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Fourier transform as a triangular matrix


Author: G. Lusztig
Journal: Represent. Theory 24 (2020), 470-482
MSC (2010): Primary 20G99
DOI: https://doi.org/10.1090/ert/551
Published electronically: October 3, 2020
MathSciNet review: 4156779
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Abstract: Let $V$ be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let $[V]$ be the vector space of complex valued functions on $V$, and let $[V]_{\mathbf {Z}}$ be the subgroup of $[V]$ consisting of integer valued functions. We show that there exists a $\mathbf {Z}$-basis of $[V]_{\mathbf {Z}}$ consisting of characteristic functions of certain isotropic subspaces of $V$ and such that the matrix of the Fourier transform from $[V]$ to $[V]$ with respect to this basis is triangular. We show that this is a special case of a result which holds for any two-sided cell in a Weyl group.


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Additional Information

G. Lusztig
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
MR Author ID: 117100
Email: gyuri@mit.edu

Received by editor(s): February 15, 2020
Received by editor(s) in revised form: February 22, 2020
Published electronically: October 3, 2020
Additional Notes: This work was supported by NSF grant DMS-1855773.
Article copyright: © Copyright 2020 American Mathematical Society