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On the restriction of cuspidal representations to unipotent elements

Published online by Cambridge University Press:  31 January 2002

DIPENDRA PRASAD
Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad, India-211019. e-mail: dprasad@mri.ernet.in and nilabh@mri.ernet.in
NILABH SANAT
Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad, India-211019. e-mail: dprasad@mri.ernet.in and nilabh@mri.ernet.in

Abstract

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

Type
Research Article
Copyright
2002 Cambridge Philosophical Society

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