}$ are Hermitian symmetric. So, without loss of generality, we may restrict to the case that $\Delta _i(w)$ is nonempty for all $i \in I$.

Step 2: Suppose that $i_s+1 = i_{s+1}$ for some $1 \le s \le t-1$. By (7.3), any pair of roots $\alpha \in \Delta _{i_s}(w)$ and $\beta \in \Delta _{i_{s+1}}(w)$ is of the form

$$\begin{aligned} \alpha&= \sigma _j + \cdots + \sigma _{i_s} , \quad \hbox { for some } i_{s-1} < j \le i_s ,\\ \beta&= \sigma _{i_{s+1}} + \cdots + \sigma _k , \quad \hbox { for some } i_{s+1} \le k < i_{s+2}. \end{aligned}$$

Therefore, $\alpha +\beta$ is a root, contradicting (7.1). Therefore,

$$\begin{aligned} i_s + 1 \, < \, i_{s+1} . \end{aligned}$$

Step 3: Suppose that $i_{s-1}<j < i_s\le k < i_{s+1}$, and $\beta = \sigma _j + \cdots + \sigma _k \in \Delta (w)$. I claim that $\beta -\sigma _j \in \Delta (w)$ as well. To see this, suppose the converse. Then $\beta -\sigma _j \in \Delta ^+\backslash \Delta (w)$. Also, $\sigma _j \in \Delta ^+({\mathfrak {g}}_0) \subset \Delta ^+\backslash \Delta (w)$. Since $\Delta ^+\backslash \Delta (w)$ is closed, cf. Remark 3.7(c), this implies $\beta \not \in \Delta (w)$, a contradiction. Therefore, $\beta -\sigma _j\in \Delta (w)$. Similarly, if $i_s < k$, then $\beta -\sigma _k = \sigma _j + \cdots + \sigma _{k-1} \in \Delta (w)$. It follows by induction that, if $\sigma _j + \cdots + \sigma _k \in \Delta (w)$, then $\sigma _{j'} + \cdots + \sigma _{k'} \in \Delta (w)$ for all $j \le j' \le i_s \le k' \le k$.

Step 4: With respect to the expression (7.3), let $j_s$ be the minimal $j$, and $k_s$ the maximal $k$, as $\alpha$ ranges over $\Delta _{i_s}(w)$. Suppose that $k_s + 1 \ge j_{s+1}$. Let $\alpha \in \Delta _{i_s}(w)$ be a root realizing the maximum value $k_s$, and $\beta \in \Delta _{i_{s+1}}(w)$ be a root realizing the minimum value $j_{s+1}$. Then $\alpha = \sigma _a + \cdots + \sigma _{k_s}$ and $\beta = \sigma _{j_{s+1}} + \cdots + \sigma _b$ for some $i_{s-1} < a \le i_s$ and $i_{s+1} \le b < i_{s+2}$. By Step 3, $\gamma = \sigma _{k_s+1} + \cdots +\sigma _b \in \Delta (w)$. However, $\alpha +\gamma$ is a root, contradicting (7.1). We conclude that

$$\begin{aligned} k_s + 1 \, < \, j_{s+1} . \end{aligned}$$

(7.4)

Step 5: Set $\Phi _s = \{ \sigma _j + \cdots + \sigma _k \, | \, j_s \le j \le i_s \le k \le k_s \}$. Then $\Delta _{i_s}(w) \subset \Phi _s$ and $\Delta (w) \subset \Phi = \cup _{1\le s \le t} \Phi _s \subset \Delta ({\mathfrak {g}}_1)$. Moreover, (7.2) and (7.4) imply that both $\Phi$ and $\Delta ^+\backslash \Phi$ are closed. Therefore, by Remark 3.7(c), $\Phi = \Delta (w')$ for some $w' \in W^\varphi$. By (3.19), $w' \in W^\varphi _{\mathcal J}$. It follows from the maximality of $\Delta (w)$ (cf. Sect. 7.1) that $\Delta (w) = \Phi$. That is,

$$\begin{aligned} \Delta _{i_s}(w) = \{ \sigma _j + \cdots + \sigma _k \, | \, j_s \le j \le i_s \le k \le k_s \}. \end{aligned}$$

(7.5)

Step 6: Let $s < t$. Steps 3 and 4 imply $\alpha = \sigma _{i_s} + \cdots + \sigma _{k_s+1} \in \Delta ({\mathfrak {g}}_1)\backslash \Delta (w)$. Suppose that $k_s + 2 < j_{s+1}$. Let $\Phi = \Delta (w) \cup \{ \alpha \}$. I claim that $\Phi = \Delta (w')$ for some $w' \in W^\varphi _{\mathcal J}$. This will contradict the maximality of $\Delta (w)$, and taken with (7.4) allows us to conclude that

$$\begin{aligned} k_s + 2 = j_{s+1} \quad \hbox { for all } \quad 1 \le s \le t-1 . \end{aligned}$$

(7.6)

To prove the claim, first note that $\Phi \subset \Delta ({\mathfrak {g}}_1)$. So, if $\Phi = \Delta (w')$ for some $w' \in W^\varphi$, then it follows immediately from (3.19) that $w' \in W^\varphi _{\mathcal J}$. By Remark 3.7(c), $\Phi = \Delta (w')$ for some $w' \in W^\varphi$ if and only if both $\Phi$ and $\Delta ^+{\backslash }\Phi$ are closed. Since $\Delta (w)$ is closed itself, $\Phi$ can fail to be closed only if $\alpha +\beta \in \Delta$ for some $\beta \in \Delta (w)$. However, it follows from (7.2) and the hypothesis $k_s + 2 < j_{s+1}$, that $\alpha +\beta$ is not a root for any $\beta \in \Delta (w)$. Thus, $\Phi$ is closed.

Similarly, since $\Delta ^+{\backslash }\Delta (w)$ is closed [Remark 3.7(c)], to see that $\Delta ^+\backslash \Phi = \Delta ^+\backslash ( \Delta (w) \cup \{\alpha \} )$ is closed, it suffices to show that there exist no $\beta \in \Delta ({\mathfrak {g}}_1){\backslash }\Phi$ and $\gamma \in \Delta ^+({\mathfrak {g}}_0)$ such that $\beta +\gamma = \alpha$. By (7.2), any such pair would necessarily be of the form

$$\begin{aligned} \beta = \sigma _{i_s} + \cdots + \sigma _b \quad \hbox {and}\quad \gamma \,=\, \sigma _{b+1} + \cdots + \sigma _{k_s+1} , \end{aligned}$$

for some $i_s \le b \le k_s$. By (7.5), $\sigma _{i_s} + \cdots + \sigma _b \in \Delta (w)\subset \Phi$ for all $i_s \le b \le k_s$. Thus, there exists no such $\beta \in \Delta ({\mathfrak {g}}_1) \backslash \Phi$, and we may conclude that $\Delta ^+\backslash \Phi$ is closed.

Step 7: Define

$$\begin{aligned} A = \{ a_{s-1} = j_s - 1 \, | \, 2 \le s \le t\} \, \mathop {=}\limits ^{(C.6)} \, \{ a_s = k_s + 1 \, | \, 1 \le s \le t-1 \}. \end{aligned}$$

From (7.5) and (7.6), we deduce that $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_a)=0 \, \forall \, a \in A \} = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_A)=0 \}$.⁹ From this, it follows that

$$\begin{aligned} \begin{array}{lll} X_w &{} = &{} \mathrm {Gr}(i_1 , a_1) \, \times \mathrm {Gr}( i_2-a_1 , a_2 - a_1) \, \times \, \cdots \\ &{} &{} \cdots \, \times \, \mathrm {Gr}(i_{t-1} - a_{t-2} , a_{t-1}-a_{t-2}) \, \times \, \mathrm {Gr}( i_t - a_{t-1} , r-a_{t-1} ) . \end{array} \end{aligned}$$

(7.7)

This establishes the proposition in the case that $G_\mathbb {C}= \mathrm {SL}_{r+1}\mathbb {C}$.

Remark 7.1

It is can be checked that the maximal Schubert VHS $X_w$ satisfying the assumption of Step 1 (that the $\Delta _i(w) \not =\emptyset$ for all $i$) are indexed by subsets $A = \{ a_1 < \cdots < a_{t-1} \}$ satisfying $i_s < a_s < i_{s+1}$ for all $1 \le s \le t-1$.

1.3 The case $G_\mathbb {C}= \mathrm {Sp}_{2r}\mathbb {C}$

We will use throughout the proof the standard representation theoretic result that the positive roots of $\mathfrak {sp}_{r+1}\mathbb {C}$ are of the form

$$\begin{aligned} \Delta ^+&= \{ \sigma _j + \cdots + \sigma _k \, | \, 1 \le j \le k \le r \} \\&\quad \cup \, \{ \sigma _j + \cdots + \sigma _{k-1} + 2(\sigma _k + \cdots +\sigma _{r-1}) + \sigma _r \ | \, 1 \le j \le k \le r-1 \}\\&= \{ \sigma _{j,k} \, | \, 1 \le j \le k \le r \} \, \cup \, \{ \sigma _{j,r-1} + \sigma _{k,r} \, | \, 1 \le j \le k \le r-1 \} ,\\&\hbox {where} \quad \sigma _{j,k} = \sigma _j + \cdots + \sigma _k , \, 1 \le j \le k \le r , \end{aligned}$$

and we employ the convention that $\sigma _{r,r-1}=0$; cf. [26]. In particular,

$$\begin{aligned} \begin{array}{llll} \hbox {if }s<t, &{} \Delta _{i_s}(w) &{} \subset &{} \{ \sigma _{j,k} \, | \, i_{s-1} \,<\, j \,\le \, i_s \,\le \, k \,<\, i_{s+1} \} \,;\\ \hbox {if }i_t<r, &{} \Delta _{i_t}(w) &{} \subset &{} \{ \sigma _{j,k} \, | \, i_{t-1} \,<\, j \,\le \, i_t \,\le \, k \,\le \, r \} \\ &{} &{} &{} \quad \cup \, \ \{ \sigma _{j,r-1} + \sigma _{k,r} \, | \, i_{t-1} \,<\, j \,\le \, i_t \,<\, k \,\le \, r-1\} \,;\\ \hbox {if }i_t=r, &{} \Delta _{i_t}(w) &{} \subset &{} \{ \sigma _{j,r-1} + \sigma _{k,r} \, | \, i_{t-1} \,<\, j , k \,\le \, r\} . \end{array} \end{aligned}$$

(7.8)

Step 1: Suppose $\Delta _i(w)$ is empty, for some $i \in I$. Then $\Delta (w) \subset \Delta (\mathfrak {a}_{i-1} \times {\mathfrak {c}}_{r-i}) = \Delta (\mathfrak {a}_{i-1}) \cup \Delta ( {\mathfrak {c}}_{r-i})$, where $\mathfrak {a}_{i-1} \times {\mathfrak {c}}_{r-i} = \mathfrak {sl}_i\mathbb {C}\times \mathfrak {sp}_{2(r-i)}\mathbb {C}$ is the semisimple subalgebra of ${\mathfrak {g}}_\mathbb {C}$ generated by the simple roots $\Sigma \backslash \{\sigma _i\}$. Therefore, $X_w = X_{w^1} \times X_{w^2} \subset (\mathrm {SL}_i\mathbb {C}/P_{I^1}) \times (\mathrm {Sp}_{2(r-i)}/P_{I^2})$, where $\Delta (w^1) = \Delta (w) \cap \Delta (\mathfrak {a}_{i-1})$ and $\Delta (w^2) = \Delta (w) \cap \Delta ({\mathfrak {c}}_{r-i})$, and $I^1 = \{ j \in I \, | \, j < i \}$ and $I^2 = \{ j \in I \, | \, i < j \}$. The Schubert variety $X_w$ is homogeneous if and only if $X_{w^1}$ and $X_{w^2}$ are homogeneous. So, without loss of generality, we may restrict to the case that $\Delta _i(w)$ is nonempty for all $i \in I$.

Step 2: Suppose that $i_s+1 = i_{s+1}$ for some $1 \le s \le t-1$. Then (7.8) implies that, for any pair of roots $\alpha \in \Delta _{i_s}(w)$ and $\beta \in \Delta _{i_{s+1}}(w)$, we have $\alpha +\beta \in \Delta$. This contradicts (7.1). Therefore,

$$\begin{aligned} i_s + 1 \, < \, i_{s+1} . \end{aligned}$$

Step 3: Let $\alpha \in \Delta _{i_s}(w)$ be as given in (7.8). Suppose that $j < i_s$ and $\beta = \alpha -\sigma _j \not \in \Delta (w)$. Then $\sigma _j \in \Delta ^+({\mathfrak {g}}_0)$ so that $\beta ,\sigma _j \in \Delta ^+\backslash \Delta (w)$ while $\beta +\sigma _j \in \Delta (w)$. This contradicts the closure of $\Delta ^+\backslash \Delta (w)$, cf. Remark 3.7(c). Therefore, $\beta \in \Delta (w)$. Similarly, if $i_s < k$, then $\beta =\alpha -\sigma _k \in \Delta (w)$. It follows by induction that,

• (a) if $\sigma _{j,k} \in \Delta _{i_s}(w)$, then $\sigma _{j',k'} \in \Delta _{i_s}(w)$ for all $j \le j' \le i_s \le k' \le k$;

• (b) if $i_t < r$ and $\sigma _{j,r-1} + \sigma _{k,r} \in \Delta _{i_t}(w)$, then $\sigma _{j',r-1} + \sigma _{k',r}\in \Delta _{i_t}(w)$, for all $j \le j' < i_t$ and $k \le k' \le r$;

• (c) if $i_t = r$ and $\sigma _{j,r-1} + \sigma _{k,r} \in \Delta _r(w)$, then $\sigma _{j',r-1} + \sigma _{k',r}$ for all $j \le j'$ and $k \le k'$.

Consider the case that $i_t < r$. If $\alpha =\sigma _{j,r}\in \Delta _{i_t}(w)$, then (a) implies $\beta =\sigma _{j,r-1} \in \Delta _{i_t}(w)$. But $\alpha +\beta \in \Delta$, contradicting (7.1). Therefore, $\sigma _{j,r}\not \in \Delta _{i_t}(w)$, if $i_t < r$.

Continuing with $i_t < r$, suppose that $\alpha =\sigma _{j,r-1} + \sigma _{k,r} \in \Delta _{i_t}(w)$. Then (b) implies $\sigma _{j,r-1} + \sigma _{r,r} = \sigma _{j,r}\in \Delta _{i_t}(w)$; we have just seen that this is not possible. Therefore, $\sigma _{j,r-1} + \sigma _{k,r} \not \in \Delta _{i_t}(w)$, if $i_t < r$.

These two observations allow us to update (7.8) to

$$\begin{aligned} \hbox {if }i_t<r, \quad \hbox {then}\quad \Delta _{i_t}(w) \, \subset \, \{ \sigma _{j,k} \, | \, i_{t-1} \,<\, j \,\le \, i_t \,\le \, k \,< \, r\} . \end{aligned}$$

In particular, if $i_t < r$, then $\Delta (w) \subset \{ \alpha \in \Delta ^+ \, | \, \alpha (\mathtt {T}_r) = 0\} = \Delta ^+(\mathfrak {a}_{r-1})$, where $\mathfrak {a}_{r-1} = \mathfrak {sl}_r\mathbb {C}$ is the simple subalgebra of ${\mathfrak {g}}_\mathbb {C}= \mathfrak {sp}_{2r}\mathbb {C}$ generated by the simple roots $\Sigma \backslash \{\sigma _r\}$. That is, we are reduced to the case that $X_w$ is a maximal Schubert VHS of a homogeneously embedded $\mathrm {SL}_{r}\mathbb {C}/P \subset \check{D} = G_\mathrm {ad}/P_\varphi$. This is precisely the case addressed in Sect. 7.2, and $X_w$ is necessarily of the form (7.7).

For the remainder of the proof, assume that $i_t = r$.

In particular, the roots of $\Delta (w) = \sqcup _{i\in I} \Delta _i(w)$ are of the form

$$\begin{aligned} \begin{array}{llll} \hbox {if }s<t, &{} \Delta _{i_s}(w) &{} \subset &{} \{ \sigma _{j,k} \, | \, i_{s-1} \,<\, j \,\le \, i_s \,\le \, k \,<\, i_{s+1} \} \,;\\ \hbox {and} &{} \Delta _{r}(w) &{} \subset &{} \{ \sigma _{j,r-1} + \sigma _{k,r} \, | \, i_{t-1} \,<\, j , k \,\le \, r\} . \end{array} \end{aligned}$$

(7.9)

Step 4: Fix $s < t$. With respect to (7.9), let $j_s$ be the minimal $j$, and $k_s$ the maximal $k$, among all $\alpha \in \Delta _{i_s}(w)$. If $s < t-1$, then the argument of Sect. 7.2, Step 4, yields $k_s + 1 < j_{s+1}$. It remains to consider the case $s=t-1$.

With respect to (7.9), let $j_t$ be the minimal $j$ over all $\alpha \in \Delta _r(w)$. Suppose that $k_{t-1}+1 \ge j_t$. Let $\alpha \in \Delta _{i_{t-1}}(w)$ be a root realizing the maximum value $k_{t-1}$, and $\beta \in \Delta _r(w)$ a root realizing the minimum value $j_t$. Then $\alpha = \sigma _{a,k_{t-1}}$ for some $a \le i_{t-1}$, and $\beta = \sigma _{j_t,r-1} + \sigma _{k,r}$ for some $j_t \le k \le r$. Step 3(c) applied to $\beta$ yields $\gamma = \sigma _{k_{t-1}+1,r-1} + \sigma _{k,r} \in \Delta _r(w)$. Then $\alpha +\gamma = \sigma _{a,r-1} + \sigma _{k,r} \in \Delta$, contradicting (7.1). Thus,

$$\begin{aligned} k_s + 1 \, < \, j_{s+1} \quad \hbox {for all}\quad 1\le s \le r-1. \end{aligned}$$

Step 5: Given $s < t$, define $\Phi _s = \{ \sigma _{j,k} \, | \, j_s \le j \le i_s \le k \le k_s\}$. Set $\Phi _t = \{ \sigma _{j,r-1} + \sigma _{k,r} \, | \, j_t \le j,k \le r \}$. Then $\Delta _{i_s}(w) \subset \Phi _s$ for all $1 \le s \le t$. Thus, $\Delta (w) \subset \Phi = \cup _s \Phi _s$. An argument analogous to that of Sect. 7.2, Step 5, yields $\Delta (w) = \Phi$. That is,

$$\begin{aligned} \begin{array}{lll} \Delta _{i_s} &{} = &{} \{ \sigma _{j,k} \, | \, j_s \le j \le i_s \le k \le k_s \} , \quad \hbox {for all} \quad s < t ,\\ \Delta _r(w) &{} = &{} \{ \sigma _{j,r-1} + \sigma _{k,r} \, | \, j_t \le j , k \le r \}. \end{array} \end{aligned}$$

(7.10)

Details are left to the reader.

Step 6: Let $s < t$. Arguing as in Step 6 of Sect. 7.2, we may show that

$$\begin{aligned} k_s+2 = j_{s+1} \quad \hbox {for all}\quad 1\le s \le t-1. \end{aligned}$$

(7.11)

Step 7: Define

$$\begin{aligned} A = \{ a_{s-1} = j_s - 1 \, | \, 2 \le s \le t\} \, \mathop {=}\limits ^{(C.12)} \, \{ a_s = k_s + 1 \, | \, 1 \le s \le t-1 \}. \end{aligned}$$

Then (7.10) and (7.11) are equivalent to $\Delta (w) = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_a)=0 \, \forall \, a \in A \} = \{ \alpha \in \Delta ({\mathfrak {g}}_1) \, | \, \alpha (\mathtt {T}_A)=0 \}$.¹⁰ Thus,

$$\begin{aligned} \begin{array}{lll} X_w &{} = &{} \mathrm {Gr}(i_1 , a_1) \, \times \mathrm {Gr}( i_2-a_1 , a_2 - a_1) \, \times \, \cdots \\ &{} &{} \cdots \, \times \, \mathrm {Gr}(i_{t-1} - a_{t-2} , a_{t-1}-a_{t-2}) \, \times \, {\mathrm {LG}}( r - a_{t-1} , 2( r-a_{t-1} ) ) . \end{array} \end{aligned}$$

(7.12)

Above, ${\mathrm {LG}}(d,2d) \simeq C_d/P_d$ is the Lagrangian Grassmannian of $d$-planes in $\mathbb {C}^{2d}$ that are isotropic with respect to a nondegenerate, skew-symmetric bilinear form ${\varvec{\varsigma }}$.

This establishes the proposition in the case that $G_\mathbb {C}= \mathrm {Sp}_{2r}\mathbb {C}$.

Appendix D: Bruhat order

Let $w,w'\in W$. Given a root $\alpha \in \Delta ^+$, let $r_\alpha \in W$ denote the associated reflection. Write $w \mathop {\rightarrow }\limits ^{\alpha } w'$ if $|w'| = |w|+1$ and $w' = r_\alpha w$. The Bruhat order is a partial order on $W$ defined by $w \le w'$ if either $w = w'$ or there is a chain $w \mathop {\rightarrow }\limits ^{\alpha _1} w_1 \mathop {\rightarrow }\limits ^{\alpha _2} \cdots \mathop {\rightarrow }\limits ^{\alpha _n} w'$. The following is well known; see, for example, [3] and the references therein.

Lemma 8.1

$w \le w'$ if and only if $X_w \subset X_{w'}$.

Lemma 8.2

Let $w,w' \in W^\varphi _{\mathcal J}$. Then $w \mathop {\rightarrow }\limits ^{\alpha } w'$ if and only if $\Delta (w') =\Delta (w) \cup \{\alpha \}$.

Together (3.15) and Lemma 8.2 yield Corollary 8.3.

Corollary 8.3

Let $w,w' \in W^\varphi _{\mathcal J}$. Then $w \le w'$ if and only if $\Delta (w) \subset \Delta (w')$; equivalently, ${\mathfrak {n}}_w \subset {\mathfrak {n}}_{w'}$.

Proof of Lemma D.2

Suppose that $\Delta (w') =\Delta (w) \cup \{\alpha \}$. So $\varrho _{w'} = \varrho _w + \alpha$, cf. (3.14). Noting that the $\Phi _w$ and $\langle \Phi _w\rangle$ of [8] are our $\Delta (w)$ and $\varrho _w$, respectively, we see that $w \mathop {\rightarrow }\limits ^{\alpha } w'$ by [8, Proposition 3.2.14(5)].

Conversely, suppose that $w \mathop {\rightarrow }\limits ^{\alpha } w'$. Then $|w'| = |w| + 1$. By [8, (3.9)], we have

$$\begin{aligned} \Delta (w') = \{\alpha \} \,\cup \, \left( \Delta (w) \cap \Delta (r_\alpha )\right) \,\cup \, r_\alpha \left( \Delta (w) \backslash \Delta (r_\alpha ) \right) . \end{aligned}$$

(8.1a)

Therefore, to establish $\Delta (w') = \Delta (w) \cup \{\alpha \}$, it suffices to show that

$$\begin{aligned} r_\alpha \left( \Delta (w) \backslash \Delta (r_\alpha ) \right) = \Delta (w) \backslash \Delta (r_\alpha ) . \end{aligned}$$

(8.1b)

To that end, suppose that $\beta \in \Delta (w) \backslash \Delta (r_\alpha )$, so that $r_\alpha \beta = \beta - n\alpha \in \Delta (w')$, for some $n\in \mathbb Z$. Since $\alpha ,\beta$ and $r_\alpha \beta$ are elements of $\Delta (w) \cup \Delta (w') \subset \Delta ({\mathfrak {g}}_1)$, we have $1 = \alpha (\mathtt {T}_\varphi ) = \beta (\mathtt {T}_\varphi )$ and $1 = (r_\alpha \beta )(\mathtt {T}_\varphi ) = 1 - n$, by (2.5). Thus, $n=0$ and $r_\alpha \beta = \beta$. This establishes (8.1b), whence (8.1a) yields $\Delta (w') = \Delta (w) \cup \{\alpha \}$. $\square$

Remark

In general, $w \le w'$ is not equivalent to $\Delta (w) \subset \Delta (w')$.

Proposition 3.10 is an immediate corollary to Lemmas 8.2 and 8.4.

Lemma 8.4

Assume $w \in W^\varphi _{\mathcal J}$ is maximal with respect to the Bruhat order. Then ${\mathfrak {n}}_w$ is a maximal IVHS.

Proof

By Lemma 3.3, IVHS are subspaces $\mathfrak {e}\subset {\mathfrak {g}}_{-1}$ such that $[\mathfrak {e},\mathfrak {e}]=0$. Suppose that ${\mathfrak {n}}_w \subset \mathfrak {e}$. Let $\zeta \in \mathfrak {e}$, and write $\zeta = \sum _{\alpha \in \Delta ({\mathfrak {g}}_1)} \zeta _{-\alpha }$ with $\zeta _{-\alpha } \in {\mathfrak {g}}_{-\alpha }$. Since $\mathfrak {e}$ is abelian, we have $[\zeta ,{\mathfrak {n}}_w]=0$. Since ${\mathfrak {n}}_w$ is a direct sum of root spaces, this is possible if and only if $[\zeta _{-\alpha },{\mathfrak {n}}_w] = 0$ for all $\alpha$. Equivalently, if $\zeta _{-\alpha }\not =0$, then the set $\Phi _\alpha = \{\alpha \} \cup \Delta (w) \subset \Delta ({\mathfrak {g}}_1)$ is closed, c.f. Remark 3.7(c). Therefore, the lemma holds if and only if the set $A = \{ \alpha \in \Delta ({\mathfrak {g}}_1)\backslash \Delta (w) \, | \, \Phi _\alpha \hbox { is closed}\}$ is empty.

Define a partial order on $A$ by declaring $\alpha _1 < \alpha _2$ if there exists $\beta \in \Delta ^+({\mathfrak {g}}_0)$ such that $\alpha _1+\beta =\alpha _2$. If $A$ is nonempty, then there exists an element $\alpha \in A$ that is minimal with respect to this partial order. I claim that $\Delta ^+\backslash \Phi _\alpha$ is also closed. Assuming the claim holds, Remark 3.7(c) yields $w' \in W^\varphi$ such that $\Phi _\alpha = \Delta (w')$. It follows from (3.15) that ${\mathfrak {n}}_w \subset {\mathfrak {n}}_{w'}$, contradicting the maximality of ${\mathfrak {n}}_w$. This establishes the lemma.

To prove the claim, suppose that $\beta ,\gamma \in \Delta ^+\backslash \Phi _\alpha$ and that $\beta +\gamma$ is a root. Note that $\Delta ^+\backslash \Phi _\alpha \subset \Delta ^+\backslash \Delta (w)$. By Remark 3.7(c), $\Delta ^+\backslash \Delta (w)$ is closed. So either $\beta +\gamma \in \Delta ^+\backslash \Phi _\alpha$ or $\beta +\gamma = \alpha$. In the latter case, $1 = \alpha (\mathtt {T}_\varphi ) = \beta (\mathtt {T}_\varphi ) + \gamma (\mathtt {T}_\varphi )$. So, either $\beta (\mathtt {T}_\varphi )=0$ and $\gamma (\mathtt {T}_\varphi ) = 1$, or vice versa. Without loss of generality, we assume the former; that is, $\beta \in \Delta ^+({\mathfrak {g}}_0)$ and $\gamma \in \Delta ^+({\mathfrak {g}}_1)$, contradicting the minimality of $\alpha$. Therefore, $\beta +\gamma \in \Delta ^+\backslash \Phi _\alpha$, and the claim holds. $\square$