errata — Invariants and K-spectrums

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).