A proof of the Howe duality conjecture (Q2792316)

Let \(F\) be a local non-Archimedean field of characteristic different than 2 and residue characteristic \(p\). In this paper, the authors prove the Howe duality conjecture. The conjecture was known to be true for \(p \neq 2\) by the work of \textit{J. L. Waldspurger} [in: Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Pt. I: Papers in representation theory, Pap. Workshop \(L\)-Functions, Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 2, 267--324 (1990; Zbl 0722.22009)].NEWLINENEWLINEThe Howe duality conjecture concerns representations coming from the theta correspondence. Let \(E\) equal \(F\) or a quadratic extension of \(F\). For \(\varepsilon= \pm\), let \(W\) be an \(-\varepsilon\)-Hermitian space over \(E\) and \(V\) an \(\varepsilon\)-Hermitian space. If \(G(W)\) and \(H(V)\) are the corresponding isometry groups, we set \(G=G(W)\) and \(H=H(V)\), except in the case when \(E=F\) and one of the spaces, say \(V\), is odd-dimensional, where we take \(G\) to be the metaplectic double cover of \(G(W)\). Then \((G,H)\) is a dual reductive pair and the group \(G \times H\) possesses a Weil representation \(\omega_\psi\) which depends on a fixed additive character \(\psi\) of \(F\). If \(\pi\) is an irreducible admissible representation of \(G\), then the maximal \(\pi\)-isotypic quotient of \(\omega_\psi\) is of the form \(\pi \otimes \Theta(\pi)\), for some smooth finite length representation \(\Theta(\pi)\) of \(H\) (possibly zero). Let \(\theta(\pi)\) denote the maximal semisimple quotient of \(\Theta(\pi)\). The Howe duality conjecture states that NEWLINE\[NEWLINE \dim \text{Hom}_H(\theta(\pi), \theta(\pi')) \leq \begin{cases} 1, & \text{if }\pi \cong \pi'; \\ 0, & \text{if } \pi \ncong \pi'. \end{cases}NEWLINE\]NEWLINE The authors give a proof of the Howe duality conjecture, following the approach of \textit{R. Howe} [Proc. Symp. Pure Math. 33, 275--285 (1979; Zbl 0423.22016); J. Am. Math. Soc. 2, No. 3, 535--552 (1989; Zbl 0716.22006)] and \textit{S. S. Kudla} [Invent. Math. 83, 229--255 (1986; Zbl 0583.22010); Isr. J. Math. 87, No. 1--3, 361--401 (1994; Zbl 0840.22029)].