On the Satake Isomorphism

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 Ĝ. 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 = GLn by Tamagawa, but the results in this case are deceptively simple, as all of the fundamental representations of the dual group are minuscule. Lusztig 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. We will explain some of these results below, with several examples.