By George Lusztig
This booklet offers a class of all (complex)
irreducible representations of a reductive team with
connected centre, over a finite box. to accomplish this,
the writer makes use of etale intersection cohomology, and
detailed info on representations of Weyl
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Additional info for Characters of Reductive Groups over a Finite Field
LEMMA. (i) Any right coset wWL ,n CW (w cW) has a unique element w 1 of minimal length (in W ). It is characterized by the property that w R+ for any L' f Rt n. 1 L' f ' (ii) If v, v' f WL n satisfy v' S v and if w 1 ' L,n f W has minimal length in w 1WLn then w 1v'Sw 1v (forthepartialorderin W)and ' (l. 1) where I is the length function of the Coxeter group (WL n' SL n). Proof. H w 1 ' f ' W has minimal length in w 1 WL·n, then E(w 1r) ' w > E(w 1) 1L' f R+ WL n. Hence, for L' f Ri, n, we have ' ' w 1L'cR-.
X ,1• LEMMA. ,n ) are characterized by the following two properties . (1. 2) X~,i/! X~,i/! = + Z[A] - linear combination of elements X~',i/! , w'< w' the coefficient of xL,,,,(w'< w) being an element of Z[A] which W,'f' is a Z-linear combination of elements a €A , where a is represented by an algebraic number in value o; all of whose complex conjugates have absolute :S Pv,(e(w)-e(w')-1). 2), follows from the definition of the intersection cohomology complex and from the purity theorem of Gabber, see [BBD].
X~ ,i/J of rank 1 over 11- 1 (93w) CL. /f(x). 0), there is a unique <1>-structure on this sheaf extended to the whole of L that has eigenvalue a on a stalk over 11- 1 c:Bw). ,n denoted aX~ 11- 1 c:Bw)) e. ,/,. 1,. Xw ,i/J. 9 1. ,n) . ,n)) . -1 where r 'w ,1, < Z[A], (w' ~ w), and rw w ,1, = [p]e(w)-ecw'). ,, ' 1'f' denotes the image of p E in A; the relation w' 'S_ w means that we Qe have the inclusion ~w' C ~w of the Zariski closures of :Bw', :Bw in :B. 5. 2) of L, we have L. 2). We shall denote rr- 1 (:Bw) CL.
Characters of Reductive Groups over a Finite Field by George Lusztig