Gerald W. Schwarz

SMOOTH FUNCTIONS INVARIANT UNDER THE ACTION OF A COMPACT LIE GROUP

be a compact Lie group acting orthogonally on IX”. By classical theorem of Hilbert ([14], p. 274), the algebra P((w”)G of G-invariant polynomials on Iw” is finitely generated. Let p,, . . . , pk be generators and let p = (p,, . . . , pk) denote the corresponding map from [w” to Dg’. The square of the radius function on Iw” is a proper map, and it is a polynomial in the pi’s Hence p is proper. Since P((w”)’ separates the orbits of G and since the orbit space IF/G is locally compact Hausdorff ([3], p. 38), p induces a homeomorphism p’ of [W”/G with the closed semi-algebraic subset p([w”) of [w” ([l], p. 100). lR”/G can be given a “smooth structure” by defining a function on KY/G to be smooth if it pulls back to a smooth function on R”, and p(Iw”) has a smooth structure defined by restricting the C” functions 6(Rk) on DBk to p([w”). It has been conjectured that p is an isomorphism of [W”/G and p([w”) together with their smooth structures. The conjecture is known in some special cases and it has proved useful in obtaining classification theorems for certain types of smooth group actions ([3], Ch. VI; [4]). Of course, the conjecture is equivalent to THEOREM 1.

Lifting smooth homotopies of orbit spaces

© Publications mathématiques de l’I.H.É.S., 1980, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Inequalities defining orbit spaces

SummaryThe orbit space of a representation of a compact Lie group has a natural semialgebraic structure. We describe explicit ways of finding the inequalities defining this structure, and we give some applications.

The Topology of Algebraic Quotients

(0.0) This is an expanded version of my talk of the above title at the Rutgers Conference on Topological Methods in Algebraic Transformation Groups, April 4–8, 1988. I report on recent work which shows there is a strong interconnection between the theory of quotients in the algebraic category (reductive groups) and in the topological category (compact groups). It turns out that one can represent quotient spaces of reductive groups as quotient spaces of compact groups. The precise method of doing this has important topological consequences for quotients of reductive groups and also provides some new insight into quotients of compact groups. I hope that my exposition will make these results more accessible to all.

Reductive group actions with one-dimensional quotient

© Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories

Abstract We give a method for finding the inequalities defining the orbit space of a linear action of a compact Lie group. We also give several equivalent descriptions of the zeroes of the scalar potential and D -term occurring in supersymmetric gauge theories.

Finite-dimensional representations of invariant differential operators

Abstract In Part I of this paper [G.W. Schwarz, Finite-dimensional representations of invariant differential operators, J. Algebra 258 (2002) 160–204] we considered the representation theory of the algebra B := D ( g ) G , where G= SL 3 ( C ) and D ( g ) G denotes the algebra of G-invariant polynomial differential operators on the Lie algebra g of G. We also considered the representation theory of the subalgebra A of B , where A is generated by the invariant functions O ( g ) G ⊂ B and the invariant constant coefficient differential operators S( g ) G ⊂ B . Among other things, we found that the finite-dimensional representations of A and B are completely reducible, and we could reduce the study of the finite-dimensional irreducible representations of B to those of A . Irreducible finite-dimensional representations of A are quotients of “Verma modules.” We found sufficient conditions for the irreducible quotients of Verma modules to be finite-dimensional, and we conjectured that these sufficient conditions are also necessary. In this paper we establish the conjecture, giving a complete classification of the finite-dimensional representations of A and B .

Orbits and invariants associated with a pair of commuting involutions

Let σ, θ be commuting involutions of the connected reductive algebraic group G where σ, θ and G are defined over a (usually algebraically closed) field k, char k = 2. We have fixed point groups H := G and K := G and an action (H × K ) × G → G, where ((h, k), g) → hgk−1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K ) denote Spec O(G)H×K (the categorical quotient). Let A be maximal among subtori S of G such that θ(s) = σ(s) = s−1 for all s ∈ S. There is the associated Weyl group W := WH×K (A). We show: • The inclusion A → G induces an isomorphism A/W ∼ → G//(H × K ). In particular, the closed (H × K )-orbits are precisely those which intersect A. • The fibers of G → G//(H × K ) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. We investigate: • The structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of . • The relation of the orbit type stratifications of A/W and G//(H × K ). Along the way we simplify some of Richardson’s proofs for the symmetric case σ = θ, and at the end we quickly recover results of Berger, Flensted-Jensen, Hoogenboom and Matsuki [Ber57, FJ78, Hoo84, Mat97] for the case k = .

Finite automorphisms of affine n-space

It is still an open question whether or not there exist polynomial automorphisms of finite order of complex affine n-space which cannot be linearized, i.e., which are not conjugate to linear automorphisms. The second author gave the first examples of non-linearizable actions of positive dimensional groups, and Masuda and Petrie did the same for finite groups.

Smoothness of Quotients Associated With a Pair of Commuting Involutions

Let �, � be commuting involutions of the connected semisimple algebraic group G where �, � and G are defined over an algebraically closed field k, char k = 0. Let H := Gand K := Gbe the fixed point groups. We have an action (H × K) × G → G, where ((h, k), g) 7→ hgk 1 , h ∈ H, k ∈ K, g ∈ G. Let G//(H × K) denote the categorical quotient Spec O(G) H×K. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg (Ste75), Pittie (Pit72) and Richardson (Ric82) in the symmetric case where � = � and H = K.