Download Characters of Reductive Groups over a Finite Field by George Lusztig PDF

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 details on representations of Weyl

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Extra resources for Characters of Reductive Groups over a Finite Field

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2) C:' where v * GY'L y*rri"'-w,o/). (B', B") E O(s)IB' = B"l in O(s). Then L~ is in a natural way a G-equivariant line bundle, since ~ is G-stable in O(s). There is a unique G-equivariant isomorphism of line bundles over O(s): L' f. ® L ~(-m). El! The line bundle L~ has a unique G-equivariant trivialization (up to a scalar) over O(s) - ~. This gives rise to an isomorphism LIO(s) El! (-m))IO(s), hence to an embedding j :L· ... (L®LAE--m))• of line bundles with zero section removed, compatible with the imbedding O(s) ...

2) All connected components of Xw are irreducible of dimension e(w). ·- L We shall consider the cohomology sheaves }{1(xw, :f. t. ) . These sheaves are in a w,n natural way, GF'_equivariant constructible Qe-sheaves on Xw with an F a-structure (inherited from p :f ~ w,n ). Note also that for our w(w

N L -> xP. 0 to and to the action of H on L. which was just defined. The action of H on L 11- 1 c:Bw) inverse images has finitely many orbits: they are the under 11: L. _, :B of the Bruhat cells :Bw C :B. The isotropy group in H of a point in L · is a product U 1 x µn C U x k*, where U 1 is a closed connected subgroup of U. 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).

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