On the Harish-Chandra homomorphism for the Chevalley groups over \(p\)-adic field
Given a Chevalley group \(G\) (not of type \(G_2\)) defined over a \(p\)- adic field and a finite-dimensional Hilbert space \(V\), let \(C_c(G : \text{End }V)\) be the algebra of the continuous functions \(G \to \text{End }V\) with compact support. For an open compact subgroup \(J\) of \(G\) and an irreducible unitary representation \(\sigma\) of \(J\) in \(V\) define the \(\sigma\)-spherical Hecke algebra \(H(G // J, \sigma)\) consisting of the functions in \(C_c (G : \text{End } V)\) such that \[ f(k_1 gk_2) = \sigma(k_1) f(g) \sigma(k_2),\qquad k_1, k_2 \in J,\;g \in G. \] The author is interested in the case when \(J = K_q\) is the \(q\)-congruence maximal compact subgroup of \(G\). For some \(\sigma\) he constructs the Harish-Chandra homomorphism \(H(G' // J', \sigma') \to H(G // J,\sigma)\) related to the Levi component of a parabolic subgroup of \(G\) and proves that it is an algebra isomorphism and an isometry of \(L^2\)-spaces. It generalizes the results obtained for \(G = GL_n\) by \textit{R. Howe} and \textit{A. Moy} [Harish-Chandra homomorphism for \(p\)-adic groups, Reg. Conf. Ser. Math. 59 (1985; Zbl 0593.22014)].
\(q\)-congruence maximal compact subgroup
scientific article; zbMATH DE number 743972