Long-root \SL_2 dual pairs in split exceptional groups: Compact types, cone geometry, and theta correspondence

Introduction

Let $H$ be the group of real points of a split simply connected algebraic group of type $E_n$, with $n = 6, 7, 8$, and let $\Pi$ be its spherical minimal Harish–Chandra module. A long root determines a subgroup $G_1 \simeq\mathrm{SL}_2(\mathbb{R})$ whose centralizer $G_2$ is, respectively, $\mathrm{SL}_6(\mathbb{R})$, split $\mathrm{Spin}_{12}$, or split simply connected $E_7$. The multiplication map $G_1 \times G_2 \to H$ has central kernel $\mu_2$, the group of square roots of 1. All restrictions below are understood through this map. Choose compatible maximal compact subgroups $K \subset H$ and $K_i \subset G_i$, with $K_1 = \mathrm{SO}(2)$, and write $\mathfrak{h} = \mathfrak{k} \oplus\mathfrak{p}$ and $\mathfrak{g}_i = \mathfrak{k}_i \oplus\mathfrak{p}_i$ for their complexified Cartan decompositions.

For a $(\mathfrak{sl}_2,K_1)$ Harish–Chandra module $\pi$, let $\pi^\vee$ denote its $K_1$-finite contragredient. We define its big theta lift by coinvariants:

where the subscript denotes coinvariants for the diagonal action. This construction does not require $\pi$ to be irreducible and defines a contravariant functor to $(\mathfrak{g}_2,K_2)$-modules. If $\pi$ is irreducible, then $\pi\boxtimes\Theta(\pi)$ is the largest $\pi$-isotypic quotient of $\Pi$, recovering the usual definition; see Bakić–Loke–Savin [1].

Our purpose is to prove that, for every irreducible $\pi$, the big lift $\Theta(\pi)$ is a nonzero finite-length $(\mathfrak{g}_2,K_2)$-module with a unique irreducible quotient, using the ping-pong method of [11]. To that end we compute the lifts of types of the maximal compact subgroups of $G_1$ and $G_2$.

Let $v_\omega\in\mathfrak{p}^*$ be a nonzero highest-weight vector for $K$. The highest-weight vector cone is

2020 Mathematics Subject Classification. Primary 22E47; Secondary 11F27, 14M15. where $K_{\mathbb{C}}$ is the complexification of $K$. Throughout, we identify $\mathfrak{p}^{*}$ with $\mathfrak{p}$ using the Killing form. Write $X=\mathbb{P}(\widehat{X})$ for the projectivization of the cone. With its standard filtration, $\Pi$ has associated graded module

The quotient $R/(\mathfrak{p}_2)$ is the coordinate ring of the scheme-theoretic linear section $\widehat{X}\cap\mathfrak{p}_2^{\perp}$. Computing this quotient is the algebraic-geometric problem that underlies the lifting of types.

Two calculations drive the argument. First, $R/(\mathfrak{p}_2)$ describes finite generating sets for the lifts of the characters of $\mathrm{SO}(2)$. More precisely, as an $\mathrm{SO}(2)\times K_2$-module this quotient decomposes as

where each $Q_k$ is a finite-dimensional graded $K_2$-module. Let $N(k)=\operatorname{Hom}_{K_1}(\chi_k,R)$ be the lift of $\chi_k$ in the graded model. Since $\mathfrak{p}_2$ has circle weight zero, $Q_k=N(k)/\mathfrak{p}_2N(k)$. The graded Nakayama lemma shows that homogeneous $K_2$-equivariant lifts of $Q_k$ to $N(k)$ generate $N(k)$ over $\operatorname{Sym}(\mathfrak{p}_2)$. We then lift these generators to the corresponding filtration degrees in $\operatorname{Hom}_{K_1}(\chi_k,\Pi)$. Induction on the filtration shows that they generate this lift over $U(\mathfrak{g}_2)$. Thus, in the filtered setting, the lift of $\chi_k$ is a quotient of the universal module induced from $Q_k$:

The $K_2$-types occurring in these finite-dimensional spaces $Q_k$ are the relevant types. As described in § 4, they form two ordered sequences $\sigma_0=1,\sigma_{\pm1},\sigma_{\pm2},\ldots$, with

Ordering each sequence by its nonnegative index, $Q_k$ has $\sigma_k$ as its largest $K_2$-type.

The second calculation describes reverse lifts of the relevant types $\sigma_k$. We have

Here $\operatorname{wt}(a)$ and $\operatorname{wt}(b)$ denote the weights for the circle $K_1=\mathrm{SO}(2)$. Since $R$ is a domain, it is a torsion-free $R^{K_2}$-module. The multiplicity space $M_k=\operatorname{Hom}_{K_2}(\sigma_k,R)$ is therefore torsion-free as well. The calculation in § 5 gives its Poincaré series:

where $t$ records degree and $z$ records circle weight. A nonzero element of $M_k$ in degree $\lvert k\rvert$ and weight $k$ generates a free rank-one $R^{K_2}$-submodule, by torsion-freeness. The displayed series shows that this submodule exhausts $M_k$, so $M_k$ is itself free of rank one over $R^{K_2}$. Passing back to $\Pi$, the reverse lift of $\sigma_k$ is the universal module induced from the $\mathrm{SO}(2)$-type $\chi_k$:

With this indexing, $\sigma_k$ has its primitive occurrence in degree $\lvert k\rvert$ and circle weight $k$. The representations $\sigma_k$ are listed in § 4. Our main conclusion is the following.

The theorem concerns the irreducible quotient of the big lift; it does not assert that the big lift itself is irreducible. A minimal weight is allowed to have either sign. If both $1$ and $-1$ are minimal weights of $\pi$, either can be used: its small lift contains both $\sigma_1$ and $\sigma_{-1}$ with multiplicity one and is generated by either.

We use the branching rules of Loke–Savin [8 Proposition 5.1] to prove the invariant and multiplicity identities needed for the present pairs. The minimal representation and its compact spectrum are supplied by Brylinski–Kostant [2]; see also [5]. For related exceptional dual pairs and their rational forms, see [7].

Section 2 fixes the groups and cone models. Sections 3 and 4 give the forward and reverse compact-type lifts. Section 5 proves the theorem. Sections 6–8 establish the three linear-section calculations, including their scheme structures. In each case the projective scheme of $R/(\mathfrak{p}_2)$ has two disjoint components, each a projective cone. The bases of these cones vary with the exceptional group. They are the projectivizations of the affine cones of symmetric $4\times4$ matrices of rank at most one, arbitrary $4\times4$ matrices of rank at most one, and skew-symmetric $8\times8$ matrices of rank at most two, respectively. These give the quadratic Veronese variety $\nu_2(\mathbb{P}^3)$, the Segre variety $\mathbb{P}^3\times\mathbb{P}^3$, and the Grassmannian $\operatorname{Gr}(2,8)$ in its Plücker embedding. The three matrix spaces realize the simple complex Jordan algebras of rank four occurring in the Brylinski–Kostant construction [2]: the complexifications of the Jordan algebras of $4\times4$ Hermitian matrices over $\mathbb{R}$, $\mathbb{C}$, and the quaternions, respectively. In each case the affine cone is the locus of elements of Jordan rank at most one; in the skew-symmetric realization, Jordan rank one corresponds to matrix rank two.

A $p$-adic variant of the one-to-one correspondence in Theorem 1.1 is established by Hanzer–Savin [6]. For the same long-root dual pairs in split $E_6$, $E_7$, $E_8$ over a $p$-adic field, they determine the exceptional theta correspondence and its reverse lifts; see in particular Sections 5 and 6 of that paper.

Acknowledgments

I thank Wee Teck Gan for reading a draft of this paper. I gratefully acknowledge support from the Simons Foundation through gift No. 946504, and thank the University of Utah for providing access to OpenAI. The paper was written with the help of ChatGPT-6 Astra. In addition to harmless assistance with writing, the software was used to compute the ring $R/(\mathfrak{p}_2)$. The write up of this result is self contained: calculating $R/(\mathfrak{p}_2)$ involves restricting the Plücker relations to $\mathfrak{p}_2$, a pleasant exercise easily done by hand, in this particular case.

The exceptional groups, dual pairs, and graded models

We describe the three dual pairs, their Cartan decompositions, and the graded models of the minimal representations.

Cartan decompositions and compact groups

All Lie algebras and algebraic varieties in the calculations are over $\mathbb{C}$. The real groups are split. We use $\operatorname{Sp}(4)$ for the compact symplectic group of rank four, with standard complex representation of dimension eight. The ambient Cartan decompositions are as follows.

$H$	$K$	$\mathfrak{k}$	$\mathfrak{p}$
$E_{6}$	$\operatorname{Sp}(4)/\mu_{2}$	$\mathfrak{sp}_{8}$	$\Lambda^{4}_{0}\mathbb{C}^{8}$
$E_{7}$	$\operatorname{SU}(8)/\mu_{2}$	$\mathfrak{sl}_{8}$	$\Lambda^{4}\mathbb{C}^{8}$
$E_{8}$	$\operatorname{Spin}(16)/\mu_{2}$	$\mathfrak{so}_{16}$	$S^{+}$

The subscript 0 means the kernel of symplectic contraction. In the last row $\mu_{2}$ is the kernel of the chosen half-spin representation $S^{+}$; the compact group is the corresponding semispin quotient. The dimensions of $\mathfrak{p}$ are 42, 70, 128.

For $G_{1}\simeq\operatorname{SL}_{2}(\mathbb{R})$, fix its maximal compact subgroup $K_{1}=G_{1}\cap K\simeq\operatorname{SO}(2)$. Choose $H_{0}\in\mathfrak{k}_{1}$ so that $iH_{0}$ is the real rotation generator, with period $2\pi$, and put

Thus $H_{0}$ acts on the character $\chi_{m}$ by $m$. The complexified Cartan decomposition of $\mathfrak{g}_{1}$ is

Thus $\mathfrak{p}_{1}=\chi_{2}\oplus\chi_{-2}$ with the usual integer weights of $\operatorname{SO}(2)\subset\operatorname{SL}_{2}(\mathbb{R})$. For the centralizers we have

$H$	$G_{2}$	$K_{2}$	$\mathfrak{p}_{2}$
$E_{6}$	$\operatorname{SL}_{6}(\mathbb{R})$	$\operatorname{SO}(6)\simeq\operatorname{SU}(4)/\mu_{2}$	$\operatorname{Sym}^{2}_{0}\mathbb{C}^{6}$
$E_{7}$	split $\operatorname{Spin}_{12}$	$(\operatorname{SU}(4)\times\operatorname{SU}(4))/\mu_{2}^{\operatorname{diag}}$	$\Lambda^{2}\mathbb{C}^{4}\otimes\Lambda^{2}\mathbb{C}^{4}$
$E_{8}$	split $E_{7}$	$\operatorname{SU}(8)/\mu_{2}$	$\Lambda^{4}\mathbb{C}^{8}$

The corresponding $\mathfrak{k}_{2}$ are $\mathfrak{so}_{6}$, $\mathfrak{so}_{6}\oplus\mathfrak{so}_{6}$, and $\mathfrak{sl}_{8}$. The dimensions of $\mathfrak{p}_{2}$ are 20, 36, 70. The adjoint branching under $\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ is

Here $S_{12}$ is a half-spin module of dimension 32, and $V_{56}$ is the 56-dimensional fundamental $E_{7}$-module. These descriptions identify the long-root centralizers directly from the adjoint branching.

Under the inclusion $K_{1}\subset K$, the element $k_{\theta}$ is $\exp_{K}(i\theta H_{0})$, with $H_{0}$ viewed in $\mathfrak{k}$. For example, in $E_{7}$ write $U=V\oplus W$ with $\dim V=\dim W=4$. The images in $\mathfrak{k}=\mathfrak{sl}(U)$ and $K=\operatorname{SU}(8)/\mu_{2}$ are

When $\theta$ increases by $2\pi$, the representative in $\operatorname{SU}(8)$ changes by $-I$, which is trivial in $K$. This realizes the period $2\pi$ of $K_{1}$. The auxiliary spaces $V,W$ have $H_{0}$-weights $1/2$, $-1/2$ on the cover, while the actual $K_{1}$-modules have integer weights. In particular the weights of $\mathfrak{p}_{1}$ are 2, $-2$, as in (2.1).

The minimal representation and its filtration

Let $v_0$ be a nonzero spherical vector in $\Pi$, and let $\omega$ be the highest weight of $\mathfrak{p}$ as a $K$-module. The compact spectrum is $\left.\Pi\right|_K = \bigoplus_{d\geq0} V_K(d\omega)$, with multiplicity one. Its standard filtration has

The associated graded action of $\operatorname{Sym}(\mathfrak{p})$ is multiplication on the cone ring, using the Killing-form identification of $\mathfrak{p}$ with $\mathfrak{p}^*$. These are the graded models of the spherical minimal representations in [2]. Explicitly:

$H$	$X$	$R_d$	$\dim X$
$E_6$	$\operatorname{LG}(4,8)$	$V_{C_4}(d\omega_4)^*$	10
$E_7$	$\operatorname{Gr}(4,8)$	$\mathbf{S}^{(d^4)}(\mathbb{C}^8)^*$	16
$E_8$	$\operatorname{OGr}^{+}(8,16)$	$V_{D_8}(d\omega_8)^*$	28

The first two embeddings are Plücker embeddings. For $E_8$ we label the selected half-spin weight by $\omega_8$ and use the spinor embedding. In each case $R$ is generated in degree one and is an integral domain.

The circle decomposition and the Jordan algebra

Let $J$ be the rank-four complex Jordan algebra associated with $H$ in the introduction. Write $\mathbb{C}_m$ for a one-dimensional space on which $K_1$ acts by $\chi_m$, and write $J_1$ and $J^*_{-1}$ for $J$ and its dual with circle weights $1$ and $-1$, respectively. The circle decomposition of $\mathfrak{p}$ is

The middle term has circle weight zero, and the two $K_2$-trivial scalar lines form $\mathfrak{p}_1$. The Killing form pairs opposite weight spaces and identifies $\mathfrak{p}_2$ with linear functions on the cone; $(\mathfrak{p}_2)$ denotes the ideal they generate in $R$.

Lifts of the SO(2)-types: the common pattern

We determine generators for the lifts of SO(2)-types in $R$ and $\Pi$ using the quotient $R/(\mathfrak{p}_2)$.

Cone bases and relevant types

Put $Q = R/(\mathfrak{p}_2)$. We call a $K_2$-type relevant if it occurs in $Q$. By (3), setting $\mathfrak{p}_2$ equal to zero cuts the highest-weight vector cone by

Sections 6–8 show that this section is reduced and that $\operatorname{Proj} Q = X_+ \sqcup X_-$, where

Here the ranks are Jordan ranks in the two dual matrix realizations. The vertex of $X_+$ is $[1:0]$, representing the line $\mathbb{C}_2$. Its base is the rank-one locus in $\mathbb{P}(J_1) = \mathbb{P}(J)$, embedded by $[u] \mapsto[0:u]$. The cone $X_+$ is the union of the projective lines $[\lambda:\mu u]$, with $[\lambda:\mu] \in\mathbb{P}^1$, joining the vertex to these base points. Similarly, $X_-$ has vertex the line $\mathbb{C}_{-2}$ and base the rank-one locus in $\mathbb{P}(J^*_{-1})$. The vertices belong to the two components, while the affine cones meet only at the origin. The bases, with their projective embeddings, are as follows.

Here $\dim V = 4$ in $E_{6}$, $\dim V = \dim W = 4$ in $E_{7}$, and $\dim V = 8$ in $E_{8}$. The embeddings are the second Veronese, the Segre, and the Plücker embedding, respectively. Accordingly, the $K_{2}$-module $J$ is $\operatorname{Sym}^{2}V^{*}$, $V^{*} \otimes W$, or $\Lambda^{2}V^{*}$, respectively.

Let $S^{+} = \bigoplus_{j\geq0} S^{+}_{j}$ and $S^{-} = \bigoplus_{j\geq0} S^{-}_{j}$ be the coordinate rings of the rank-at-most-one cones in $J$ and $J^{*}$, respectively, that is, the affine cones over the two bases in the table. By the standard homogeneous coordinate-ring decompositions [3, 4], each degree-$j$ piece is irreducible under $K_{2}$; denote its $K_{2}$-type by $\tau^{\pm}_{j}$. Including the circle action, we have

The coordinate weights are opposite to the vector weights on $J_{1}$ and $J^{*}_{-1}$. Besides the common trivial type, these are the relevant types; for $j \geq1$ they are listed below.

The notation $\tau^{\pm}_{j}$ records only the $K_{2}$-type; the circle weights are recorded separately above. All these types descend through the compact quotients in § 3. Geometrically, they are the irreducible constituents of $H^{0}(X_{\pm}, \mathcal{O}_{X_{\pm}}(d))$, for $d \geq0$, where $\mathcal{O}(1)$ is the ambient Plücker line bundle in $E_{6}$, $E_{7}$ and the spinor line bundle in $E_{8}$. Sections 6–8 establish the ring descriptions. The section interpretation follows from projective normality of the Veronese, Segre, and Grassmannian embeddings and their projective cones.

In the formulas below, $a,b \in R^{K_{2}}$ are coordinates on the scalar summands $\mathbb{C}_{2},\mathbb{C}_{-2}$, extended by zero on the other summands; we use the same notation for their images in $Q$. They have degree one and weights $-2,2$, since coordinate weights are opposite to vector weights.

The scheme and graded-ring calculations proving this theorem are given in Sections 6–8. To make the circle action entirely explicit, forgetting the degree, (3.2) means

The degree in the first sum is $|q|$, and in the other sums it is $j+q$. In particular the weight-zero lift in this quotient is just $1$.

Passage to the minimal representation

For $k\in\mathbb{Z}$ set

The graded $\operatorname{Sym}(\mathfrak{p}_2)$-module $N(k)$ retains the grading of $R$: its degree-$d$ part is $\operatorname{Hom}_{K_1}(\chi_k,R_d)$. The filtered module $A(k)$ is a $(\mathfrak{g}_2,K_2)$-module. Define the signed notation

Thus the index of $\sigma_k$ is its primitive circle weight.

For $k\ne0$, put

When $|k|$ is even the last summand is $\sigma_0$, included only once.

Proof. Taking a fixed compact type commutes with associated graded because compact representations are semisimple. Thus $\operatorname{gr} A(k)=N(k)$. Since $\mathfrak{p}_2$ has circle weight zero, $N(k)/\mathfrak{p}_2N(k)=Q_k=F(k)$ with the degrees specified above. Choose homogeneous lifts of a basis of this quotient. Induction on degree shows that they generate $N(k)$ over $\operatorname{Sym}(\mathfrak{p}_2)$: after subtracting their constant coefficients, an element lies in $\mathfrak{p}_2N(k)$, and its coefficients have smaller degree. Lift these generators $K_2$-equivariantly to $A(k)$ and apply the same filtration induction using $U(\mathfrak{g}_2)$. This proves the claim.

Reverse lifts of the relevant types

For a $K_2$-type $\tau$ define

Their gradings and filtrations retain the circle action. We compute only the reverse lifts needed for the relevant types.

For every signed relevant type $\sigma_k$, its graded multiplicity space $M(\sigma_k)$ is free of rank one over this invariant ring, with generator in degree $|k|$ and weight $k$. Its Poincaré series is

Moreover, as a $(\mathfrak{sl}_2,\mathrm{SO}(2))$-module,

with its generating line in filtration degree $|k|$.

Branching and the Poincaré series

We use the restriction of $R_d$ to $K_1\times K_2$, retaining both its circle weights and its $K_2$-types. The branching rules are given by Loke–Savin [8 Proposition 5.1], with $m=4$ in each of its three cases. We state only their consequence for the relevant types, in the notation and integer circle-weight normalization of Section 3.

For every $k\in\mathbb{Z}$ and $d\geq0$, the $K_1$-module of multiplicities is

The sum is empty when $d<|k|$. Thus $\sigma_k$ first occurs in degree $|k|$, with circle weight $k$. In degree $|k|+n$, its occurrences have the distinct weights

each with multiplicity one. Equivalently, the $\sigma_k$-isotypic part of $R_d$, as a $K_1\times K_2$-module, is the sum of $\chi_{k-2r+2s}\boxtimes\sigma_k$ over the same indices. This statement includes $k=0$, for which $\sigma_0=1$.

Writing $d=|k|+r+s$, we obtain at once

This proves (11) uniformly in the three cases.

From the series to free and universal modules

Taking $k=0$ in (13) gives

The elements $a,b$ are nonzero in the domain $R$. Their monomials are nonzero and have distinct pairs of degree and weight. They are therefore linearly independent and realize every occurrence in this series. This proves (10), including the algebraic independence of $a,b$.

Choose a nonzero homogeneous equivariant map $g_k:\sigma_k\to R_{|k|}$ of circle weight $k$. Multiplication defines

It is injective: a nonzero polynomial cannot annihilate the nonzero image of $g_k$ in the domain $R$. The source and target have the same degree-and-weight series (11), so the map is surjective. This proves the rank-one assertion. A Poincaré series by itself would not establish freeness; the injective multiplication map is essential.

Now lift $g_k$ $K_1$-equivariantly to $B(\sigma_k)$. Filtered generation gives a surjection $I(k) \to B(\sigma_k)$. PBW gives

with the generating line placed in degree $|k|$. The associated graded map is the rank-one isomorphism just proved, so the filtered map is an isomorphism. This finishes the proof of Theorem 4.1.

Write $\pi_k = \operatorname{Hom}_{K_1}(\chi_k,\pi)$ for the weight-$k$ multiplicity space. The universal property we need is

Remark 4.2. The same argument applies to every $K_2$-type $\tau$ occurring in $R$. Indeed, fixing its highest-weight differences in [8 Proposition 5.1] leaves two nonnegative parameters. There is a unique first occurrence, in degree $d_\tau$ and circle weight $w_\tau$, and

Since $R$ is a domain, multiplication by a nonzero first-occurrence map identifies $M(\tau)$ with a free rank-one $R^{K_2}$-module. The filtered argument then gives $B(\tau)\simeq I(w_\tau)$, with its generating line in degree $d_\tau$. For $\tau=\sigma_k$ these parameters are $d_\tau=|k|$ and $w_\tau=k$; in general $d_\tau$ need not equal $|w_\tau|$.

Ping-pong and the uniform theta correspondence

We prove finite length of the big theta lifts and use the compact-type calculations to establish Theorem 1.1.

Finiteness and compact-type reciprocity

For an irreducible $(\mathfrak{g}_1,K_1)$-module $\pi$, recall that $\pi\boxtimes\Theta(\pi)$ is the maximal $\pi$-isotypic quotient of $\Pi$. Define the reverse big lift of an irreducible $(\mathfrak{g}_2,K_2)$-module $\rho$ similarly, with quotient $\Theta(\rho)\boxtimes\rho$. Compact-type reciprocity reads

The first isomorphisms use the maximal isotypic quotients, and the second use the compact isotypic components. For a locally finite $K$-module $M$ and an irreducible $K$-type $\eta$, $\operatorname{Hom}_K(M,\eta)\simeq\operatorname{Hom}_K(\eta,M)^*$, with algebraic dual, so finite-dimensionality is equivalent to finite multiplicity.

Proposition 5.1. Both big theta lifts have finite length whenever the input is irreducible. Moreover, for every relevant type,

In particular $\Theta(\pi)\ne0$ for every irreducible $\pi$.

Proof. For any $K_2$-type $\tau$ occurring in $R$, Remark 4.2 gives $B(\tau)\simeq I(w_\tau)$. Thus (5.1) and (4.5) identify $\operatorname{Hom}_{K_2}(\Theta(\pi),\tau)$ with $\pi_{w_\tau}$, which is finite-dimensional. Types absent from $R$ have zero multiplicity. Likewise, Corollary 3.2 and (5.2) give

The preceding duality now proves that both big lifts are admissible. To obtain finite length, we also verify finite generation using the compact-type lifts.

Every circle weight space of an irreducible $(\mathfrak{sl}_2,\mathrm{SO}(2))$-module is at most one-dimensional. For any occurring $k$, taking weight $k$ in $\Pi\twoheadrightarrow\pi\boxtimes\Theta(\pi)$ gives $A(k) \twoheadrightarrow\Theta(\pi)$, proving finite generation. For the reverse direction, choose a $K_2$-type $\tau$ occurring in $\rho$. Taking its isotypic multiplicity space gives a surjection $B(\tau) \twoheadrightarrow\Theta(\rho) \otimes\operatorname{Hom}_{K_2}(\tau,\rho)$; projection to a nonzero factor proves finite generation of $\Theta(\rho)$. Admissible finitely generated modules have finite length [12]. Finally (17) follows from (15), Theorem 4.1, and (14). Some $\pi_l$ is nonzero, proving nonvanishing.

The two passes

For an arbitrary pair $(\mathfrak{g},K)$, let $M$ be a nonzero $(\mathfrak{g},K)$-module of finite length as a $U(\mathfrak{g})$-module. If $M$ is generated over $U(\mathfrak{g})$ by one copy of an irreducible $K$-type $\eta$ and that type has multiplicity one in $M$, then $M$ has a unique irreducible quotient. Indeed every nonzero quotient must contain the image of its generating copy of $\eta$. The semisimple head therefore has only one irreducible summand, with multiplicity one.

Proof of Theorem 1.1. Choose a circle weight $k$ of $\pi$ with minimal absolute value. Equation (17) gives

The surjection $A(k) \twoheadrightarrow\Theta(\pi)$ and Corollary 3.2 show that the big lift is generated by the image of $F(k)$. All its summands except $\sigma_k$ vanish by (18). Thus $\Theta(\pi)$ is generated by one copy of $\sigma_k$, and the observation gives its unique irreducible quotient $\rho=\theta(\pi)$. Its generating type survives and still has multiplicity one. Since maps to $\rho$ are determined by their restriction to this generating type, $\dim\operatorname{Hom}_{\mathfrak{g}_2,K_2}(\Theta(\pi),\rho)=1$, which gives the multiplicity-one assertion of the theorem. The smaller relevant types are absent.

For the return pass, the quotient $\Pi\twoheadrightarrow\pi\boxtimes\rho$ makes $\pi$ a quotient of $\Theta(\rho)$. By (16),

The inequality follows by restricting a homomorphism to the chosen generators. The quotient onto $\pi$ makes the dimension at least one, so it equals one. On the other hand, taking the $\sigma_k$-multiplicity space of the reverse maximal quotient gives

Therefore $\Theta(\rho)$ is generated by its single weight-$k$ line. Every nonzero quotient contains that line, so the same observation proves that its unique irreducible quotient is $\pi$.

The linear section in type $E_6$

We compute $Q=R/(\mathfrak{p}_2)$ for the long-root $\mathrm{SL}_2\times\mathrm{SL}_6$ pair in $E_6$. We identify its two Veronese cones and their scheme structure, then read off the graded $\mathrm{SO}(2)\times K_2$-decomposition from their coordinate rings to prove Theorem 3.1.

The two Veronese cones and their equations

Let $U = V \oplus V^*$, $\dim V = 4$, with symplectic form

A point $L \in\operatorname{LG}(4,U)$ is a four-dimensional Lagrangian subspace of $U$. The ambient Cartan space is $\mathfrak{p} = \Lambda^4_0 U$, the kernel of symplectic contraction, and it contains the Plücker vectors of $\operatorname{LG}(4,U)$. As an $\operatorname{SU}(V)$-module it decomposes as

The summands are ordered by the number $4,3,2,1,0$ of $V$ factors in $\Lambda^4(V \oplus V^*)$. Their dimensions are $1,10,20,10,1$, and their circle weights are $2,1,0,-1,-2$. Determinant characters have been suppressed in this display because they are trivial on $\operatorname{SU}(V)$.

To see the decomposition, the two extreme terms are $\det V$ and $\det V^*$. The term $\Lambda^3 V \otimes V^*$ identifies with $\det V \otimes(V^* \otimes V^*)$; contraction vanishes on the symmetric part and is injective on the alternating part. Its kernel is therefore $\det V \otimes\operatorname{Sym}^2 V^*$, and the opposite term is dual. On the middle term $\Lambda^2 V \otimes\Lambda^2 V^*$, contraction maps onto $V \otimes V^*$ and has the irreducible 20-dimensional kernel $S_{(2,2)}V$. Equivalently, this is $\operatorname{Sym}^2_0(\Lambda^2 V) \simeq\operatorname{Sym}^2_0 \mathbb{C}^6$, the Cartan space $\mathfrak{p}_2$ of the centralizer.

The coordinate functions have the opposite circle weights. Thus $\mathfrak{p}_2 = 0$ sets the middle Plücker coordinates to zero, leaving the four spaces with weights $-2,-1,1,2$. To describe the resulting four-planes, put $r = \dim(L \cap V)$ and $s = \dim(L \cap V^*)$. Complete bases of these intersections by vectors $v_i + \phi_i$ in $L$. The classes of the $v_i$ and of the $\phi_i$ are independent in the respective quotient spaces. Expanding their wedge shows that the nonzero components have precisely $r,r+1,\ldots,4-s$ factors from $V$. The middle component therefore vanishes exactly when $r \ge3$ or $s \ge3$. We obtain the two loci

They are disjoint: the two indicated intersections cannot both have dimension at least three inside a four-dimensional space.

To understand $X_+$, take $L \ne V$ and put $A = L \cap V$. Then $\dim A = 3$, and $L/A$ is a line in $(V/A) \oplus V^*$ with nonzero projection to $V^*$. Isotropy forces this projection to annihilate $A$: for $a \in A$ and $v+\phi\in L$, we have $\langle a,v+\phi\rangle= \phi(a) = 0$. Since the annihilator

is itself a line, the projected line must be $D$. The choice of $A$ therefore determines $D$; there is no additional choice of a line in $V^*$. In particular the base will be $\operatorname{Gr}(3,V) = \mathbb{P}(V^*)$.

Conversely, fix $A \in\operatorname{Gr}(3,V)$, and put $Q_A = V/A$, so $D = Q_A^*$. The orthogonal complement for the symplectic form is $A^\perp= V \oplus D$, and

is a two-dimensional symplectic space. Every line in this space is isotropic, and its inverse image in $A^\perp$ is a Lagrangian four-plane containing $A$. These four-planes form a $\mathbb{P}^1$. The line $Q_A$ gives $V$, and the line $Q_A^*$ gives $A \oplus D$. Every other member also has intersection exactly $A$ with $V$. In the Plücker embedding this $\mathbb{P}^{1}$ is the line joining $V$ to $A\oplus D$. Indeed, if $u_{1},u_{2},u_{3}$ is a basis of $A$, $v\in V\setminus A$, and $0\ne\phi\in D$, its points are

To see the embedding of the base, use $\Lambda^{3}V\simeq\det V\otimes V^{*}$. Under this identification $\Lambda^{3}A$ corresponds to $\det V\otimes D$. The Plücker line of $A\oplus D$ is consequently $\det V\otimes D^{\otimes2}$. As $D$ varies through the lines in $V^{*}$, this is exactly the second Veronese embedding $\nu_{2}:\mathbb{P}(V^{*})\to\mathbb{P}(\operatorname{Sym}^{2}V^{*})$, with the harmless common factor $\det V$. Thus $X_{+}$ is the projective cone over this Veronese variety, with vertex $V$. Interchanging $V$ and $V^{*}$ gives $X_{-}$ and its vertex $V^{*}$.

We now check this description scheme-theoretically. Start with the Plücker ideal of $\operatorname{Gr}(4,V\oplus V^{*})$, generated by its degree-two part $\overline{I}_{2}$. Passing to $\operatorname{LG}(4,U)$ adds the linear contraction equations just described. On the surviving coordinates they make the two $4\times4$ arrays $x,y$ symmetric, with the usual cofactor signs. Killing the middle summand $\mathfrak{p}_{2}$ amounts to restricting the equations that kill the full middle coordinate space in the ordinary Grassmannian.

Thus the surviving coordinate spaces are

as $\operatorname{SU}(V)$-modules, with circle weights $-2,-1,1,2$. Here $x_{ji}=x_{ij}$ and $y_{ji}=y_{ij}$. Write $\overline{I}_{2}$ for the image of the Plücker quadratic relation space after imposing these linear equations. It lies in $S=\mathbb{C}[a,b,x,y]$, and the defining ideal of $Q$ is $(\overline{I}_{2})$. Let $\overline{I}_{2}(x)$ and $\overline{I}_{2}(y)$ denote its subspaces involving only $x$ or only $y$, respectively. By the calculation in § 8, they are spanned by the $2\times2$ minors of the corresponding symmetric matrices. These are spaces of quadratic polynomials; parentheses will denote the ideals they generate.

The $\operatorname{SU}(V)$-module structure of $\overline{I}_{2}(x)$ follows from multiplication of quadratic polynomials:

Indeed, the standard Schur decomposition is $\operatorname{Sym}^{2}(\operatorname{Sym}^{2}V)=\operatorname{Sym}^{4}V\oplus\mathbf{S}_{(2,2)}V$ [9 Section 1.2]. The kernel is spanned by $x_{ij}x_{kl}-x_{il}x_{kj}$, which vanish under $x_{ij}\mapsto s_{i}s_{j}$. It has dimension $55-35=20$ and highest weight $2\omega_{2}$. The dual space $\overline{I}_{2}(y)=\mathbf{S}_{(2,2)}V^{*}$ is also isomorphic to this irreducible $\operatorname{SU}(4)$-module.

Specializing the generator decomposition of § 8 now gives

To display its irreducible constituents, let $V_{\lambda}$ denote the $\operatorname{SU}(4)$-module of highest weight $\lambda$, with fundamental weights $\omega_{1},\omega_{2},\omega_{3}$. The tensor product in the middle is

with dimensions $1,15,84$. Grouping by the total $V$-degree of the original Plücker indices, the quadratic generators are:

total $V$-degree	generator space	$\operatorname{SU}(4)$-module	dimension
6	$\overline{I}_{2}(x)$	$V_{2\omega_{2}}$	20
5	$a\mathcal{Y}$	$V_{2\omega_{3}}$	10
4	$\mathbb{C}ab\oplus\mathcal{X}\mathcal{Y}$	$\mathbb{C}^{\oplus2}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}$	101
3	$b\mathcal{X}$	$V_{2\omega_{1}}$	10
2	$\overline{I}_{2}(y)$	$V_{2\omega_{2}}$	20

The two copies of the trivial representation in degree 4 are $Cab$ and the invariant contraction in $XY$. All dimensions follow from the displayed modules; their sum is 161. The degree-two part of $Q$ consequently has dimension $\binom{23}{2}-161=92$.

The relations in total $V$-degrees 5, 4, 3 are exactly $ay_{ij}$, $ab$, $x_{ij}y_{kl}$, $bx_{ij}$. They generate $(a,x)(b,y)$, so they separate the two coordinate blocks, while the remaining relations impose rank at most one on the symmetric matrix in each block. Consequently,

Here $\mathbb{C}[x]=\mathbb{C}[x_{ij}:i\leq j]$ and similarly for $y$; each map to $\mathbb{C}$ evaluates at the origin. The substitution $x_{ij}=s_i s_j$ identifies the first matrix ring with

where a quadratic monomial in the $s_i$ has degree one in the matrix ring. This is the homogeneous coordinate ring of the quadratic Veronese variety $\nu_2(\mathbb{P}(V^*))$. Its affine spectrum consists of symmetric matrices of rank at most one; its projectivization consists of their nonzero rank-one classes. The second factor is dual. Adjoining the free vertex coordinates $a,b$ gives the two projective cones. The fibre product identifies their constant terms: the affine cones meet only at the origin, while their projectivizations are disjoint. Both factors are polynomial extensions of reduced Veronese rings, which proves the scheme assertion.

The degree-$j$ piece of the first matrix ring is therefore $S_j^+\simeq\chi_{-j}\boxtimes\operatorname{Sym}^{2j}V$, and the second is $S_j^-\simeq\chi_j\boxtimes\operatorname{Sym}^{2j}V^*$. Multiplication by $a^q$ gives degree $j+q$ and circle weight $-j-2q$; multiplication by $b^q$ on the dual side gives weight $j+2q$. The fibre product in (6.1) retains one common constant. This proves the graded $K_1\times K_2$-decomposition and completes Theorem 3.1 for $E_6$.

The linear section in type $E_7$

Here we compute $Q=R/(p_2)$ for the long-root $\mathrm{SL}_2\times\mathrm{Spin}_{12}$ pair in $E_7$. The geometry of four-planes in a direct sum gives two Segre cones; their equations and graded coordinate rings prove Theorem 3.1 in this case.

The two Segre cones

Let $U=V\oplus W$, with $\dim V=\dim W=4$. A point $L\in\operatorname{Gr}(4,U)$ is a four-dimensional subspace of $U$. The vector decomposition

has weights $2,1,0,-1,-2$; its coordinate functions have the opposite weights. Here $p_2=\Lambda^2V\otimes\Lambda^2W$ is precisely the middle term. Under the Killing-form identification with linear functions, the equations $p_2=0$ set all middle Plücker coordinates to zero. Thus the middle component of a Plücker vector for $L$ must vanish. We first describe the resulting points geometrically; afterwards we check the scheme structure with equations.

For a four-plane $L\subset V\oplus W$, put $r=\dim(L\cap V)$ and $s=\dim(L\cap W)$. Choose bases of these two intersections and complete them to a basis of $L$ by vectors $v_i+w_i$. The classes of the $v_i$ in $V/(L\cap V)$ are linearly independent, as are the classes of the $w_i$ in $W/(L\cap W)$. Expanding the wedge of this basis, its nonzero components have exactly the $V$-degrees $r,r+1,\ldots,4-s$: the terms of each such degree are distinct basis wedges and cannot cancel. In particular the middle component is nonzero exactly when $r \le2$ and $s \le2$. Its vanishing therefore gives

These loci are disjoint, since a four-plane cannot contain both a three-plane in $V$ and a three-plane in $W$.

Now let $L \in X_{+}$, with $L \ne V$. Then $A = L \cap V$ has dimension three, and $L/A$ is a line in

Its projection to $W$ is nonzero, so its image is a line $D \subset W$. We have therefore associated to $L$ a pair

Conversely, fix such a pair. The four-planes containing $A$ and contained in $V \oplus D$ are parameterized by $\mathbb{P}((V/A) \oplus D) \simeq\mathbb{P}^{1}$. One of them is $V$; all the others have intersection $A$ with $V$ and image $D$ in $W$. Thus fixing $(A,D)$ gives a whole line of four-planes, with $V$ as one endpoint and $A \oplus D$ as the other distinguished point.

This is also a projective line in the Plücker embedding. Indeed, choose a basis $u_{1},u_{2},u_{3}$ of $A$, a vector $v \in V \setminus A$, and a nonzero $w \in D$. The four-plane $A + \mathbb{C}(\lambda v + \mu w)$ has Plücker point

The first endpoint is the fixed point $[\Lambda^{4}V]$, representing $V$. The second endpoint, representing $A \oplus D$, varies over the Segre embedding of $\operatorname{Gr}(3,V) \times\operatorname{Gr}(1,W)$ in $\mathbb{P}(\Lambda^{3}V \otimes W)$. Hence $X_{+}$ is the projective cone over this product, with vertex $V$. Projection from the vertex is exactly $L \mapsto(A,D)$. Interchanging $V$ and $W$ gives $X_{-}$, the cone over $\mathbb{P}(V) \times\mathbb{P}(W^{*})$ with vertex $W$.

We now verify that these two cones give the entire linear section scheme-theoretically, with no additional nilpotents.

Index bases of $V,W$ by $1,\ldots,4$ and $5,\ldots,8$. Let $I$ be the Plücker ideal in $\mathbb{C}[p_{J}]$, and let $I_{2}$ be its homogeneous degree-two part, the vector space of Plücker quadrics. These quadrics generate $I$. Set to zero the coordinates with two indices in each block, and denote the image of $I_{2}$ by $\overline{I}_{2}$. The surviving coordinates are $a=p_{1234}$, $b=p_{5678}$, and two $4 \times4$ arrays: $x$ has three $V$ indices and one $W$ index, while $y$ has one $V$ index and three $W$ indices. Thus $\overline{I}_{2}$ is a space of quadratic polynomials in $S=\mathbb{C}[a,b,x,y]$, and $Q=S/(\overline{I}_{2})$, where parentheses mean the ideal generated by this space.

Write $X=\operatorname{span}\{x_{ij}\}$ and $Y=\operatorname{span}\{y_{ij}\}$. After the cofactor sign conventions, these are the $\operatorname{SU}(V) \times\operatorname{SU}(W)$-modules $V \otimes W^{*}$ and $V^{*} \otimes W$, respectively; $a,b$ are invariant. Let $\overline{I}_{2}(x)$ be the vector space spanned by the quadratic minors

and define $\overline{I}_{2}(y)$ similarly. In particular,

each of dimension $6 \cdot6 = 36$. The decomposition of the specialized quadratic relation space is

Here $\mathcal{X}\mathcal{Y}$ denotes the span of all products $x_{ij}y_{kl}$, naturally isomorphic to $\mathcal{X} \otimes\mathcal{Y}$. All summands in (7.1) are spaces of quadratic generators. They are grouped by total $V$-degree, rather than into irreducible representations: the middle summand is reducible.

To verify the decomposition, substitute the middle-coordinate equations in the generating Plücker quadrics

Repeated indices give zero and permutation of indices changes signs. Grouping by the total number of $V$ indices gives the following summands:

total $V$-degree	quadratic generators	$SU(V)\times SU(W)$-module	dimension
6	$\overline{I}_{2}(x)$	$\Lambda^{2}V\otimes\Lambda^{2}W^{*}$	36
5	$aY$	$V^{*}\otimes W$	16
4	$\mathbb{C}ab\oplus XY$	$1\oplus(\operatorname{End}V\otimes\operatorname{End}W)$	257
3	$bX$	$V\otimes W^{*}$	16
2	$\overline{I}_{2}(y)$	$\Lambda^{2}V^{*}\otimes\Lambda^{2}W$	36

The dimensions follow immediately from the displayed modules: $6\cdot6$, $4\cdot4$, $1+16\cdot16$, $4\cdot4$, and $6\cdot6$. We verify directly that these spaces are exactly the specialized relations.

The minors of $x,y$ and the products $ay_{ij},bx_{ij}$ follow at once from (21) after setting the middle coordinates to zero. For the mixed products, write $v_{1},\ldots,v_{4}$ and $w_{1},\ldots,w_{4}$ for the two indexed bases, with index sets $V=\{v_{1},\ldots,v_{4}\}$ and $W=\{w_{1},\ldots,w_{4}\}$. Choose the cofactor signs

If $i\ne k$, choose a third $V$ index $m$. In (21) take the three-index list to be $(V\setminus\{v_{i},v_{m}\})\cup\{w_{j}\}$ and the five-index list to be $\{v_{m},v_{k}\}\cup(W\setminus\{w_{l}\})$. Only the term $x_{ij}y_{kl}$ survives, so this product vanishes. Interchanging the blocks gives the same conclusion when $j\ne l$. For the remaining products, fix $j$ and put $c_{i}=x_{ij}y_{ij}$. Using the same three-index list and the five-index list $\{v_{m},v_{i}\}\cup(W\setminus\{w_{j}\})$ gives $c_{i}+c_{m}=0$ for every $i\ne m$. Taking three distinct indices yields $c_{i}=0$. Thus all $xy$ products are relations. Finally, the three-index list $V\setminus\{v_{i}\}$ and the five-index list $\{v_{i}\}\cup W$ give $ab$ plus a linear combination of these products, so $ab$ is a relation as well.

Conversely, the proposed cones consist of decomposable four-vectors, so every specialized Plücker quadric vanishes on them. Their quadratic vanishing space is precisely the right side of (20): within each coordinate block the relations are the minors, and between the blocks they are all mixed products. This proves (20). Since $I$ is generated by $I_{2}$, its specialization is generated by $\overline{I}_{2}$.

In particular, the relations of total $V$-degrees 5, 4, 3 are exactly

Together they generate the product ideal $(a,x)(b,y)$. Thus a point has either $(a,x)=0$ or $(b,y)=0$; the remaining minors cut out the corresponding rank-one cone. The two affine components meet only at the origin, and their projectivizations are disjoint.

Let $(\overline{I}_{2}(x))\subset\mathbb{C}[x]$ and $(\overline{I}_{2}(y))\subset\mathbb{C}[y]$ be the ideals generated by the two spaces of minors. Adjoining the vertex coordinates and imposing the mixed quadratic generators gives

Both factors are polynomial extensions of rank-one matrix rings, hence reduced. This proves the scheme assertion and the cone description.

The substitution $x_{ij}=s_it_j$ identifies the degree-$j$ piece of the first matrix ring with the polynomials of bidegree $(j,j)$:

The second ring gives the dual $K_2$-type with circle weight $j$. Adjoining $a,b$, of degree one and weights $-2,2$, gives the terms $a^qS_j^+$ and $b^qS_j^-$ in degree $j+q$, with weights $-j-2q$ and $j+2q$. The fibre product in (7.3) retains one common constant, proving Theorem 3.1 for $E_7$.

The linear section in type $E_8$

This section computes $Q=R/(p_2)$ for the long-root $\mathrm{SL}_2\times E_7$ pair in $E_8$. We identify two cones over Grassmannians inside the spinor variety, verify their scheme structure, and read off the graded coordinate rings to prove Theorem 3.1.

Pure spinors and the two Grassmannian cones

Write the orthogonal space as $U=V\oplus V^*$, $\dim V=8$, with the hyperbolic symmetric form

The variety $\operatorname{OGr}^+(8,U)$ parameterizes one of the two components of maximal isotropic eight-planes; we choose the component containing $V$. It also contains $V^*$, since $\dim V=8$ is even. We write $L\in\operatorname{OGr}^+(8,U)$ for such an eight-dimensional maximal isotropic subspace of $U$. We use the spinor embedding of $\operatorname{OGr}^+(8,U)$. The subgroup of $\operatorname{SO}(U)$ acting by $(v,\phi)\mapsto(gv,(g^{-1})^*\phi)$, with $g\in\operatorname{GL}(V)$, is a Levi subgroup of the parabolic stabilizing $V$. By the covering Levi we mean its inverse image in $\operatorname{Spin}(U)$. Explicitly, this group is

Projection $(g,z)\mapsto g$ is a double cover with kernel $\{(1,1),(1,-1)\}$. Its compact form is the double cover of $U(V)$ obtained by requiring $g$ unitary and $|z|=1$. Here $V$ has dimension eight, whereas $U=V\oplus V^*$ has dimension sixteen. The one-dimensional representation $\delta(g,z)=z$ is an actual character of this cover and satisfies $\delta^2=\det V$. The exterior-algebra construction gives the ambient Cartan space $\mathfrak p=S^+=\delta\otimes\Lambda^{\mathrm{even}}V^*$. On the lifted copy $g\mapsto(g,1)$ of $\operatorname{SU}(V)$, the character $\delta$ is trivial; a volume form identifies $\Lambda^6V^*\simeq\Lambda^2V$ and $\Lambda^4V^*\simeq\Lambda^4V$. Thus, as an $\operatorname{SU}(V)$-module,

These five summands come from exterior degrees $0,2,4,6,8$ in the spinor model. Their dimensions are $1,28,70,28,1$, and their circle weights are $2,1,0,-1,-2$: the factor $\delta$ has weight $2$, while $V^*$ has weight $-1/2$ on the cover. The half-spin action descends to the semispin quotient used for $K$ in Section 2.

Dually, the degree-one coordinate space is $R_1=(S^+)^*=\delta^{-1}\otimes\Lambda^{\mathrm{even}}V$, whose five pieces have circle weights $-2,-1,0,1,2$. The middle piece is the 70-dimensional Cartan space $\mathfrak p_2$ of the centralizer, identified with linear functions by the Killing form. Thus $\mathfrak p_2=0$ sets all exterior-degree-four spinor coordinates to zero. Let $a,x,y,b$ denote the remaining coordinate pieces, of exterior degrees $0,2,6,8$, respectively. Using the fixed volume form, identify the $y$ coordinates with complementary two-index coordinates on the dual side.

We first examine the open chart $a \ne0$, consisting of eight-planes transverse to $V^*$. Such a plane is the graph of a map $B: V \to V^*$. Isotropy says $B(v)(w)+B(w)(v)=0$, so $B$ is an alternating form, an element of $\Lambda^2 V^*$. The spinor coordinates on this chart are its principal Pfaffians; equivalently its spinor line is represented by

We use the usual Pfaffian chart conventions, as in [10 Section 2]. Killing the middle coordinates says $B\wedge B=0$. An alternating form has this property exactly when its rank is at most two. Thus $B=0$, giving $L=V$, or $B$ is a decomposable two-form with six-dimensional kernel. Since the graph meets $V$ in $\ker B$, this calculation leads to the locus $X_+$ below; the opposite chart gives $X_-$:

They are disjoint, since $6+6>8$. We next describe each entire locus, including the boundary of its graph chart. The equations in the next subsection will show that these are exactly the two components of the linear section, with no further components or nilpotents.

Take $L\in X_+$ with $L\ne V$. In our chosen orthogonal component $\dim(L\cap V)$ is even, so it must be six. Set

Both $Q_A$ and $D$ have dimension two. Isotropy forces $L$ to lie in $A^\perp=V\oplus D$, so $L/A$ is a maximal isotropic two-plane in the four-dimensional orthogonal space

It belongs to the same component as $Q_A$.

For fixed $A$, this component is a projective line. To see this directly, its two-planes transverse to $Q_A^*$ are graphs of alternating maps $Q_A\to Q_A^*$. These maps form the one-dimensional space $\Lambda^2Q_A^*=\det Q_A^*$. The only remaining two-plane in this component is $Q_A^*$ itself: intersection with $Q_A^*$ has even dimension, so a nontransverse plane must equal it. Thus the affine line of graphs acquires one point at infinity, giving

The zero map gives $Q_A$, hence $L=V$; the point at infinity gives $Q_A^*$, hence $L=A\oplus D$. Every other member has intersection exactly $A$ with $V$.

This projective line is a line in the spinor embedding as well. Choose $0\ne\eta\in\det D\subset\Lambda^2V^*$. Since $\eta\wedge\eta=0$, the graphs of multiples of $\eta$ have spinor points $[\delta(1+c\eta)]$. Their closure is the line

joining the vertex $[\delta]$, representing $V$, to $[\delta\eta]$, representing $A\oplus D$. As $A$ varies, its annihilator $D$ varies through $\operatorname{Gr}(2,V^*)$, and $[\eta]=[\det D]$ is precisely its Plücker point. Consequently $X_+$ is the projective cone over the Plücker-embedded $\operatorname{Gr}(2,V^*)$, with vertex $V$. Projection from the vertex sends $L\ne V$ to $D=\operatorname{Ann}(L\cap V)$. The description of $X_-$ is dual, with base $\operatorname{Gr}(2,V)$ and vertex $V^*$.

The specialized quadratic generators

Put $N = \{1,\ldots,8\}$ and label spinor coordinates $p_I$ by even subsets $I \subset N$. Let $I$ be the pure-spinor ideal and $I_2$ its homogeneous degree-two part. This space is spanned by the Wick relations and generates $I$ [10 Section 2]. For odd subsets $S,T$, write $S \mathbin{\triangle} T = \{i_1 < \cdots< i_m\}$; the relation is

Set all $p_I$ with $|I|=4$ to zero, and denote the image of $I_2$ by $\overline{I}_2$. The surviving coordinates are $a=p_{\varnothing}$, $b=p_N$, and two skew arrays, which we normalize by

The sign on $y$ is the complement sign for the fixed volume form. As $\operatorname{SU}(V)$-modules, the coordinate spans are $\mathcal{X}\simeq\Lambda^2V$ and $\mathcal{Y}\simeq\Lambda^2V^*$, each of dimension 28. The full specialized ideal in $\mathbb{C}[a,b,x,y]$ is $(\overline{I}_2)$.

Write $\overline{I}_2(x)$ for the space spanned by the quadratic Pfaffians

and define $\overline{I}_2(y)$ similarly. These are the Plücker quadrics for $\operatorname{Gr}(2,V^*)$ and $\operatorname{Gr}(2,V)$, respectively. Their module structures are

Indeed, the Pfaffian map identifies the first space with the $\Lambda^4V$ summand in $\operatorname{Sym}^2(\Lambda^2V)=S_{(2,2)}V\oplus\Lambda^4V$; the second is dual. Each has dimension $\binom{8}{4}=70$.

The full space of specialized quadratic generators is

Here $\mathcal{X}\mathcal{Y}\simeq\Lambda^2V\otimes\Lambda^2V^*$ is the span of all $xy$ products. With $V_\lambda$ denoting the irreducible $\operatorname{SU}(8)$-module of highest weight $\lambda$, it decomposes as

Consequently the summands of (8.2) are as follows:

circle weight	generator space	SU(8)-module	dimension
$-2$	$\overline{I}_2(x)$	$V_{\omega_4}$	70
$-1$	$a\mathcal{Y}$	$V_{\omega_6}$	28
0	$\mathbb{C}ab\oplus\mathcal{X}\mathcal{Y}$	$\mathbb{C}^{\oplus2}\oplus V_{\omega_1+\omega_7}\oplus V_{\omega_2+\omega_6}$	785
1	$b\mathcal{X}$	$V_{\omega_2}$	28
2	$\overline{I}_2(y)$	$V_{\omega_4}$	70

The two scalar summands at weight zero are $\mathbb{C}ab$ and the invariant contraction in $\mathcal{X}\mathcal{Y}$. The dimensions follow from the displayed modules and sum to 981.

We prove (8.2) directly from (8.1). Taking $S=\{i\}$ and $T=\{j,k,l\}$, with four distinct indices, gives a Pfaffian in $x$, since the term involving $p_{\{i,j,k,l\}}$ vanishes. For a six-element set $C$, taking $S=\{i\}$ and $T=C\setminus\{i\}$ with $i\in C$ leaves only $ap_C$; all other terms contain a four-index coordinate. This gives $a\mathcal{Y}$. Complementing all index sets gives the dual Pfaffians and $b\mathcal{X}$.

For the mixed products there are three cases. If $i,j,k,l$ are distinct, choose $r$ outside these four indices and take $S=\{i,j,r\}$ and $T=N\setminus\{k,l,r\}$. Only the term proportional to $x_{ij}y_{kl}$ survives, so it is a relation. If the pairs share one index, fix distinct $i,l$ and put $q_j=x_{ij}y_{jl}$ for $j\notin\{i,l\}$. The choices $S=\{i,j,k\}$ and $T=N\setminus\{j,k,l\}$ give $q_j+q_k=0$ for distinct $j,k$ outside $\{i,l\}$. Using three such indices yields $q_j=0$. Finally, put $c_{ij}=x_{ij}y_{ij}=c_{ji}$. Taking $S=\{i,j,k\}$ and $T=N\setminus S$ gives

Comparing the four triangle equations on four distinct indices yields $c_{ij}=c_{kl}$ for disjoint pairs. Any two pairs have a third pair disjoint from both, so all $c_{ij}$ are equal. The triangle equation then forces them all to vanish. Thus every $xy$ product is a relation. Taking $S=\{i\}$ and $T=N\setminus\{i\}$ now gives $ab$ plus mixed products already shown to vanish, proving that $ab$ is a relation.

Conversely, the cone in the $(a,x)$ coordinates consists of pure spinors: for $a\ne0$ it is represented by $a\exp(B)$ with $B$ decomposable, and the remaining points lie in its closure. The other cone is dual. Hence every specialized Wick quadric vanishes on these two cones. Their quadratic vanishing space is the right side of (24): the Pfaffians within each block and all products between the blocks. This proves equality of the generator spaces. Since $I=(I_2)$, these quadrics generate the entire specialized ideal.

The generators at circle weights $-1,0,1$ generate $(a,x)(b,y)$. They separate the two components; the remaining generators impose decomposability on each skew array. Writing parentheses for the ideals generated by the pure quadratic spaces, we obtain

Here $\mathbb{C}[x]=\mathbb{C}[x_{ij}:i<j]$, and similarly for $y$. The two matrix rings are the reduced homogeneous coordinate rings of the Plücker-embedded $\operatorname{Gr}(2,V^*)$ and $\operatorname{Gr}(2,V)$. Adjoining $a,b$ gives their projective cones with vertices $V,V^*$. Both maps in the fibre product evaluate at the origins. Thus the two affine components meet only at the origin, while their projectivizations are disjoint. This proves the scheme assertion.

The standard graded decomposition of the Plücker coordinate ring of $\operatorname{Gr}(2,V^*)$ gives

Here each $x$-coordinate has degree one and circle weight $-1$. The second ring gives $(\mathbb{S}_{(j,j)}V)^*$ with weight $j$. Thus $a^qS_j^+$ and $b^qS_j^-$ have degree $j+q$ and weights $-j-2q$ and $j+2q$, respectively. The fibre product in (25) retains one common constant. This proves Theorem 3.1 for $E_8$. Mathematics Department, University of Utah, Salt Lake City, UT 84112

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