Foruma

Abstract

Let $(G,G')$ be a type I irreducible reductive dual pair in $\mathrm{Sp}(W_{\mathbb{R}})$ . We assume that $(G,G')$ is in the stable range where $G$ is the smaller member. Let $K$ and $K'$ be maximal compact subgroups of $G$ and $G'$ respectively. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and $\mathfrak{g}' = \mathfrak{k}' \oplus \mathfrak{p}'$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G'$ respectively. Let ${\widetilde{K}}$ and ${\widetilde{K}}'$ be the inverse images of $K$ and $K'$ in the metaplectic double cover $\widetilde{\mathrm{Sp}}(W_\mathbb{R})$ of ${\mathrm{Sp}}(W_\mathbb{R})$ . Let $ρ$ be a genuine irreducible $(\mathfrak{g},{\widetilde{K}})$ -module. Our first main result is that if $ρ$ is unitarizable, then except for one special case, the full local theta lift $ρ' = Θ(ρ)$ is equal to the local theta lift $θ(ρ)$ . Thus excluding the special case, the full theta lift $ρ'$ is an irreducible and unitarizable $(\mathfrak{g}',{\widetilde{K}}')$ -module. Our second main result is that the associated variety and the associated cycle of $ρ'$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation $ρ^*$ respectively. Finally we obtain some interesting $(\mathfrak{g},{\widetilde{K}})$ -modules whose ${\widetilde{K}}$ -spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_\mathbb{C}$ -orbits in $\mathfrak{p}^*$ .

Author's errata for Invariants and K-spectrums of local theta lifts. Page and equation numbers refer to the published paper in Compositio Mathematica 151 (2015), 179–206 (opens in a new tab).

Correction 1

The condition $(\dagger)$ (on page 180) should also include the case when $(G,G') = (\mathrm{Sp}_{2n}(\mathbb C), \mathrm{O}_{4n}(\mathbb C))$ and $\rho$ is the trivial representation. In fact, in this case $\Theta(\rho)$ has two irreducible constituents (See “Lee Soo Teck, Zhu Chenbo, Degenerate principal series and local theta correspondence III”).

Correction 2

Page 183, the exclusion list $(\dagger\dagger)$ . Retain the existing exclusions and add the following cases:

 $\begin{aligned} &(G,G')=(\mathrm O(p,q),\mathrm{Sp}(2n,\mathbb R)), && n=p+q;\\ &(G,G')=(\mathrm U(n_1,n_2),\mathrm U(p,q)), && p=n_1+n_2\ \text{or}\ q=n_1+n_2;\\ &(G,G')=(\mathrm{Sp}(p,q),\mathrm O^*(2n)), && n=2(p+q). \end{aligned}$

All references to $(\dagger\dagger)$ use this enlarged list. The stable range in Table 1 is unchanged.

Correction 3

Page 189, Lemma 3.1. Replace $\mathbf E_{2j_0+2j+1}$ with $\mathbf E_{j_0+2j+1}$ , so that the statement reads

 $\eta(\mathbf E_{j_0+2j})=\eta(\mathbf E_{j_0+2j+1})=V'_j.$

Correction 4

Page 190, following (15). The $S(\mathfrak p)$ -module structure on the source of $\epsilon$ should be understood as pulled back along the automorphism induced by $X\mapsto-X$ on $\mathfrak p$ .

Correction 5

Page 198, the proof of Lemma 6.2. Replace $F'=\phi^{-1}(x')$ with $F'=(\phi')^{-1}(x')$ .

Correction 6

Page 198, after (22). Replace “at every point $w\in W$ ” with “at every point $w\in W^\circ$ ”.

Source: Author's errata page (opens in a new tab).

Invariants and K-spectrums of local theta lifts