Published online by Cambridge University Press: 24 October 2008
1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading . LetI be the ideal in S generated by the w-invariant elements in . The quotient algebra inherits the W-action and also a grading
is the image of Sk under S → S̅. It is well known that the W-module S̅ is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, S̅k = 0 for large k. (More precisely, S̅k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module S̅k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in q
can sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.
To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.
To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.