The exotic inverted Kloosterman sum

Let $B$ be a product of finitely many finite fields containing ${\mathbb{F}}_q$, $\psi :{\mathbb{F}}_q\to {\bar{\mathbb{Q}}}_{\mathrm{\ell }}^{\ast }$ a nontrivial additive character, and $\chi :B^{\ast }\to {\bar{\mathbb{Q}}}_{\mathrm{\ell }}^{\ast }$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} $\mathrm{E}\mathrm{I}\mathrm{K}$(\mathbb F_q, a):=\sum_{\substack{x\in B^* \ $\mathrm{T}\mathrm{r}${B/\mathbb F_q}(x)\not =0\ $\mathrm{N}${B/\mathbb F_q}(x)=a}} χ(x)ψ\Big(\frac{1}{$\mathrm{T}\mathrm{r}$_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb F_q^. \end{eqnarray} We estimate this sum using $\mathrm{\ell }$-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1={\dim }_{{\mathbb{F}}_q}B$. Up to a trivial main term, this gives the expected square root cancellation.