The Casselman--Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $GL_n$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y. Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

Keywords:

Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs

MSC Classifications:

22E50, 11F70, 11F85 show english descriptions Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Representation-theoretic methods; automorphic representations over local and global fields
$p$-adic theory, local fields [See also 14G20, 22E50] 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05]
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]

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