AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras

Given a complex semisimple Lie algebra is a compact real form of g), let be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra , where t is a maximal abelian subalgebra of . Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to , and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range f (Ad(K)x), where f is a linear functional on g. We establish the star-shapedness of f (Ad(K)x) for simple Lie algebras of type B.

Equidimensional immersions of locally compact groups

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestrictionis a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

Closure Theorem for Analytic Subgroups of Real Lie Groups

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.

Character formulas for discrete series on semisimple Lie groups

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .