Harmonic analysis on \(\mathrm{GL}_n\) over finite fields (Q2071013)

In the paper under the review, the authors study and further develop several invariants of representations which provide strong information regarding the character ratios, in the case of the general linear group over the finite field. Let \(\omega\) stand for the permutation representation of \(\mathrm{GL}_n\) over the finite field \(\mathbb{F}_q\), on the space \(L^2(\mathbb{F}_q)\) given by \([\omega(g)(f)](x) = f(g^{-1}x)\), for \(g \in \mathrm{GL}_n\), \(f \in L^2(\mathbb{F}_q)\), and \(x \in \mathbb{F}_q\). Note that \(\omega\) is the restriction of the oscillator representation of \(\mathrm{Sp}_{2n}\) to \(\mathrm{GL}_n\), up to a sign. We denote the trivial representation by \(\mathbf{1}\) and by \(\widehat{\mathrm{GL}}_n(\omega^{\otimes^k})\) the set of all irreducible representations of \(\mathrm{GL}_n\) appearing in the \(k\)-fold tensor product of \(\omega\). An irreducible representation \(\rho\) of \(\mathrm{GL}_n\) is said to be of strict tensor rank \(k\) if it appears in \(\widehat{\mathrm{GL}}_n(\omega^{\otimes^k})\), but it does not appear in \(\widehat{\mathrm{GL}}_n(\omega^{\otimes^{k-1}})\), where for \(i = 0\) we let \(\widehat{\mathrm{GL}}_n(\omega^{\otimes^i}) = \mathbf{1}\). Further, we say that an irreducible representation \(\rho\) of \(\mathrm{GL}_n\) is of tensor rank \(k\) if it can be written as a tensor product of a character and an irreducible representation of strict tensor rank \(k\), but not less. For a fixed \(0 \leq k \leq n\) and an irreducible representation \(\rho\) of \(\mathrm{GL}_n\) of tensor rank \(k\), the authors provide an estimate for the character ratio \(\frac{\chi_{\rho(T)}}{\mathrm{dim}(\rho)}\) in terms of a certain integer combinatorially associated to \(\rho\). The provided estimate gives a significant improvement of known results and can also be used to induce the results for the irreducible representations of the special linear group \(\mathrm{SL}_n\) over \(\mathbb{F}_q\). The paper under the review contains several illustrative numerical examples and an application of the obtained results to certain random walks.