Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module Article Swipe
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· 2023
· Open Access
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· DOI: https://doi.org/10.1007/s00029-023-00848-9
· OA: W2999568849
The equivariant K -theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q -Heisenberg algebra. The action 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 $$|W|\times |W|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>W</mml:mi> <mml:mo>|</mml:mo> <mml:mo>×</mml:mo> <mml:mo>|</mml:mo> <mml:mi>W</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> matrices over the q -Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric q -Toda system introduced by Cherednik and the author.
