Nham V. Ngo

Commuting varieties of r-tuples over Lie algebras

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p and let g be the Lie algebra of G. It is well known that for p large enough the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of elements in the nilpotent cone of g (Suslin et al., 1997, [27]). In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen–Macaulayness of the commuting varieties Cr(gl2), Cr(sl2) and Cr(N) where N is the nilpotent cone of sl2. Our calculations lead us to state a conjecture on Cohen–Macaulayness for commuting varieties of r-tuples. Furthermore, in the case when g=sl2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of sl3, we are able to verify the aforementioned properties for Cr(u). Finally, applying our calculations on the commuting variety Cr(Osub¯) where Osub¯ is the closure of the subregular orbit in the nilpotent cone of sl3, we prove that the nilpotent commuting variety Cr(N) has singularities of codimension ⩾2.

Cohomology for Frobenius kernels of SL2

Abstract Let ( SL 2 ) r be the r -th Frobenius kernels of the group scheme SL 2 defined over an algebraically closed field of characteristic p > 0 . In this paper we give for r ⩾ 1 a complete description of the cohomology groups for ( SL 2 ) r . We also prove that the reduced cohomology ring H • ( ( SL 2 ) r , k ) red is Cohen–Macaulay. Geometrically, we show for each r ⩾ 1 that the maximal ideal spectrum of the cohomology ring for ( SL 2 ) r is homeomorphic to the fiber product G × B u r . Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobenius–Lusztig kernels of quantized enveloping algebras of type SL 2 .

Second cohomology for finite groups of Lie type

Let

On varieties of commuting nilpotent matrices

G

On nilpotent commuting varieties and cohomology of Frobenius kernels

be a simple, simply-connected algebraic group defined over

Reducibility of nilpotent commuting varieties

\mathbb{F}_p

Rational singularities of commuting varieties over small rank matrices

. Given a power

Mixed Commuting Varieties

q = p^r

Cohomology of

of

SL_2

p