Hecke algebras of classical groups over \(p\)-adic fields. II (Q2743769)

Let \(G\) denote the group of rational points of a classical \(p\)-adic group. In an earlier paper [Am. J. Math. 121, No. 5, 967-1029 (1999; Zbl 0933.22024)], the author constructed a family of pairs \((J,\rho)\), where \(J\) is a compact open subgroup of \(G\) and \(\rho\) is an irreducible representation of \(J\). (These pairs might be ``types'', in the sense of \textit{C. J. Bushnell} and \textit{Ph. C. Kutzko} [Proc. Lond. Math. Soc., III. Ser. 77, No. 3, 582-634 (1998; Zbl 0911.22014)].) For each pair, one can consider the support of the corresponding Hecke algebra \(H(G//J,\rho)\), and thus construct a large family of supercuspidal representations. NEWLINENEWLINENEWLINEOne approach toward a classification of the admissible dual of \(G\) involves two steps. First, find a family of pairs \((J,\rho)\) as above such that every irreducible admissible representation of \(G\) must contain some \(\rho\) upon restriction to the corresponding \(J\). Second, find the structures of the resulting Hecke algebras. NEWLINENEWLINENEWLINEThe present paper does the latter for a subset of the class of pairs \((J,\rho)\) constructed before. To describe this subset, recall that the pairs are constructed from triples \(\Sigma = (\Gamma,P'_0,\varrho)\), where \(\Gamma\) is an element of the Lie algebra of \(G\) having a certain form, \(P_0'\) is a parahoric subgroup of the centralizer \(C_G(\Gamma)\), and \(\varrho\) is an irreducible cuspidal representation of the finite reductive quotient of \(P_0'\). In the special case, where \(P_0'\) is a minimal parahoric subgroup and \(\varrho\) is trivial, the corresponding representation \(\rho\) will sometimes be one dimensional. This is the case that the author treats. NEWLINENEWLINENEWLINEAs a consequence, one obtains (in this case) a \(*\)-preserving, support-preserving, \(L^2\)-isomorphism from \(H(G//J,\rho)\) to an explicitly given Hecke algebra coming from the group \(C_G(\Gamma)\), thus advancing a program initiated by \textit{R. Howe}. For an overview of this program, see [Proc. Symp. Pure Math. 68, 321-332 (2000; Zbl 0957.22021)].