Mathematics – Representation Theory
Scientific paper
2011-05-15
Mathematics
Representation Theory
42 pages
Scientific paper
The strong Macdonald theorems state that, for L reductive and s an odd variable, the cohomology algebras H^*(L[z]/z^N) and H^*(L[z,s]) are freely generated, and describe the cohomological, s-, and z-degrees of the generators. The resulting identity for the z-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate H^*(p / z^N p) and H^*(p[s]) for p a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist. In particular we show that H^*(p / z^N p) contains (a parabolic subalgebra of) the coinvariant algebra of the fixed-point subgroup of the Weyl group of L, and thus is no longer free. When p is a parahoric in an untwisted loop algebra and N=1 our calculation gives a Lie algebraic proof of Borel's theorem that the coinvariant algebra is isomorphic to the cohomology algebra of the corresponding flag variety. Our calculation also implies that while H^*(p / (z^N-t) p) is independent of t as a vector space when p is a twisted arc algebra, this breaks when p has a non-trivial parabolic component. Finally, we prove a strong Macdonald theorem for H^*(b; S^* n^*) and H^*(b / z^N n) when b and n are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking z-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to L, proving an analogue of the Kostant slice theorem.
No associations
LandOfFree
Twisted strong Macdonald theorems and adjoint orbits does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Twisted strong Macdonald theorems and adjoint orbits, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Twisted strong Macdonald theorems and adjoint orbits will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-76627