A finiteness result for split exceptional dual pairs

To Ranee Brylinski, in memoriam.

Introduction

Let $\mathfrak{e}$ be a split real Lie algebra of type $E_6$, $E_7$, or $E_8$, and let $\Omega$ be the Harish–Chandra module of its spherical minimal representation. A reductive dual pair in $\mathfrak{e}$ will mean a pair of reductive subalgebras

which are preserved by a Cartan involution and are mutual centralizers, up to the harmless center issues. Write $K_1$ and $K_2$ for the corresponding maximal compact subgroups. If $\pi_2$ is an irreducible $(\mathfrak{g}_2,K_2)$-module, its big theta lift is the $(\mathfrak{g}_1,K_1)$-module

Our purpose is to prove that these algebraic big theta lifts have finite length for a broad family of split exceptional dual pairs.

Each split algebra $\mathfrak{e}$, for $n=6,7,8$, contains a nested chain of split subalgebras of types

Taking the centralizer of each member gives the pairs displayed in (35); see also [10]. There are five entries for each of the three ambient algebras. In $E_8$, however, the centralizer of $G_2$ is $F_4$ and the centralizer of $F_4$ is $G_2$. Those two entries therefore describe the same unordered pair, leaving fourteen distinct pairs in all.

Our main result treats all but the symmetric $D_4$ pair in $E_8$.

We describe the argument. For an irreducible $K_2$-module $\tau$, consider the compact-type lift

The central point is that $\Theta_1(\tau)$ is finitely generated over $U(\mathfrak{g}_1)$. Three ingredients enter the proof. First, give $\Omega$ its natural $K$-invariant filtration and put

Key words and phrases. Exceptional theta correspondences, minimal representations, Harish–Chandra modules.

This work is supported by the Croatian Science Foundation under the project IP-2022-10-4615, a gift No. 946504 from the Simons Foundation and an NUS MOE grant A-8002493-00-00.

Then $R$ is the coordinate ring of the highest-weight cone associated with the minimal representation: these are the Plücker cones over $\operatorname{LG}(4,8)$ and $\operatorname{Gr}(4,8)$ in types $E_{6}$ and $E_{7}$, and the pure-spinor cone over $\operatorname{OGr}^{+}(8,16)$ in type $E_{8}$. Second, the Hilbert basis theorem and averaging over $K_{2}$ imply that the covariant space

is a finitely generated module over $R^{K_{2}}$. Third, with $\mathfrak{g}_{1}$ chosen as the smaller member of the dual pair, the natural graded map

is an isomorphism in the thirteen cases of the theorem. More precisely, it arises from a strict filtered isomorphism between the universal spherical $(\mathfrak{g}_{1},K_{1})$-module and $\Omega^{K_{2}}$. The injectivity of the filtered action map follows from restriction estimates placing the matrix coefficients of $\Omega$ in $L^{2-\epsilon}(G_{1})$; the surjectivity then follows from the calculation of the graded invariant dimensions.

The first two ingredients and (2) show that the associated graded compact-type lift is finitely generated over $\operatorname{Sym}(\mathfrak{p}_{1})$. A filtered-generation argument then lifts its homogeneous generators and proves finite generation over $U(\mathfrak{g}_{1})$. Taking a $K_{2}$-type in an irreducible module $\pi_{2}$ exhibits $\Theta(\pi_{2})$ as a quotient of such a compact-type lift. Consequently, the big theta lift is finitely generated in one direction. Applying the same argument from the other side, through the usual see-saw adjunction, proves admissibility in the reverse direction. The known correspondence of infinitesimal characters, together with the standard finiteness theorems for Harish–Chandra modules, then gives finite length in both directions.

The paper is organized as follows. Section 2 proves the elementary filtered-generation lemma. Sections 3 and 4 recall the three minimal representations and their homogeneous cones. Section 5 proves finite generation of compact-type lifts by averaging over the compact group, and Section 6 passes from these modules to big theta lifts. In Section 7 we verify the required invariant dimensions for the split centralizer pairs and reduce Theorem 1.1 to the injectivity of the filtered action maps. Section 8 proves the matrix-coefficient estimates that supply this injectivity and thereby completes the argument.

This paper was written with assistance of ChatGTP. The organization of the paper and the proofs are due to the authors.

A filtered-generation lemma

Let $(\mathfrak{g},K)$ be a Harish–Chandra pair, with complexified Cartan decomposition

Let $V$ be a $(\mathfrak{g},K)$-module with an exhaustive increasing filtration

such that every $V_{n}$ is $K$-stable and

Equivalently, $\mathfrak{g}V_{n}\subset V_{n+1}$; the sharper degree-zero statement for $\mathfrak{k}$ follows from $K$-stability. The $K$-stability implies $\mathfrak{k}V_{n}\subset V_{n}$. Consequently $\mathfrak{p}$ acts in degree one on

If $X,Y\in\mathfrak{p}$, then $[X,Y]\in\mathfrak{k}$ preserves the filtration degree. Hence $XY-YX$ has degree at most one less than $XY$ on the associated graded module. The degree-one operators coming from $\mathfrak{p}$ therefore commute, and $\operatorname{gr}V$ is naturally a graded $(\operatorname{Sym}(\mathfrak{p}),K)$-module.

The three split groups of type $E$

Let $H_i$ be the split real group of type $E_i$, for $i=6,7,8$, and let $\Omega_i$ be its spherical minimal Harish–Chandra module. It is the irreducible spherical module whose annihilator is the Joseph ideal. The construction of Brylinski–Kostant [2] gives a uniform description of its $K_i$-types.

Up to the indicated finite central quotients, the complexified Cartan decompositions and the noncompact modules are as follows. We number the fundamental weights in the standard Bourbaki convention.

$H_i$	$K_i$	$\mathfrak{p}_i$	highest weight of $\mathfrak{p}_i$
$E_{6(6)}$	$\operatorname{Sp}(4)/\{\pm1\}$	$\Lambda^4_0\mathbb{C}^8$	$\omega_4$ ($C_4$)
$E_{7(7)}$	$\operatorname{SU}(8)/\{\pm I\}$	$\Lambda^4\mathbb{C}^8$	$\omega_4$ ($A_7$)
$E_{8(8)}$	$\operatorname{Spin}(16)/\mu_2$	$S^+$	$\omega_8$ ($D_8$)

Here $\operatorname{Sp}(4)$ is the compact symplectic group of rank four, $\Lambda^4_0\mathbb{C}^8$ is the kernel of symplectic contraction, and $S^+$ is one of the two 128-dimensional half-spin modules. In the last row, $\mu_2$ is the central subgroup acting trivially on the chosen $S^+$; thus $K_8$ is the corresponding semispin group. Interchanging $S^+$ and $S^-$ interchanges $\omega_8$ and $\omega_7$ and changes none of the statements.

The $K_i$-spectra are multiplicity free:

In each case the summand with $n = 0$ is the trivial $K_i$-type and contains the spherical vector.

The Joseph ideal and its quadratic generators

We recall the part of the construction of the Joseph ideal that will be needed below. All Lie algebras and modules in this paragraph are complexified. Let $\beta_i$ be the highest root of $\mathfrak{e}_i$. There is a multiplicity-free decomposition

where $W_i$ is irreducible in the three exceptional cases under consideration. Its dimensions are

More intrinsically, if $\mathfrak{m}_i$ is the centralizer of $h_{\beta_i}$ and $\beta_i'$ is the highest root of $[\mathfrak{m}_i,\mathfrak{m}_i]$, then

Let $C_i \in U(\mathfrak{e}_i)$ be the Casimir element. The copy of $W_i$ in (3) has a canonical lift $\widetilde{W}_i \subset U^2(\mathfrak{e}_i)$, obtained by symmetrization. Garfinkle’s description [4], in the form recalled by Gan–Savin [3], says that the Joseph ideal $J_i$ is generated by

where $c_i$ is the Casimir eigenvalue of the minimal representation. Let $\mathcal{O}_{\min} \subset\mathfrak{e}_i$ denote the minimal nonzero nilpotent adjoint orbit. Then

Equivalently, the kernel of the Cartan-product map

is precisely $\mathbf{1} \oplus W_i$. Thus both the classical equations of the minimal nilpotent orbit and their quantization in the Joseph ideal are generated in degree two.

The three highest-weight cones

Here we prove that $R_i = \operatorname{gr}\Omega_i$ is the coordinate ring of the highest-weight cone in $\mathfrak{p}_i$. Let $v_i$ be a highest-weight vector in $\mathfrak{p}_i$ and put

The projective variety $X_i$ is the closed $K_{i,\mathbb{C}}$-orbit in $\mathbb{P}(\mathfrak{p}_i)$. The three cases are

$i$	$X_i$	homogeneous embedding
6	$\operatorname{LG}(4,8)$	$\operatorname{LG}(4,8) \hookrightarrow\mathbb{P}(\Lambda^4\mathbb{C}^8)$
7	$\operatorname{Gr}(4,8)$	$\operatorname{Gr}(4,8) \hookrightarrow\mathbb{P}(\Lambda^4\mathbb{C}^8)$
8	$\operatorname{OGr}^{+}(8,16)$	$\operatorname{OGr}^{+}(8,16) \hookrightarrow\mathbb{P}(S^{+})$

Their dimensions are 10, 16, and 28, respectively; the corresponding affine cones have dimensions 11, 17, and 29. The first two maps are the Plücker embeddings; in the symplectic case the image lies in the primitive summand $\Lambda^4_0\mathbb{C}^8$. The third variety is one of the two connected components of the Grassmannian of maximal isotropic eight-planes in $\mathbb{C}^{16}$. Its embedding in $\mathbb{P}(S^{+})$ is the spinor, or pure-spinor, embedding. It is important that this is the minimal homogeneous embedding, not the ordinary Plücker embedding into $\mathbb{P}(\Lambda^8\mathbb{C}^{16})$: the Plücker line bundle is the square of the spinor line bundle.

Use the restriction of the Killing form to identify $\mathfrak{p}_i \simeq\mathfrak{p}_i^\vee$, and put $\varpi_i = \omega_4,\omega_4,\omega_8$, respectively, so that $\mathfrak{p}_i = V_{K_{i,\mathbb{C}}}(\varpi_i)$. The degree-two equations of the three cones are the kernels of the Cartan-product maps

More explicitly, one has

$i$	$S^{2}(\mathfrak{p}_{i})$	$I_{2}(\widehat{X}_{i})$
$6$	$V_{C_{4}}(2\omega_{4}) \oplus V_{C_{4}}(2\omega_{2}) \oplus\mathbf{1}$	$V_{C_{4}}(2\omega_{2}) \oplus\mathbf{1}$
$7$	$V_{A_{7}}(2\omega_{4}) \oplus V_{A_{7}}(\omega_{2}+\omega_{6}) \oplus\mathbf{1}$	$V_{A_{7}}(\omega_{2}+\omega_{6}) \oplus\mathbf{1}$
$8$	$V_{D_{8}}(2\omega_{8}) \oplus V_{D_{8}}(\omega_{4}) \oplus\mathbf{1}$	$V_{D_{8}}(\omega_{4}) \oplus\mathbf{1}$

The dimensions of the nontrivial quadratic-relation modules in the last column are 308, 720, and 1820. These are the symplectic Plücker, ordinary Plücker, and pure-spinor quadrics, respectively. Together with the invariant quadratic equation in each row, they generate the homogeneous ideal of $\widehat{X}_{i}$.

We now relate these equations directly to the Joseph ideal. The Cartan decomposition gives a $K_{i,\mathbb{C}}$-equivariant projection

Under the Killing-form identifications, this projection is the restriction of quadratic polynomials from $\mathfrak{e}_{i}$ to $\mathfrak{p}_{i}$.

Proof. Put $Q_{i}=\operatorname{pr}_{\mathfrak{p}_{i}}(\mathbf{1}\oplus W_{i})$. Since $\mathbf{1}\oplus W_{i}$ generates the ideal of the minimal orbit closure by (5), the common zero set of $Q_{i}$ is

Here the last equality is the geometric identification of the minimal orbit’s intersection with $\mathfrak{p}_{i}$ as the highest-weight orbit, with the origin added; see [1 Section 3]. At this stage we only use this identity as an equality of closed sets.

By (7),

where $q_{i}$ is the restricted Killing form and $E_{i}$ is the nontrivial irreducible module of dimension 308, 720, or 1820, respectively. The restriction of the invariant quadratic form on $\mathfrak{e}_{i}$ to $\mathfrak{p}_{i}$ is nonzero, so $\mathbb{C}q_{i}\subset Q_{i}$. Since $Q_{i}$ is $K_{i,\mathbb{C}}$-stable and $E_{i}$ is irreducible, either $Q_{i}=\mathbb{C}q_{i}$ or $Q_{i}=\mathbb{C}q_{i}\oplus E_{i}$. The first possibility would make $\mathcal{V}(Q_{i})$ a quadric hypersurface in $\mathfrak{p}_{i}$, whereas $\widehat{X}_{i}$ has codimension greater than one in all three cases. Thus $Q_{i}=I_{2}(\widehat{X}_{i})$, proving (8).

Finally, $I(\widehat{X}_{i})$ is generated by its degree-two part. Hence the ideal generated by the restrictions of $\mathbf{1}\oplus W_{i}$ is exactly $I(\widehat{X}_{i})$, which also proves the assertion about the scheme-theoretic intersection. □

By the Borel–Weil Theorem, the homogeneous coordinate ring of each cone is

These representations are self-dual, so the dual signs may be suppressed when one records only the $K_{i}$-spectrum.

Let $v_{0}\in\Omega_{i}$ be a spherical vector and give $\Omega_{i}$ its cyclic PBW filtration

Since $\mathfrak{k}_i v_0=0$, commuting every $\mathfrak{k}_i$-factor to the right shows that the associated graded module is cyclic over $S(\mathfrak{p}_i)$. Thus there is a natural surjection

If $z\in U_2(\mathfrak{e}_i)$ annihilates $\Omega_i$, its leading action on the spherical vector is the projection to $S^2(\mathfrak{p}_i)$ of its ordinary PBW symbol. Indeed, this is simply the quotient

The constant $c_i$ in $C_i-c_i$ disappears on passing to the associated graded. Proposition 4.1 and (4) therefore show that every element of $I_2(\widehat{X}_i)$ annihilates the cyclic vector in $\operatorname{gr}_{F}\Omega_i$. Since the ideal of $\widehat{X}_i$ is generated by these quadrics, (11) factors through a graded surjection

The $K_i$-spectrum of an associated graded module agrees with that of the filtered module. By the Brylinski–Kostant $K_i$-type formula in the preceding section and the Borel–Weil formula (10), the two sides of (12) contain precisely the same irreducible $K_i$-modules, each with multiplicity one. The surjection is therefore an isomorphism. We have proved, without assuming a geometric model for the associated graded module, that

In particular, $V_{K_i}(n\varpi_i)$ first occurs in filtration degree $n$, so the cyclic PBW filtration agrees with the $K_i$-type filtration

This also proves $\mathfrak{p}_i(\Omega_i)_{\leq n}\subset(\Omega_i)_{\leq n+1}$ and gives the claimed $S(\mathfrak{p}_i)$-action intrinsically. Since every ring in (10) is generated in degree one, it is cyclic over $\operatorname{Sym}(\mathfrak{p}_i)$, with generator $1$ in degree zero. Proposition 2.1 therefore gives the familiar conclusion: each $\Omega_i$ is generated, as a $U(\mathfrak{e}_i)$-module, by a lift of its trivial $K_i$-type, namely by its spherical vector.

Finite generation of compact-type lifts

Fix one of the three Lie algebras of type $E_n$ and now suppress the index $i$. Thus

Let

be a reductive subalgebra preserved by the Cartan involution. Write

for the induced Cartan decompositions and maximal compact groups. The two factors commute, so $K_1$ acts on every $K_2$-invariant or $K_2$-covariant space.

Let $v_0\in\Omega$ be the spherical vector. We impose the following invariant-cyclicity hypothesis: the map

is a strict filtered surjection. Here $U(\mathfrak{g}_1)$ has the relative PBW filtration, in which $\mathfrak{k}_1$ has degree zero and $\mathfrak{p}_1$ has degree one, and $\Omega^{K_2}$ has the filtration induced from $\Omega$. Strictness is precisely what permits passage to the associated graded. Since $v_0$ is $K_1$-fixed, PBW reduces the cyclic associated graded source to $\operatorname{Sym}(\mathfrak{p}_1)$. Compactness of $K_2$ gives $\operatorname{gr}(\Omega^{K_2})\cong(\operatorname{gr}\Omega)^{K_2}=R^{K_2}$; moreover, under $\operatorname{gr}\Omega\simeq R$, the symbol action of $\mathfrak{p}$ is multiplication by linear functions. Thus (14) gives a surjection of graded algebras

Equivalently, one may take (15) as the graded form of the invariant-cyclicity hypothesis. In applications it is usually this map that is calculated explicitly.

Let $\tau$ be an irreducible $K_2$-module and define its graded covariant space by

Because $K_2$ is compact, averaging identifies coinvariants and invariants, and hence

Thus $M_\tau$ is the multiplicity, or covariant, space of $\tau$ in $R$. For the trivial representation $\mathbf{1}$ this reduces to

We prove the finiteness statement needed here by an elementary averaging argument. For general invariant-theoretic background, see [11 Chapter 1].

Proof. Let $B = R \otimes\operatorname{Sym}(\tau^\vee)$, graded by symmetric degree, so that $B_0 = R$ and $B_1 = R \otimes\tau^\vee$. Let $I \subset B$ be the ideal generated by $B_1^{K_2}$. Since $B$ is a finitely generated complex algebra, the Hilbert basis theorem implies that $I$ is generated by a finite subset $f_1,\ldots,f_m$ of $B_1^{K_2}$.

For $f \in B_1^{K_2}$, express $f$ in this ideal and take symmetric degree one to obtain $f = \sum_i r_i f_i$ with $r_i \in B_0 = R$. Averaging over the compact group $K_2$, since $f$ and the $f_i$ are invariant, gives

where $dk$ is normalized Haar measure. The averaged coefficients belong to $R^{K_2}$, so the $f_i$ generate $M_\tau\simeq B_1^{K_2}$ over $R^{K_2}$. □

Combining Proposition 5.1 with (15), we see that $M_\tau$ is finitely generated over $\operatorname{Sym}(\mathfrak{p}_1)$. Since this module is $K_1$-finite and $K_1$ commutes with $K_2$, its homogeneous generators may be chosen to form a finite-dimensional graded $K_1$-module; denote one such generating module by $F_\tau$.

Now form the compact-type lift

It is naturally a $(\mathfrak{g}_1,K_1)$-module. Give it the quotient filtration induced from the filtration of $\Omega$. Exactness of $K_2$-coinvariants for a compact group gives

Proof. Proposition 5.1 and (15) show that the associated graded module in (19) is generated over $\operatorname{Sym}(\mathfrak{p}_1)$ by the finite-dimensional homogeneous $K_1$-module $F_\tau$. Proposition 2.1, applied to the filtered $(\mathfrak{g}_1,K_1)$-module $\Theta_1(\tau)$, now gives (20). □

From compact-type lifts to big theta lifts

Let $\pi_2$ be an irreducible $(\mathfrak{g}_2,K_2)$-module. We use the algebraic definition

Equivalently, $\Theta(\pi_2)\boxtimes\pi_2$ is the maximal $\pi_2$-isotypic quotient of $\Omega$. In particular, there is a natural surjection

Choose a Vogan-minimal $K_2$-type $\tau$ of $\pi_2$, and put

This is a nonzero finite-dimensional vector space. Since taking a fixed $K_2$-isotypic component is exact, applying $\operatorname{Hom}_{K_2}(\tau,-)$ to (22) gives a surjection of $\mathfrak{g}_1$-modules

Choose a nonzero linear functional $E_\tau\to\mathbb{C}$. Composing it with (23) gives a surjection

Thus only the occurrence of $\tau$ in $\pi_2$ is needed for this argument; choosing a Vogan-minimal type is the natural choice for applications to theta correspondence.

Assume the graded invariant-cyclicity hypothesis

By Theorem 5.2, the compact-type lift in (23) is a quotient of the induced module

Together with (24), this gives

In particular, the big theta lift is finitely generated over $U(\mathfrak{g}_1)$. Admissibility of $\Theta(\pi_2)$ comes from applying the compact-type argument in the opposite direction.

Assume also the reverse graded invariant-cyclicity hypothesis

For an irreducible $K_1$-type $\sigma$, put

Theorem 5.2, with the indices interchanged, shows that $\Theta_2(\sigma)$ is finitely generated over $U(\mathfrak{g}_2)$. The see-saw adjunction gives

Indeed, the left-hand multiplicity space before taking its dual is

and dualizing the coinvariants gives the right-hand Hom space in (28).

Choose a finite-dimensional $K_2$-stable generating space $D_\sigma$ for $\Theta_2(\sigma)$. Restriction to $D_\sigma$ gives an injection

The space on the right is finite dimensional because $\pi_2$ is admissible. Equations (28) and (29) therefore show that every $K_1$-type has finite multiplicity in $\Theta(\pi_2)$.

We have proved the following.

The two hypotheses in Theorem 6.1 are symmetric, but they are not both available for all the exceptional dual pairs considered below. The point needed for the main theorem is that a single invariant map controls the two opposite lifting directions.

Proof. Let $\pi_2$ be an irreducible $(\mathfrak{g}_2,K_2)$-module. In the direction from $\mathfrak{g}_2$ to $\mathfrak{g}_1$, (29) is the hypothesis of Theorem 6.1(1), so $\Theta(\pi_2)$ is finitely generated over $U(\mathfrak{g}_1)$. By the infinitesimal-character correspondence it has an infinitesimal character. The standard Harish–Chandra finiteness theorem therefore makes it admissible and of finite length; see [12 Theorems 3.4.1 and 4.2.1].

Now let $\pi_1$ be an irreducible $(\mathfrak{g}_1,K_1)$-module and lift in the opposite direction. After interchanging the indices in Theorem 6.1(2), the very same map (29) is the reverse invariant map. Hence the lift of $\pi_1$ to $\mathfrak{g}_2$ is admissible. It again has an infinitesimal character. There are only finitely many irreducible $(\mathfrak{g}_2,K_2)$-modules with that infinitesimal character [12 Theorem 5.5.6]. Choose one $K_2$-type from each of them. Admissibility bounds the sum of the multiplicities of these finitely many types, and hence uniformly bounds the length of every finitely generated submodule of the lift. A finitely generated submodule of maximal length is the whole lift. Thus the reverse lift also has finite length. □

Checking invariant cyclicity for split dual pairs

The hypothesis in Theorem 6.2 is usually most accessible when $\mathfrak{g}_1$ is the smaller member of the dual pair. Let $v_0$ again denote the spherical vector in $\Omega$. The action map factors through the universal spherical module

Give $I_1$ the relative PBW filtration. PBW identifies it, as a filtered vector space, with $\operatorname{Sym}(\mathfrak{p}_1)$. Compactness of $K_2$ gives

and the associated graded map of (30) is the graded algebra homomorphism

where $1\in R_0$ is the symbol of $v_0$.

The injectivity required below is supplied by the following analytic input. It is here that matrix-coefficient decay enters the proof.

Consequently the filtered action map $\beta_1$ in (30) is injective.

Proof. For the trivial group both assertions are immediate. Otherwise, the restriction estimate is proved in Proposition 8.1 below. We explain why it implies injectivity. The minimal representation is unitary, so its matrix coefficients are bounded. After increasing the exponent in (32) if necessary, we may therefore take it strictly between 1 and 2. The matrix-coefficient pairing of [9 Section 3.3 and Proposition 3.2], together with Hölder’s inequality, then gives a family of $\mathfrak{g}_1$-intertwining maps

to spherical principal-series modules, for $\lambda$ in a neighborhood of zero, such that $m_\lambda(v_0)$ is a nonzero multiple of the spherical vector $v_\lambda$.

If a nonzero $X\in\operatorname{Sym}(\mathfrak{p}_1)$ belonged to the kernel of $\beta_1$, then

for every $\lambda$ in that neighborhood. The generic nonvanishing lemma [9 Lemma 4.1] says that $\pi_\lambda(X)v_\lambda\ne0$ for generic $\lambda$, a contradiction. This is precisely the argument of [9 Corollary 4.2]; it applies without change to the reductive groups occurring here (and is elementary for the split torus $T_2$). Hence $\beta_1$ is injective. □

This filtered argument is the method used by Loke and Savin in the exceptional theta correspondences in which one member is the split group $G_{2(2)}$ [9]. For the pairs of types

they prove the restriction estimate and injectivity for the smaller member and then establish (33); see [9 Corollary 4.2 and Proposition 4.3]. Thus the smaller-side map is a filtered isomorphism in all three cases. Proposition 7.1 supplies the same injectivity input for the additional smaller members treated below. Loke and Savin also prove the opposite identity in the $E_6$ and $E_7$ cases [9 Proposition 4.7]; no opposite identity is needed here, because Theorem 6.2 shows that the smaller-side identity already suffices in both directions.

The split centralizer table

The same discussion can be made uniformly in the three split groups of type $E$. Up to finite central quotients, the connected centralizers relevant here are

$G$	$C_{E_6}(G)^\circ$	$C_{E_7}(G)^\circ$	$C_{E_8}(G)^\circ$
$A_1$	$A_5$	$D_6$	$E_7$
$A_2$	$A_2 \times A_2$	$A_5$	$E_6$
$G_2$	$A_2$	$C_3$	$F_4$
$D_4$	$T_2$	$A_3$	$D_4$
$F_4$	$\{1\}$	$A_1$	$G_2$

Here all groups are split and $T_2 \simeq(\mathbb{G}_m)^2$ is the split two-dimensional torus. Although we assumed semisimplicity in § 5 to simplify the notation, its proofs use only reductivity and therefore apply without change to the $T_2 \times D_4$ pair. The $A_1$ row records the regular $A_1$ embedding. The $A_1$ occurring as the centralizer of $F_4$ in $E_7$ is a different embedding and belongs to the $F_4$ row. Finally, the two $E_8$ entries $G_2 \times F_4$ and $F_4 \times G_2$ are the same unordered pair. Thus the fifteen entries in the table represent fourteen distinct dual pairs.

The $G_2$ row of (35) is the result of Loke and Savin recalled above. Its reverse orientation is the $E_8$ member of the $F_4$ row. We now check the $A_1$, $A_2$, $F_4$, and $D_4$ rows; in particular, we include the two remaining $F_4$ centralizers in $E_6$ and $E_7$.

The $A_1$ and $A_2$ rows

Lang’s thesis treats the split $A_2$ cases and the corresponding complex $A_1$ cases; see [5 Sections 4.3–4.4 and 6.7]. In particular, the $A_2$ dimension count below is [5 Proposition 4.4.1]. We include the proofs for completeness.

Write $R_n=V(n\omega)$ for the degree-$n$ type in the minimal representation. Proposition 5.1 of [9] gives a multiplicity-free restriction of $V(n\omega)$ to $\mathrm{GL}_4$, $\mathrm{GL}_4 \times\mathrm{GL}_4$, and $\mathrm{GL}_8$ in the $E_6$, $E_7$, and $E_8$ cases, respectively. Taking invariants of the maximal compact subgroup of the indicated centralizer gives the following uniform count.

Proof. For the $A_1$ row, a fixed vector for the maximal compact of the centralizer forces the relevant highest weight in Proposition 5.1 of [9] to be a determinant character. The possible determinant exponents are

$H$	parameter $a$
$E_6$	$-n,-n+2,\ldots,n$
$E_7$	$0,1,\ldots,n$
$E_8$	$-n/2,-n/2+1,\ldots,n/2$

Thus there are $n+1$ fixed lines in each case. For the $A_2$ row, the fixed lines are parametrized, respectively, by

Indeed, in the three cases the complexified compact group is, respectively, $\mathrm{SO}_4$, diagonally embedded $\mathrm{SL}_4$, and $\mathrm{Sp}_8$; the fixed-line conditions are exactly the parity, contragredience, and paired weight conditions occurring in Proposition 5.1 of [9]. Each set $\mathcal{P}_i(n)$ is the set of multisets of size four chosen from $n+1$ values, and hence has cardinality $\binom{n+4}{4}$.

Proposition 7.1 gives the injectivity of $\beta_1$ for both $A_1$ and $A_2$. The dimension equalities just proved and Proposition 7.2 therefore give the two strict filtered isomorphisms. □

The $F_4$ row

For this row it is convenient to orient the dual pair as

and to let $K_{F_4}$ denote a maximal compact subgroup of the split group $F_4$. Thus $K_2=K_{F_4}$ in the notation of § 5 and § 6. The three cases are

In particular, in every case the invariant ring is the symmetric algebra on the noncompact space of the opposite member:

These are the graded maps induced by the action of the opposite member, not merely abstract isomorphisms of polynomial algebras.

For completeness, the first equality says that the only $K_{F_4}$-fixed line in the $E_6$ coordinate ring occurs in degree zero:

For $E_7$, write the standard eight-dimensional space as $U\oplus W$, where $\dim U=6$, $\dim W=2$, and the complexified compact group of $F_4$ acts through $\mathrm{Sp}(U)\times\mathrm{SL}(W)$. Since

the Littlewood–Richardson rule, together with the usual symplectic fixed-vector criterion, shows that a fixed line occurs precisely for

Indeed, $\lambda'$ has even parts, so $S_\lambda(U)$ contains one $\mathrm{Sp}(U)$-fixed line, $S_{(a,a)}(W)=\det(W)^a$ is trivial on $\mathrm{SL}(W)$, and

There are no other fixed summands. Hence

For the $E_7$ entry, Proposition 7.1 gives injectivity for the diagonal $\mathrm{SL}_2$, and the equality of Hilbert functions together with Proposition 7.2 gives the middle isomorphism in (35). The $E_8$ equality is exactly the smaller-side identity for $G_2 \times F_4$ recalled before (35); the $E_6$ entry is tautological. This proves (34) and (35) in all three groups of type $E$.

The $D_4$ row

Let $K_D$ be a maximal compact subgroup of the split group of type $D_4$. Up to a finite central quotient,

The three centralizers and the relevant dimensions are

$H$	$C_H(D_4)^\circ$	$\dim\mathfrak{p}_1$	$\dim\widehat{X}$
$E_6$	$T_2$	2	11
$E_7$	$\mathrm{SL}_2^3$	6	17
$E_8$	$D_4$	16	29

Here $\widehat{X}$ is the appropriate highest-weight cone from Section 4.

For the first two rows, the invariant calculation is again elementary. Decompose the standard eight-dimensional space as a sum of four symplectic planes. The standard symplectic branching rule in the $E_6$ case, and the fourfold Littlewood–Richardson rule in the $E_7$ case, give

Equivalently,

For example, in the $E_7$ calculation one writes $R_n=S_{(n^4)}(\mathbb{C}^2\oplus\mathbb{C}^2\oplus\mathbb{C}^2\oplus\mathbb{C}^2)$; an $\mathrm{SL}_2^4$-invariant can occur only in a term indexed by four rectangular partitions $(a_i,a_i)$, and the resulting Littlewood–Richardson sum is

Proposition 7.1 gives injectivity for $T_2$ and $\mathrm{SL}_2^3$. Since their noncompact spaces have dimensions 2 and 6, respectively, (39) and Proposition 7.2 give strict filtered isomorphisms

Here $R_{E_i}$ denotes the coordinate ring of the corresponding $E_i$ Plücker cone.

The $E_8$ case is different: the desired surjection is impossible for dimension reasons. Let $K_{D,\mathbb{C}}\simeq\mathrm{SL}_2^4$ be the complexification of $K_D$. The Plücker cone has dimension 29, whereas $\dim K_{D,\mathbb{C}}=12$. Since this cone is factorial and $K_{D,\mathbb{C}}$ is semisimple, its field of rational invariants is the fraction field of $R^{K_D}$. The dimension of a generic orbit is at most 12, so

On the other hand, $\dim\mathfrak{p}_1=16$ for the other split $D_4$. A quotient of $\operatorname{Sym}(\mathfrak{p}_1)$ has Krull dimension at most 16, and hence there can be no surjection

Thus Theorem 6.2 applies to the $D_4$ row for $E_6$ and $E_7$, but its invariant-surjection hypothesis cannot hold for the symmetric $D_4\times D_4$ pair in $E_8$.

Combining these calculations with the known correspondences of infinitesimal characters completes the proof of the main result.

Proof of Theorem 1.1. The graded dimension calculations above, together with Propositions 7.1 and 7.2, verify the one-sided invariant-surjection hypothesis of Theorem 6.2 for the $A_1$ and $A_2$ rows in all three groups of type $E$, for the $G_2$ row, for the entire $F_4$ row, and for the $D_4$ row in $E_6$ and $E_7$.

The required matching of infinitesimal characters is given by Li [6] for the pairs appearing in his table. The one relevant omission from that table is

It is supplied by Loke’s earlier calculation [8]. In the quaternionic group $E_{7,4}$, Loke considers the dual pair

and determines its discrete correspondence. His calculation of the parameters gives the corresponding matching of infinitesimal characters. Since the homomorphism between the centers of the enveloping algebras is determined by the complexified dual-pair embedding, this is precisely the missing infinitesimal-character relation for the split $D_4 \times\mathrm{SL}_2^3$ pair. In fact, several of the entries recorded later in Li’s table can already be read from the correspondences in [8]. Theorem 6.2 therefore applies to all the nondegenerate cases listed above.

The degenerate pair $\{1\} \times F_4 \subset E_6$ may be handled also in the same way. Here $R_{K^2}=\mathbb{C}$ and the enveloping algebra of $F_4$ acts by an infinitesimal character. This follows from quaternionic groups, that is, the restriction of the minimal representation of the quaternionic $E_{6,4}$ to $F_4$ is a direct sum of two irreducible representations with the same infinitesimal character [7].

The only remaining pair is $D_4 \times D_4 \subset E_8$. Equation (39) shows that the invariant-surjection method used in this section cannot apply to it. In the $p$-adic case, Weissman [13] exhibits irreducible quotients of the form $\pi\boxtimes\pi$ of the minimal representation with infinite multiplicity. Consequently, the corresponding big theta lifts are not of finite length. We expect the same phenomenon for the split real pair, but do not prove it here. As observed after (35), the table contains fourteen distinct unordered pairs. Removing this one symmetric pair leaves the thirteen pairs asserted in the theorem.

Restriction estimates and injectivity

It remains to prove the restriction assertion used in Proposition 7.1. This is an essential step: the estimates below give the injectivity of $\beta_1$, which combines with the invariant counts of Section 7 to produce the required one-sided isomorphisms. Let $M$ be a $K_H$-finite matrix coefficient of the minimal representation of the split group $H$ of type $E_6$, $E_7$, or $E_8$. We call $p_S$ a critical exponent for a split subgroup $S \subset H$ if

As usual, no assertion at the endpoint is intended. The pointwise estimates underlying [9 Section 3.2] give the following values.

Proposition 8.1. The following restriction estimates hold for the smaller members of the split dual pairs in (35), excluding the symmetric $D_4 \times D_4$ pair in $E_8$.

(1) For the long-root $A_2 \subset G_2$ in the chain (1) and the highest-root $A_1 \subset A_2$, one may take the following critical exponents $p_S$:

(2) For the $G_2$ row, the smaller members have critical exponents

where $A_2^c=C_{E_6}(G_2)^\circ$.

(3) For the two additional subgroups of $C_3\simeq\operatorname{Sp}_6$ in $E_7$, one may take

Here $A_1^\Delta$ is the diagonal subgroup of the three mutually orthogonal long-root $A_1$'s; its centralizer in $E_7$ is $F_4$, while the centralizer of $A_1^3$ is $D_4$.

(4) For $T_2=C_{E_6}(D_4)^\circ$, the restriction $M|_{T_2}$ belongs to $L^q(T_2)$ for every $q>0$.

In particular, for every nontrivial smaller member used in the proof of Theorem 1.1, one can choose an exponent strictly below $2$. Thus (32) holds for some $\varepsilon>0$.

Proof. It is convenient to separate the pointwise decay parameter from the critical integrability exponent. Put $q_S=p_S/2$. Thus a bound of the form

on a positive chamber implies $M|_S\in L^{p_S+\varepsilon}(S)$ for every $\varepsilon>0$, by the Cartan integration formula; see [12 Section 2.4]. Indeed, the radial Haar density is bounded by $a^{2\rho_S}$, while raising (46) to the power $p_S+\varepsilon$ leaves, after multiplication by this density, the exponential factor $a^{-2\varepsilon\rho_S/p_S}$. For the semisimple groups considered here this dominates the polynomial factor on the positive chamber. The $K_H$-finiteness makes the estimate uniform over the compact factors; the torus case is treated separately below. The pointwise estimates for the restriction to $G_2$ in [9 Section 3.2] have this form, with

for $H=E_6,E_7,E_8$, respectively.

The estimates in (41) are also exactly those of [9 Section 3.2]. In $E_6$ the smaller member of the $A_2^c\times G_2$ pair is $A_2^c$, for which the first family of marked diagrams gives $p_{A_2^c}=1$. In $E_7$ and $E_8$ the smaller member is $G_2$, and the second family gives, respectively, $3/2$ and $1.

Let $\alpha$ and $\gamma$ be the short and long simple roots of $G_2$. The dominant cone of the long-root $A_2\subset G_2$ is the union of two adjacent $G_2$ cones. On these two cones the relevant half sums are

The orthogonal projection of either $\rho_i$ onto the $\rho_{A_2}$-line is $\frac{3}{2}\rho_{A_2}$, and on the corresponding half-cone the remaining component is nonnegative. Consequently $\rho_i\geq\frac{3}{2}\rho_{A_2}$ there. Restricting (46) therefore replaces $q_{G_2}$ by $q_{A_2}=\frac{2}{3}q_{G_2}$, and hence

Now put $A_1$ at the highest root of this $A_2$. On its split Cartan,

Thus $q_{A_1}=q_{A_2}/2$, or equivalently

It remains to consider the two subgroups of $C_3$ in $E_7$. The $C_3$ pointwise estimate is $a^{-\rho_{C_3}}$, so $q_{C_3}=1$ and $p_{C_3}=2$. In the standard coordinates, the dominant cone of $A_1^3$ is the positive octant, the union of the six $C_3$ cones, and

The orthogonal projection of every permutation of $(3,2,1)$ onto the diagonal line is $(2,2,2)=2\rho_0$. More explicitly, on the cone $x_{\sigma(1)} \ge x_{\sigma(2)} \ge x_{\sigma(3)} \ge0$ one has

Therefore $a^{-\rho_\sigma}\le a^{-2\rho_0}$, so $q_{A_1^3}=1/2$ and $p_{A_1^3}=1$.

Finally, for the diagonal $A_1^\Delta\subset A_1^3$ its half sum, in these coordinates, is

Restricting $a^{-2\rho_0}$ to its split Cartan gives $a^{-6\rho_{A_1^\Delta}}$. Hence $q_{A_1^\Delta}=1/6$ and $p_{A_1^\Delta}=1/3=2/6$. Compact Weyl representatives transfer the estimates between the finitely many cones, and the strict $\varepsilon>0$ absorbs all polynomial factors.

Finally, since $G_2\subset D_4\subset E_6$, one has

Both $T_2$ and $A_2^c$ have split rank two, so $T_2$ is a maximal split torus of $A_2^c$. The pointwise estimate corresponding to $p_{A_2^c}=1$ is, on every Weyl chamber,

In logarithmic coordinates this is exponential decay in every nonzero direction, whereas Haar measure on $T_2$ is Lebesgue measure. It follows that $M|_{T_2}\in L^q(T_2)$ for every $q>0$.

Every smaller-side critical exponent listed in the proposition is strictly less than 2. Choose $r$ with

(and any $1<r<2$ for $T_2$). Then $M|_S\in L^r(S)$; writing $r=2-\varepsilon$ proves (32) and completes the proof of Proposition 7.1. □

References

Department of Mathematics, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077

Email address: [email protected]

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Email address: [email protected]