Research of G. Schwarz

My early work was concerned with properties of C functions [1, 7, 8, 9] and with properties of smooth actions of compact groups [2]: Let K be a compact Lie group and let W be a real K-module. Then R[W]K is a finitely generated, say by the polynomials p1,...,pd. In [2] we show that any f ε C[W]K, a smooth invariant functions on W, is of the form h(p1,...,pd), where h(y1,...,yd) is a C function. This reduces many problems in the analysis of smooth actions of K to problems in algebra and the invariant theory of K and its associated complex algebraic group KC. We exploited this fact in [6], where we extended Palais' celebrated homotopy lifting theorem for topological actions of K to the smooth category.

The first sections of [6] are analytic in nature, but soon the subject matter becomes concerned with problems of the representation theory of KC. We had to develop many methods to compute algebras of invariants C[V]G, where G is a complex reductive group (e.g., KC) and V is a (complex) G-module. We exploited these techniques in the papers [4] and [5], where we were able to classify the G-modules V where C[V]G is a polynomial ring (or where C[V] is a free module over C[V]G). We went on to generalize results of Classical Invariant Theory to the cases of the fundamental representations of G2 and Spin7 [10, 13, 14]. Our work with Wehlau [27] is along these lines.

Returning to our real roots, we went on to consider the problem of how to describe the orbit space W/K, where K is compact and W is a real K-module. Using techniques from the complex case, we were able to find several characterizations of these inequalities [11, 12]. One consequence is the solution to an equivariant version of Hilbert's 17th problem. Other works along these lines are the expository works [16, 28] and [31].

Inspired by work of Luna and a collaboration with Kraft, we turned to considering the possible ways that a complex reductive group G could act on Cn. In case the algebraic quotient Cn//G is zero dimensional, Luna's celebrated slice theorem shows that the action of G is algebraically conjugate to a linear action. Some had hypothesized that all G-actions on Cn had to be linearizable, but in [18] we announced several counterexamples. Kraft and I gave an (almost) complete accounting of the case where dimV//G = 1 in [17, 20]. Related work is in [25].

Recently we have been looking at the algebras of G-invariant (algebraic) differential operators D(V)G on a complex G-module V [19, 21, 22, 23, 26, 30, 34, 35]. In [19, 21, 23, 24] we enlarge upon the work in [6], which relied on finding out which differential operators of order 1 on the quotient V//G lifted to D(V)G, to differential operators of all orders. We have complete knowledge of what goes on for irreducible representations of the simple groups [21], In particular, we show that all differential operators on V//G lift as long as V//G is not smooth.

In another direction, one can investigate properties of D(V)G itself. If G is finite, then D(V)G is a simple ring. If G is positive dimensional, this is no longer the case, and little is known about the irreducible or finite dimensional representations of D(V)G, except when G is a torus. We have begun to investigate what happens in general, starting with the cases of adjoint representations of low rank simple algebraic groups. Here there are direct connections to the representation theory of the groups themselves. We have succeeded in the case G = SL3 using a combination of techniques from Verma module theory together with identities involving several interesting series of differential operators [34, 35].

Finally, together with Helminck, we have been considering some generalizations of symmetric varieties [29, 33, 36]. Let s and q be commuting involutions of the reductive algebraic group G, where our base field is algebraically closed of characteristic not 2. We study the geometry and invariant theory of the double coset space H\G/K where H = Gs and K = Gq. Our methods so far have been mostly characteristic free. The closed double cosets in H\G/K can be viewed as the quotient of a maximal (s, q)-split torus A of G by a curious type of Weyl group W*, whose structure, although partially clear, needs further investigation. We intend to investigate several applications to the real case, where the extremes (q is a Cartan involution, or G(R) is compact) are already known to us.