Sören Böhm
Address:
Ruhr-Universität Bochum
Fakultät für Mathematik
Universitätsstraße 150
44801 Bochum
Room:
IB 2/135
Phone:
(+49)(0)234 / 32-23302
Publications
- On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups
- Abstract: Suppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result under an extra hypothesis using so-called good A1 subgroups of G, introduced by Seitz. In the process we prove some new results about good A1 subgroups of G and their properties. We also formulate a counterpart of Korhonen's theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen's theorem for Lie algebras.
arXiv.2407.16379; accepted by: Pacific Journal of Mathematics
- G-complete reducibility and saturation
- Abstract: Let H⊆G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p>0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G=GL(V). Our proof uses R.W. Richardson's notion of reductive pairs to reduce to the GL(V) case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G=GL(V). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case G=GL(V).
- arXiv.2401.16927
- Complete reducibility for Lie subalgebras and semisimplification
- Abstract: Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra h of the Lie algebra g=Lie(G) of G and prove some results when h is solvable or char(k)=0. We introduce the concept of a k-semisimplification h′ of h; h′ is a Lie subalgebra of g associated to h which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that h′ is unique up to Ad(G(k))-conjugacy in g. Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and H′ a k-semisimplification of H, the Lie algebra Lie(H′) is a k-semisimplification of Lie(H).
- arXiv.2305.0084; published in: Eur.J.Math. 9 (2023), no. 4, Paper No. 116, 27 p
