Rudolf Tange

Complete reducibility and separability

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serres concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and G is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztigs deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnicks result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.

THE ZASSENHAUS VARIETY OF A REDUCTIVE LIE ALGEBRA IN POSITIVE CHARACTERISTIC

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand–Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G2.

The symplectic ideal and a double centraliser theorem

We interpret a result of Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on , where V is the natural module for the symplectic group. This result was obtained in characteristic zero by Weyl. Furthermore, we use this to extend to arbitrary connected reductive groups G with simply connected derived group the earlier result of the author that the algebra K[G] of infinitesimal invariants in the algebra of regular functions on G is a unique factorisation domain.

Infinitesimal invariants in a function algebra

Let G be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let g be its Lie algebra. First we correct and generalise a well-known result about the Picard group of G. Then we prove that, if the derived group is simply connected and g satisfies a mild condition, the algebra K[G]^g of regular functions on G that are invariant under the action of g derived from the conjugation action, is a unique factorisation domain.

HIGHEST WEIGHT VECTORS AND TRANSMUTATION

Let G = GLn be the general linear group over an algebraically closed field k, let g=gln

Bases for Spaces of Highest Weight Vectors in Arbitrary Characteristic

\mathfrak{g}={\mathfrak{gl}}_n

S

be its Lie algebra and let U be the subgroup of G which consists of the upper uni-triangular matrices. Let kg