The Twisted Satake Transform and the Casselman-Shalika Formula

Abstract: The Fourier coefficients of automorphic forms are an important object of study due to their connection to $L$-functions. In the adelic framework, constructions of $L$-functions involving Fourier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) naturally lead to spherical Whittaker functions on $p$-adic groups. Thus we would like to understand these spherical Whittaker functions to better understand $L$-functions.

Casselman-Shalika determined a formula for the spherical Whittaker functions, and basic algebraic manipulations reveal that their formula can be more succinctly expressed in terms of characters of the Langlands dual group. We will describe a new proof of the Casselman-Shalika formula that provides a conceptual explanation of the appearance of characters.

number theory

Audience: researchers in the discipline

POINT: New Developments in Number Theory

Series comments: There will be two 20-minute contributed talks during each meeting aimed at a general number theory audience, including graduate students.

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