arXiv subjects
Publications and source records attributed to Srijeet Bhattacharjee.
We give an alternate expression for the Kloosterman sums associated with the long Weyl element of \(GL(n)\) that avoides the use of Plücker coordinates. The expression is instead based on the use of the well-known Cauchy-Binet formula from linear algebra. From this expression, we prove some important properties of these Kloosterman sums.
Let $p$ be a prime. We solve two problems in the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{F}_{q})$ where $q=p^f$. We first prove a Clebsch-Gordan decomposition theorem for the tensor product of two mod $p$ representations of $\mathrm{GL}_2(\mathbb{F}_{q})$. As an application, we use this to guess the structure of quotients of symmetric power representations of $\mathrm{GL}_2(\mathbb{F}_{q})$ by submodules in the theta filtration. We then give a direct proof of this structure showing that such quotients are built out of principal series representations.