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We prove the Casselman–Shalika formula for unramified groups over a non-archimedean local field by studying the action of the spherical Hecke algebra on the space of compact spherical Whittaker functions via the twisted Satake transform. This method provides a conceptual explanation of the appearance of characters of a dual group in the Casselman–Shalika formula.

The Twisted Satake Transform and the Casselman–Shalika Formula for Quasi-Split Groups