Linear Koszul Duality and Fourier Transform for Convolution Algebras

In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in $K$-homology of \cite{MR3} and the Fourier transform isomorphism for convolution algebras in Borel--Moore homology of \cite{EM} are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori--Matsumoto involution for graded affine Hecke algebras in \cite{EM} and for ordinary affine Hecke algebras in \cite{MR3}.

2010 Mathematics Subject Classification: 18E30, 16E45, 16S37

Keywords and Phrases: Koszul duality, Fourier transform, affine Hecke algebras

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