On extended Weil representation for finite general linear group and Howe correspondence (Q2186025)

Let \(q\) be a power of a prime number. A variety over a finite field \(\mathbb{F}_q\) is constructed and it is shown that its middle cohomology realizes the Heisenberg-Weil representations of a general linear group \(\mathrm{GL}_n(q)\). If \(n\) is even, using an endomorphism of this variety, a representation of a semidirect group \(\mathrm{GL}_n(q) \rtimes (\mathbb{Z}/2\mathbb{Z})\) is constructed, which is an extension of the Weil representation of \(\mathrm{GL}_n(q)\). It leads to a representation of \(\mathrm{Sp}_n(q) \times O_2^+(q)\). This representation can be used for the Howe correspondence for \((\mathrm{Sp}_n, O_2^+)\) in any characteristic. As application, it is shown that unipotency is preserved under the correspondence for \((\mathrm{Sp}_n, O_2^+)\).