Equivariant homology and K-theory of affine Grassmannians and Toda lattices

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Perverse coherent sheaves (2009, background): “See [6] for more information on this example.”
Loop groups and twisted K‐theory I (2007, methodology): “the techniques of Bott [5] and Bott-Samelson [6] and has also been described by Bezrukavnikov, Finkelberg and Mirkovi´c [ 4 ].”
Quantum cohomology of G/P and homology of affine Grassmannian (2007, background): “More recently, Bezrukavnikov, Finkelberg and Mirkovic [ 2 ] described the equivariant K-homology of GrG and discovered a relation with the Toda lattice.”
Equivariant Satake category and Kostant-Whittaker reduction (2007, unknown): “As an application we prove a conjecture of [ 11 ] which identifies the algebra of global equivariant homology of Gr equipped with the convolution algebra structure with the quantized Toda lattice (Theorem 3). Note that the quantized Toda lattice also appears in the apparently related computations by”