Matrix Kloosterman sums (Q6622895)

Let \(\mathbb{F}_q\) be a finite field with \(q=p^f\) elements, and \(\overline{\mathbb{F}}\) be an algebraic closure of \(\mathbb{F}_q\) so that for \(m \geq 1 \mathbb{F}_{q^m} \subset \overline{\mathbb{F}}\) is the unique degree-\(m\) extension of \(\mathbb{F}_q\). Let \N\[\N\varphi_0: \mathbb{F}_p \rightarrow \mathbb{C}^*\N\]\Nbe the additive character which maps \(1 \in \mathbb{F}_p\) to \(\zeta=\exp (1 / p)=e^{2 \pi i / p}\), and fix the additive characters \N\[\N\varphi=\varphi_0 \circ \operatorname{Tr}_{\mathbb{F}_q / \mathbb{F}_p} \quad \text {and} \quad \varphi_m=\varphi_0 \circ \operatorname{Tr}_{\mathbb{F}_{q^m} / \mathbb{F}_p} \N\]\Nof \(\mathbb{F}_q\) and \(\mathbb{F}_{q^m}\). Let \(M_n\left(\mathbb{F}_{q^m}\right)\) be the algebra of \(n \times n\) matrices over \(\mathbb{F}_{q^m}\), and \(\mathrm{GL}_n\left(\mathbb{F}_{q^m}\right)=M_n^*\left(\mathbb{F}_{q^m}\right) \subset M_n\left(\mathbb{F}_{q^m}\right)\) be the general linear group. Let \(\psi\) (resp. \(\psi_m\)) be the additive character of \(M_n\left(\mathbb{F}_q\right)\) (resp. \(M_n\left(\mathbb{F}_{q^m}\right)\)) defined by \N\[\N\psi=\varphi \circ \operatorname{tr}\N\]\N(resp. \(\psi_m=\varphi_m \circ \operatorname{tr}\)), where \(\operatorname{tr}=\operatorname{tr}_n: M_n\left(\mathbb{F}_{q^m}\right) \rightarrow \mathbb{F}_{q^m}\) is the matrix trace. For \(a \in M_n\left(\mathbb{F}_{q^m}\right)\) define the sum\N\[\NK_n\left(a, \mathbb{F}_{q^m}\right)=\sum_{x \in \mathrm{GL}_n\left(\mathbb{F}_{q^m}\right)} \psi_m\left(a x+x^{-1}\right),\N\]\Ngeneralizing the classical Kloosterman sum [\textit{H. D. Kloosterman}, Acta Math. 49, 407--464 (1927; JFM 53.0155.01)]\N\[\NK_1\left(\alpha, \mathbb{F}_{q^m}\right)=K\left(\alpha, \mathbb{F}_{q^m}^*\right)=\sum_{x \in \mathbb{F}_{q^m}^*} \varphi_m\left(\alpha x+x^{-1}\right).\N\]\NThe paper gives optimal bounds for sums \(K_n\left(a, \mathbb{F}_{q^m}\right)\).