Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S (n) can be understood as zonal spherical functions of the Gelfand pair (S (n) × S (n), diag S (n)). They form an orthogonal basis in the space of the functions on the group S (n) invariant with respect to conjugations by S (n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an "unbalanced" Gelfand pair (S (n) × S (n - 1), diag S (n - 1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S (n), namely in the space of functions invariant with respect to conjugations by S (n - 1). We refer to these zonal spherical functions as normalized generalized characters of S (n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S (n).