The Twisted Satake Transform and the Casselman-Shalika Formula for Quasi-Split Groups
Nadya Gurevich, Edmund Karasiewicz
TL;DR
This paper offers a conceptual proof of the Casselman-Shalika formula for unramified groups over non-archimedean local fields by analyzing the twisted Satake transform's role in the spherical Hecke algebra's action on Whittaker functions.
Contribution
It introduces a new approach using the twisted Satake transform to explain the Casselman-Shalika formula for quasi-split groups, enhancing conceptual understanding.
Findings
Provides a proof of the Casselman-Shalika formula for unramified groups
Shows the role of the twisted Satake transform in the formula
Clarifies the appearance of dual group characters
Abstract
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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Topics in Algebra · Algebraic structures and combinatorial models