3 problems

Let $V$ be a finite-dimensional vector space over $\mathbf k$, let $G=GL(V)$, and let $\mathcal X_1=G\times V$ with $G$ acting diagonally. Let $\widehat{\mathcal X}_1'$ be the set…

Let $D$ be a vector space, and let $V$ and $W$ be vector spaces with $\dim(V)+\dim(W)=\dim(D)$. Let $\CG\in\calC(\operatorname{GL}(V))$ and $\CG'\in\calC(\operatorname{GL}(W))$, an…

Let $V={\mathsf k}^N$ and let $\operatorname{GL}_N(\Fq)$ act on $\operatorname{GL}_N(\Fq)\times\Fq^N$. Consider the mirabolic character sheaves described above and their Frobenius…