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The equivariant $K$-theory of the semi-infinite flag manifold, as developedrecently by Kato, Naito, and Sagaki, carries commuting actions of thenil-double affine Hecke algebra (nil-DAHA) and a $q$-Heisenberg algebra. Theaction of the latter generates a free submodule of rank $|W|$, where $W$ is the(finite) Weyl group. We show that this submodule is stable under the nil-DAHA,which enables one to express the nil-DAHA action in terms of $W\times W$matrices over the $q$-Heisenberg algebra. Our main result gives an explicitalgebraic construction of these matrices as a limit from the (non-nil) DAHA insimply-laced type. This construction reveals that multiplication by equivariantscalars, when expressed in terms of the Heisenberg algebra, is given by thenonsymmetric $q$-Toda system introduced by Cherednik and the author.

Equivariant $K$-theory of the semi-infinite flag manifold as a nil-DAHA module