Grassmann-Fourier type operators on \(\mathbb{I}_ U^{\text{GL}_ n} (\mathbb{F}_ q)\) (Q1125935)

We define operators \(T_1, \dots, T_{n-1}\) in the algebra of endomorphisms \({\mathcal H}\) (Hecke algebra) of the induced representations \(\text{Ind}^G_U \mathbf{1}\), where \(G\) is the linear group \(GL(n,\mathbb{F}_q)\), and \(U\) is the upper unipotent maximal subgroup of \(G\). These operators have nice properties, for example: (1) Using the operators \(T_i\) we can construct a \(G\)-module \({\mathcal S}\) that is a representation module of the Weyl group (symmetric group) \(S_n\) of \(G\). (2) We have a structure of a \((G,S_n \ltimes (\mathbb{F}^\times_q)^n)\)-bimodule over \({\mathcal S}\). Thus, we have a one-to-one correspondence between the series of generalized Steinberg representations of \(G\) and certain representations of \(S_n \ltimes (\mathbb{F}^\times_q)^n\). (3) Using these operators we can explicitly define a new embedding of \(S_n\) in \({\mathcal H}\).