It is well known that there are two geometric realizations of the affine Hecke algebra. In this talk, I will compare the two geometric realizations of the periodic modules for the affine Hecke algebra, which is due to Lusztig and Braverman—Kazhdan. One of them uses the equivariant K theory of T^*G/B. The other one involves the unramified principle series of Langlands dual p-adic group. We will compare the basis in those two spaces. With this, we can have an equivariant K-theoretic analogue of the Macdonald’s formula for the spherical functions and the Casselman—Shalika formula for the Whittaker functions. Joint work with Gufang Zhao and Changlong Zhong.

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