Howe correspondence for reductive groups over finite fields
Let \(\mathbb{F}_q\) be a finite field with odd characteristic \(p\). Let \((G, G')\) be a pair of subgroups of \(\text{Sp}_{2n}(\mathbb{F}_q)\) which are centralizers of each other and are a dual reductive (irreducible) pair. Assume \[ (G, G')= \begin{cases} (\text{Sp}_{2m}(\mathbb{F}_q),\quad O^{\pm}_{2m'}(\mathbb{F}_q))\qquad n= 2mm'\\ (\text{Sp}_{2m}(q),\quad O_{2m'+ 1}(q)\qquad n= m(2m'+ 1)\\ \text{or } (U_m(q),\;U_{m'}(q))\text{ or } (\text{GL}_m(q),\;\text{GL}_{m'}(q))\quad n= mm'.\end{cases} \] The purpose of the article under review is to describe the Howe correspondence arising from restricting the Weil representation \(\omega\) of \(\text{Sp}_{2n}(q)\) to \(G\times G'\).
scientific article; zbMATH DE number 912129
scientific article