On the Satake isomorphism. (English) Zbl 0996.11038

Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998).
From the introduction: In this paper, we present an expository treatment of the Satake transform. This gives an isomorphism between the spherical Hecke algebra of a split reductive group \(G\) over a local field and the representation ring of the dual group \(\widehat{G}\).
If one wants to use the Satake isomorphism to convert information on eigenvalues for the Hecke algebra to local \(L\)-functions, it has to be made quite explicit. This was done for \(G= \text{GL}_n\) by Tamagawa, but the results in this case are deceptively simple, as all of the fundamental representations of the dual group are minuscule. G. Lusztig [Astérisque 101/102, 208-229 (1983; Zbl 0561.22013)] discovered that, in the general case, certain Kazhdan-Lusztig polynomials for the affine Weyl group appear naturally as matrix coefficients of the transform. His results were extended by S. Kato [Invent. Math. 66, 461-468 (1982; Zbl 0498.17005)].
We explain some of these results in this paper, with several examples.
Contents: The algebraic group \(\underline{G}\), The Gelfand pair \((G,K)\), The Satake transform, Kazhdan-Lusztig polynomials, Examples, \(L\)-functions, The trivial representation, Normalizing the Satake isomorphism.
For the entire collection see [Zbl 0905.00052].


11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields