Local Theta Correspondences between Supercuspidal Representations

Author's errata for Local theta correspondences between supercuspidal representations, Annales scientifiques de l'École Normale Supérieure 51 (2018), 927–991 (opens in a new tab).

1. Corollary 5.11. $\Gamma+\mathfrak g_{x,r^+}$ should be $\Gamma+\mathfrak g_{x,-r^+}$. The proof was correct.

The page numbers below refer to the published paper.

2. Page 936, Section 3.2, Remark (2). Replace the last sentence with:

   We impose the condition $\Gamma_{-1}\in F'[\Gamma_d,\ldots,\Gamma_0]$, where $F'=Z(D)$, as in Definition 3.4(a). This ensures that $G^0=Z_G(\Gamma)$.

3. Page 940, Lemma 3.9. Add the assumption $\check G=Z_G(\gamma)$. In the proof, replace the assertion about the center of $\operatorname{Lie}(\check G)$ by the observation that the component of $\gamma$ in the orthogonal factor is zero.

4. Page 946, Lemma 4.1(iii). Add the assumption that $\Gamma$ is invertible.

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

Page 943, clarification of the proof of Theorem 3.14. For $k\in K\cap gKg^{-1}$, write

$$\widetilde g^{-1}\xi(k)\widetilde g
=c_{g,\xi}(k)\,\xi(g^{-1}kg).$$

The character $c_{g,\xi}$ need not be trivial for arbitrary $g$, so the compatibility asserted in (b) is not automatic.

Nevertheless, in the supercuspidal setting, the intertwining bound obtained by comparing the representations on the relevant pro-$p$ subgroups, using their canonical splittings, already forces $g\in K$. For such $g$, any lift $\widetilde g$ differs from $\xi(g)$ by a central scalar. Since $\xi$ is a group homomorphism, it follows that $c_{g,\xi}(k)=1$. Thus

$$I(\widetilde\eta_{\widetilde\Sigma})=\widetilde K,$$

and the conclusion of the theorem is unchanged.