Dade's conjecture for special linear groups in the defining characteristic (Q1806102)

Let \(\text{GL}(n,q)\) be the general linear group of degree \(n\) over the finite field \(\mathbb{F}_q\) with \(q\) elements, where \(q=p^e\) is a power of the prime \(p\). For a positive integer \(h\) dividing \(q-1\), let \[ L_h(n,q)=\{x\in\text{GL}(n,q)\mid\det(x)^h=1\}=\ker({\det}^h), \] where \(\det^h\) means \(\det^h(x)=\det(x)^h\). In particular, \(L_{q-1}(n,q)\) is \(\text{GL}(n,q)\) and \(L_1(n,q)\) is the special linear group \(\text{SL}(n,q)\). Moreover, let \(PL_h(n,q)\) denote the factor group of \(L_h(n,q)\) modulo its center \(Z(L_h(n,q))\). In particular, \(PL_1(n,q)\) is the projective special linear group \(\text{PSL}(n,q)\). In this paper, the author shows Dade's ordinary conjecture [see \textit{E.~C.~Dade}, Invent. Math. 109, No. 1, 187-210 (1992; Zbl 0738.20011)] for \(L_h(n,q)\) and \(PL_h(n,q)\) in the defining characteristic.