The Weil-Steinberg character of finite classical groups

Abstract

Let  G G be one of the groups GL ⁡ ( n , q ) \operatorname {GL}(n,q) or U ( n , q ) U(n,q) , and let  H H denote the subgroup GL ⁡ ( n − 1 , q ) \operatorname {GL}({n-1},q) or U ( n − 1 , q ) U(n-1,q) , respectively. We show that the restriction of the Steinberg character of  G G to  H H equals the product of the Weil character and the Steinberg character of  H H .