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 | References | Similar Articles | Additional Information
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.
- George Lusztig, Unipotent representations of a finite Chevalley group of type $E_{8}$, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 119, 315β338. MR 545068, DOI https://doi.org/10.1093/qmath/30.3.315
- George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472
- G. Lusztig, The Grothendieck group of unipotent representations: a new basis, Represent. Theory 24 (2020), 178β209. MR 4103274, DOI https://doi.org/10.1090/ert/542
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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