On the irreducible very cuspidal representations. II (Q1367069)

Supercuspidal representations of \(\text{GL}(n, F)\), where \(F\) is a non-archimedean local field, have been constructed by Carayol by induction from certain subgroups, which are compact modulo the center. The inducing representations, called very cuspidal, were investigated by the author in [\textit{T. Takahashi}, J. Math. Kyoto Univ. 29, 653-670 (1990; Zbl 0716.22008)]. If such a representation has even level, it is monomial and if it has odd level, it is not monomial, but is induced from a representation \(\eta\) defined in the author's paper cited above. In the present paper, a representation of odd level is written explicitly as a \(\mathbb{Q}\)-linear combination of monomial representations. This is done by computing the character of \(\eta\) and inverting a matrix, which is very complicated.