A remark on Gelfand pairs of finite groups of Lie type
Let \(G\) be a finite group of Lie type and \(H\) a subgroup. We say that \((G,H)\) is a Gelfand pair if \(\text{ind}_H^G1\) is multiplicity free. We say that \((G,H)\) is a cuspidal Gelfand pair if the multiplicity of a cuspidal representation of \(G\) in \(\text{ind}_H^G1\) is at most 1. Let \(\overline G=\text{SO}(\overline Q)\), where \(\overline Q\) is a nondegenerate quadratic form on an \(n\)-dimensional vector space over the finite field \(\overline F\). Let \(\overline H=\text{SO}(\overline W)\), where \(\overline W\) is a nondegenerate hyperplane of \(\overline V\). Theorem 1. Let \((\overline\pi,X)\) be a cuspidal representation of \(\overline G\) and let \((\overline\sigma,Y)\) be a cuspidal representation of \(\overline H\). Then \(\dim(\Hom_{\overline H}(\overline\pi,\overline\sigma))\leq 1\). Let \(\overline G_n=\text{GL}_n(\overline F)\), where \(\overline F\) is a finite field. Let \(m\geq 2\), \(k\geq 1\) be two integers with \(m+k=n\) and denote by \(\overline H=\overline H_{m,k}\) the subgroup \(\text{GL}_m(\overline F)\times\text{GL}_k(\overline F)\). Let \(\pi\) be an admissible irreducible representation of \(\overline G_n\). Theorem 5. \(\dim(\Hom_{\overline H}(\overline\pi,1))\leq 1\). Hence \((\text{SO}_n(q),\text{SO}_{n-1}(q))\) and \((\text{GL}_n(q),\text{GL}_m(q)\times\text{GL}_k(q))\) are cuspidal Gelfand pairs.
scientific article; zbMATH DE number 1214111
scientific article