By Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen

Allow ? be a posh th root of solidarity for a strange integer >1 . For any advanced easy Lie algebra g , permit u ? =u ? (g) be the linked "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which are realised as a subalgebra of the Lusztig (divided energy) quantum enveloping algebra U ? and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ? . It performs an incredible position within the illustration theories of either U ? and U ? in a fashion analogous to that performed via the limited enveloping algebra u of a reductive staff G in optimistic attribute p with appreciate to its distribution and enveloping algebras. usually, little is understood concerning the illustration conception of quantum teams (resp., algebraic teams) whilst l (resp., p ) is smaller than the Coxeter quantity h of the underlying root process. for instance, Lusztig's conjecture in regards to the characters of the rational irreducible G -modules stipulates that p=h . the most lead to this paper offers an incredibly uniform solution for the cohomology algebra H (u ? ,C) of the small quantum team

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Without loss of generality we can assume that G = GLn (k). The centralizer is connected so CG (x) = CG (x)0 . 4] CG (x)0 ⊆ PJ . Case 2: Φ has type Bn . Let N = 2n + 1 and write N = lm + s where 0 ≤ s ≤ l − 1 and m > 0. 1. Set η = (lm , s ) and recall that N (Φ0 ) = OηB where ηB is the B-collapse of η. For type Bn we have ηB = (lm , s ) (lm , s − 1, 1) if s is odd or s = 0, if s is even and s = 0. 3]. For x ∈ uJ (with N (Φ0 ) = G · uJ ), let Q be the parabolic subgroup obtained from a standard triple in g involving x.

1 − 2, 2 + 4} 3. 1. Richardson orbits. Let G be a complex, simple and simply connected algebraic group with root system Φ. For J ⊂ Π, the (standard) parabolic subgroup PJ = LJ UJ ⊇ B of G has a dense (open) orbit CJ in the Lie algebra uJ of UJ under the adjoint action of PJ . In particular, if J = ∅, then LJ = T , and PJ = B, the Borel subgroup corresponding to Φ− . The corresponding Richardson orbit CJ is the G-orbit G · x for any x ∈ CJ . Therefore, when J = ∅, CJ is the regular or principal nilpotent orbit.

J. PARSHALL, AND C. 10). The module M is therefore isomorphic to a “Steinberg” type module on Uζ (lJ ) that remains irreducible if viewed as a uζ (lJ )-module. The highest weight of M is −w0,J (w · 0) and the lowest weight of M is −w · 0. Note that the module M does depend on the choice of w. 2. 7, the Ad-action induces an action of Uζ (pJ ) (and hence also of uζ (pJ )) on Uζ (uJ ). This deﬁnes an action of Uζ (pJ ) on the cohomology H• (Uζ (uJ ), C). 8. 1 below, we determine Homuζ (lJ ) ((indUζζ (b)J w · 0)∗ , H• (Uζ (uJ ), C)).