aa r X i v : . [ m a t h . R T ] S e p REDUCED INVARIANT SETS
GERALD W. SCHWARZ
In honor of Dick Palais
Abstract.
Let K be a compact Lie group and W a finite-dimensional real K -module. Let X be a K -stable real algebraic subset of W . Let I ( X ) denote the ideal of X in R [ W ] and let I K ( X ) be the ideal generated by I ( X ) K . We find necessary conditions and sufficient conditionsfor I ( X ) = I K ( X ) and for p I K ( X ) = I ( X ). We consider analogous questions for actions ofcomplex reductive groups. Introduction
Let K be a compact Lie group, let W be a finite-dimensional real K -module and let X ⊂ W be K -invariant and real algebraic (the zero set of real polynomial functions on W ). Let I ( X )denote the ideal of X in R [ W ]. Let R [ W ] K denote the K -invariants in R [ W ] and let I K ( X ) be theideal generated by I ( X ) K := I ( X ) ∩ R [ W ] K . We say that X is K -reduced if I K ( X ) = I ( X ) and almost K -reduced if p I K ( X ) = I ( X ). Let Kw be an orbit in W . Then the slice representationat w is the action of the isotropy group K w on N w , where N w is a K w -complement to T w ( Kw )in W ≃ T w ( W ). An orbit Kw ⊂ W is principal (resp. almost principal ) if the image of K w inGL( N w ) is trivial (resp. finite). We denote the principal (resp. almost principal) points of W by W pr (resp. W apr ) and we set X pr := W pr ∩ X and X apr := W apr ∩ X . The strata of W arethe collections of points S ⊂ W whose isotropy groups are conjugate. There are finitely manystrata. If R [ W ] is a free R [ W ] K -module, then we say that W is cofree . In the following, whenwe talk about one set being dense in another, we are referring to the Zariski topology.Here are our main results: Theorem 1.1. If X is K -reduced (resp. almost K -reduced), then X pr (resp. X apr ) is dense in X , and conversely if W is cofree. Theorem 1.2.
Let w ∈ W . Then the orbit Kw is K -reduced (resp. almost K -reduced) if andonly if Kw is principal (resp. almost principal). To prove the results above and to obtain further results we need to complexify. Let V = W ⊗ R C and G = K C be the complexifications of W and K . We have the quotient morphism π : V → V //G where π is surjective, V //G is an affine variety and π ∗ C [ V //G ] = C [ V ] G . We havethe Luna strata of the quotient V //G whose inverse images in V are the strata of V . The strataof V are in 1-1 correspondence with those of W [Sch80, § Y = X C be the complexificationof X (the Zariski closure of X in V ). We say that Y is G -saturated if Y = π − ( π ( Y )) and that Y is G -reduced if the ideal I ( Y ) of Y is generated by I ( Y ) G . We can define Y apr and Y pr asabove (see § f , . . . , f k are functions on a complex variety, let I ( f , . . . , f k ) denote the idealthey generate. Theorem 1.3. (1) X is almost K -reduced if and only if Y is G -saturated. (2) X is K -reduced if and only if Y is G -reduced. (3) X apr (resp. X pr ) is dense in X if and only if Y apr (resp. Y pr ) is dense in Y . Mathematics Subject Classification.
Key words and phrases.
Invariant polynomials, reduced, saturated.
Theorem 1.4.
Assume that
Y //G ⊂ V //G is the zero set of f , . . . , f k . (1) Suppose that Y apr is dense in Y and that for any stratum S of V which intersects Y \ Y apr the codimension of S in V is at least k + 1 . Then Y is G -saturated.. (2) Suppose that Y pr is dense in Y and that Y is G -saturated. In addition, suppose that I ( π ( Y )) = I ( f , . . . , f k ) where Y has codimension k in V . Then Y is G -reduced. Corollary 1.5.
If (1) above holds, then X is almost K -reduced. If (2) holds, then X is K -reduced. In sections 2–4 we consider when a general G -invariant Y ⊂ V is G -saturated or G -reducedand we establish Theorem 1.4. In section 5 we treat the real case by complexifying. At the endof section 5 we establish Theorems 1.1, 1.2 and 1.3.D. ˇZ. ¯Dokovi´c posed the question of identifying the X which are K -reduced. Our results givea partial answer. We thank M. Ra¨ıs for transmitting the question to us. We thank the refereefor a careful reading of the manuscript, helpful suggestions and Lemma 3.3.2. The complex case
Let G be a complex reductive group and Y an affine algebraic set with an algebraic G -action.Dual to the inclusion C [ Y ] G ⊂ C [ Y ] we have the quotient morphism π Y : Y → Y //G . Let V bea finite-dimensional G -module and let Y be a G -stable algebraic subset of V (the zero set ofan ideal of C [ V ]). We shall denote π V simply by π . Then π Y = π | Y and π ( Y ) ≃ Y //G is analgebraic subset of
V //G . We say that Y is G -saturated if Y = π − ( π ( Y )). Let I ( Y ) denotethe ideal of Y in C [ V ] and let I G ( Y ) denote the ideal generated by I ( Y ) G . We say that Y is G -reduced if I ( Y ) = I G ( Y ). The null cone N ( V ) of V is the fiber π − ( π (0)). Then N ( V ) is(scheme theoretically) defined by the ideal I G ( { } ) so that the scheme N ( V ) is reduced if andonly if the set N ( V ) is G -reduced, in which case we say that V is coreduced . See [KS11] formore on coreduced representations.The points of V //G are in one-to-one correspondence with the closed G -orbits in V . The Lunastrata of V //G are the sets of closed orbits whose isotropy groups are all G -conjugate. Thereare finitely many strata in V //G , and we consider their inverse images in V to be the strata of V . Let v ∈ V such that Gv is closed. Then the isotropy group G v is reductive, and there is a G v -stable complement N v to T v ( Gv ) in V ≃ T v ( V ). We call the action of G v on N v the slicerepresentation at v .We start with some examples. Example . Let (
V, G ) = ( k C n , SL n ), k ≥ n . The invariants are generated by the determinantsdet i ,...,i n where the indices 1 ≤ i < · · · < i n ≤ k tell us which n copies of C n to take. Then V is coreduced since N ( V ) is the determinantal variety of ( k × n )-matrices of rank at most n − Example . Let G ⊂ GL( V ) be finite and nontrivial. Then N ( V ) is the origin which is G -saturated but not G -reduced.Part (2) of the proposition below follows from Serre’s criterion for reducedness [Mat80, Ch.7]. Part (1) also follows, using the Jacobian criterion for smoothness. Proposition 2.3.
Let Y ⊂ V be a G -saturated algebraic set. (1) If Y is G -reduced, then for every irreducible component Y k of Y there is a point of Y k where rank f = codim Y k . Here f = ( f , . . . , f d ) : V → C d and the f i generate I G ( Y ) . (2) If I G ( Y ) = I ( f , . . . , f d ) where the f i ∈ C [ V ] G and Y has codimension d , then Y is G -reduced if and only if the rank condition of (1) is satisfied. EDUCED INVARIANT SETS 3
Example . Let G = SO ( C ) acting as usual on V = 2 C . Then the invariants are generatedby inner products f ij , 1 ≤ i ≤ j ≤
2. Each copy of C has a weight basis { v , v , v − } relativeto the action of the maximal torus T = C ∗ where v j has weight j . The null cone Y := N ( V )is the G -orbit of all the vectors v = ( αv , βv ) for α , β ∈ C . But one easily calculates that therank of ( f , f , f ) : V → C at v is at most 2 while Y has codimension 3. Thus the null coneis not G -reduced. 3. The case where Y pr or Y apr is dense in Y Throughout this section we assume that V is a stable representation of G , i.e., there is anonempty open subset of closed orbits. This is always the case when ( V, G ) = ( W C , K C ) isa complexification ([Lun72] or [Sch80, Cor. 5.9]). Let Gv be a closed orbit. We say that Gv is principal if the slice representation ( N v , G v ) is a trivial representation and that Gv isalmost principal if G v → GL( N v ) has finite image. We denote the principal (resp. almostprincipal) points of V by V pr (resp. V apr ). If Y ⊂ V is G -stable, we set Y pr = Y ∩ V pr and Y apr = Y ∩ V apr . Both Y apr and Y pr are open in Y . In general, the fiber of π through a closedorbit Gv ⊂ V is G × G v N ( N v ) (the G -fiber bundle with fiber N ( N v ) associated to the G v -principal bundle G → G/G v ). Thus the fiber is set-theoretically the orbit if and only N ( N v )is a point. This happens if and only if the image G v → GL( N v ) is finite, i.e, v ∈ V apr . Hence Y apr is always G -saturated. Similarly, the fiber is scheme-theoretically the orbit if and only if N ( N v ) is schematically a point which is equivalent to G v acting trivially on N v , i.e., we have v ∈ V pr . Hence Y pr is always G -reduced. To sum up we have Proposition 3.1.
Let Gv be a closed orbit and let Y ⊂ V be a G -stable algebraic set. (1) If Y = Y apr , then Y is G -saturated. In particular, Gv is G -saturated if and only if it isalmost principal. (2) If Y = Y pr , then Y is G -reduced. In particular, Gv is G -reduced if and only if it isprincipal. (3) The fiber π − ( π ( v )) is G -reduced if and only if the slice representation ( N v , G v ) is core-duced. Corollary 3.2.
If the isotropy groups of G acting on Y are all finite, then Y is G -saturatedand if G acts freely on Y , then Y is G -reduced. Of course, it is possible that Y is G -saturated (resp. G -reduced) even if Y apr (resp. Y pr ) isempty. But in the case of a complexification Y = X C it is necessary for G -saturation (resp. G -reducedness) that Y apr (resp. Y pr ) is dense in Y (see § Y apr or Y pr is not dense in Y in the next section.Unfortunately, we do not have the analogues of Proposition 3.1(1) and (2) for X . See Example5.3 below.Recall that V is cofree if C [ V ] is a free C [ V ] G -module. Equivalently, π : V → V //G is flat, or C [ V ] G is a regular ring and the codimension of N ( V ) is dim C [ V ] G [Sch80, Proposition 17.29].We owe the following lemma to the referee. Lemma 3.3.
Let V be a cofree G -module and let U ⊂ V //G be locally closed. (1)
We have π − ( U ) = π − ( U ) . (2) If π − ( U ) is reduced, then so is π − ( U ) .Proof. For (1) set Z := π − ( U ). Then π ( Z ) is closed [Kra84, II.3.2], hence π ( Z ) = U . Since π is flat, so is π − ( U ) → U . Set S := π − ( U ) \ Z . Then S is open, hence π ( S ) is open in U (byflatness). By construction, π ( S ) does not meet U , hence we must have S = ∅ , establishing (1). GERALD W. SCHWARZ
For (2) we can assume that U = U f = { u ∈ U | f ( u ) = 0 } for f ∈ C [ U ]. Set Z := π − ( U ),the schematic inverse image of U . Since C [ U ] → C [ U ] = C [ U ] f is injective and C [ Z ] is flatover C [ U ], it follows that C [ Z ] → C [ Z ] f = C [ π − ( U )] is also injective. Since the latter ring isreduced, so is C [ Z ] and we have established (2). (cid:3) Corollary 3.4.
Suppose that ( V, G ) is cofree and that Y ⊂ V is a G -stable algebraic set suchthat Y apr is dense in Y . Then Y is G -saturated.Example . Let (
V, G ) = (4 C , SL ) and let Y = 2 C × { } . Then Y pr = Y apr is dense in Y (it is the set of linearly independent vectors in Y ) but Y is not G -saturated since it does notcontain the null cone. The G -module V is not cofree, so we don’t contradict Corollary 3.4. Notethat this example is the complexification of the case where X = C × { } ⊂ W := C ⊕ C and K = SU(2 , C ). Thus X pr = X apr is dense in X but X is not almost K -reduced. (We useTheorem 1.3.) This shows that cofreeness is also necessary in Theorem 1.1. Theorem 3.6.
Suppose that Y ⊂ V is G -stable such that (1) Y apr is dense in Y . (2) Y //G ⊂ V //G is the zero set of f , . . . , f k where the minimal codimension of a non almostprincipal stratum of V which intersects Y is at least k + 1 .Then Y is G -saturated.Proof. Let e Y denote π − ( π ( Y )). Then each irreducible component of e Y has codimension lessthan or equal to k . Let S be a non almost principal stratum of V which intersects Y . Then S ∩ e Y is nowhere dense in e Y . Thus e Y apr is dense in e Y . Now e Y apr and Y apr have the same imagein Y //G . Hence Y apr = e Y apr and Y = e Y is saturated. (cid:3) Example . Let (
V, G ) = ( k C , SL ), k ≥
2. The codimension of the null cone is k − Y where the first copy of C is zero is not saturated, but corresponds to the subsetof V //G where the determinant invariants det , . . . , det k vanish (see Example 2.1). Thus thecodimension condition in Theorem 3.6(2) is sharp.Here is an example that is a complexification. Example . Let (
V, G ) = (2 C , SO ( C )) and let Y = C × { } ∪ { } × C . Then Y apr is densein Y since any point not in N ( V ) is on a principal orbit and N ( V ) is nowhere dense in Y .However, Y is not G -saturated since it does not contain N ( V ). Note that I ( Y //G ) is generatedby det (the determinant), f and f f where the f ij are the inner product invariants. Sincedet = f f − f , I ( Y //G ) is the radical of the ideal generated by f and f f . The nullcone has codimension 2. Again this shows that the codimension condition in Theorem 3.6 issharp.We now have the following corollary of Lemma 3.3 Corollary 3.9.
Suppose that ( V, G ) is cofree and that Y ⊂ V is G -stable such that Y pr is densein Y . Then Y is G -reduced.Remark . For Y to be G -reduced, it is not sufficient that every slice representation of V is coreduced. (This is the same as saying that every fiber of π : V → V //G is reduced.)Just consider Example 3.5 again. Here Y pr is dense in Y but Y is not G -saturated, let alone G -reduced. Theorem 3.11.
Let V be a G -module and let Y ⊂ V be G -saturated such that Y pr is dense in Y . Suppose that π ( Y ) ⊂ V //G is the zero set of f , . . . , f k where the codimension of Y is k .Then Y is G -reduced. EDUCED INVARIANT SETS 5
Proof.
The rank of the differential of f = ( f , . . . , f d ) : V → C d is maximal at a point of eachirreducible component of Y since Y is reduced at all points of Y pr . Thus we can apply Serre’scriterion (Proposition 2.3). (cid:3) Example . Let (
V, G ) = (4 C , SL ( C )) and let Y be the zero set of two of the determinantinvariants det ij . Then Y pr is dense in Y since the only non-principal stratum is N ( V ) whichhas codimension 3 while Y has codimension 2. By Theorem 3.11, Y is G -reduced.4. The case where Y pr or Y apr is not dense in Y We can say something in the case that Y apr or Y pr is not dense in Y . We are certainly in thiscase if V is not stable, since then V pr and V apr are empty. Let v ∈ Y such that Gv is closed.Let ( N v , G v ) be the slice representation and S the corresponding stratum of V . We say that( N v , G v ) is a generic slice representation for Y if S ∩ Y is dense in an irreducible component of Y . We also say that S is generic for Y . Proposition 4.1.
Let ( N v , G v ) be a generic slice representation of Y corresponding to thestratum S of V . If Y is G -saturated, then Y ∩ S = π − ( π ( Y ∩ S )) . If Y is G -reduced, then ( N v , G v ) is coreduced.Proof. If Y is G -saturated, then we obviously must have that Y ∩ S = π − ( π ( Y ∩ S )). Let Z denote π ( S ). Then π − ( Z ) → Z is a fiber bundle with fiber G × G v N ( N v ). If Y is G -reduced,then the bundle is reduced, hence ( N v , G v ) is coreduced. (cid:3) Let S be a stratum of V . We say that Y is S -saturated if Y ∩ S = π − ( π ( Y ∩ S )). We saythat Y is S -reduced if Y is S -saturated and the slice representation ( N v , G v ) associated to S iscoreduced. Corresponding to Corollaries 3.4 and 3.9 and Theorems 3.6 and 3.11 we have thefollowing result whose proof we leave to the reader. Theorem 4.2.
Let Y ⊂ V be a G -stable algebraic set. (1) If V is cofree and Y is S -saturated for every stratum S which is generic for Y , then Y is G -saturated. (2) If V is cofree and Y is S -reduced for every stratum S which is generic for Y , then Y is G -reduced. (3) Suppose that Y is S -saturated for every every generic stratum S of Y . Further assumethat the minimal codimension of the strata of V which intersect Y but are not generic for Y is greater than k and that Y //G is the zero set of f , . . . , f k . Then Y is G -saturated. (4) Suppose that Y is G -saturated and that the ideal of π ( Y ) ⊂ V //G is generated by f , . . . , f k where the codimension of Y in V is k . Also assume that Y is S -reducedfor every generic stratum S of Y . Then Y is G -reduced. The real case
Let W be a real K -module where K is compact. Let X ⊂ W be real algebraic and K -stable.Now K is naturally a real algebraic group and the action on W is real algebraic. Moreover, everyorbit of K in W is a real algebraic set [Sch01]. Let Y := X C denote the complexification of X inside V := W ⊗ R C and let G denote the complexification K C of K . Then G is reductive and V is a stable G -module ([Lun72] or [Sch80, Cor. 5.9]). We say that a slice representation ( N w , K w )is a generic slice representation for X if w ∈ X and the corresponding stratum contains anonempty open subset of X . Equivalently, the complexification of ( N w , K w ) is generic for Y . Proposition 5.1. (1) X is almost K -reduced if and only if Y is G -saturated. (2) X is K -reduced if and only if Y is G -reduced. GERALD W. SCHWARZ (3)
The set X apr (resp. X pr ) is dense in X if and only if the set Y apr (resp. Y pr ) is dense in Y . (4) X is almost K -reduced implies that X apr is dense in X . (5) X is K -reduced implies that X pr is dense in X .Proof. The ideal of Y is I ( X ) ⊗ R C ⊂ R [ W ] ⊗ R C = C [ V ] and I K ( X ) ⊗ R C = I G ( Y ). Thus I ( Y ) = I G ( Y ) if and only if I ( X ) = I K ( X ), and I ( Y ) = p I G ( Y ) if and only if I ( X ) = p I K ( X ). Hence we have (1) and (2). For (3), note that X apr is open in X and that Y apr is openin Y . If a stratum S of W is dense in an irreducible component of X , then the correspondingstratum S C of V is dense in an irreducible component of Y . Thus if X apr is not dense in X , then Y apr is not dense in Y . Clearly, if X apr is dense in X , Y apr ⊃ X apr is dense in Y . The argumentfor X pr and Y pr is similar, hence we have (3). Now suppose that X is almost K -reduced. Thenfor S a generic stratum of X and x ∈ S ∩ X , the complexification Gx ≃ G/G x of Kx is Zariskidense in the fiber G × G x N ( W x ⊗ R C ) where G x = ( K x ) C . Thus N ( W x ⊗ R C ) is a point, i.e., thestratum consists of almost principal orbits. Hence we have (4), and (5) is proved similarly. (cid:3) Corollary 5.2.
Let X = Kw be an orbit. Then X is almost K -reduced if and only if Kw isalmost principal and X is K -reduced if and only if Kw is principal. Unfortunately, it is not true that X = X pr (or X = X apr ) implies the same equality for Y . Example . Let K = SU ( C ) and W = 2 C ⊕ R where K acts as usual on the copies of C andtrivially on R . We consider W to be 2 H ⊕ R where H denotes the quaternions. Then K ≃ S ,the unit quaternions, and the action on 2 H is given by k ( p, q ) = ( kp, kq ), p , q ∈ H , k ∈ S .Let p ¯ p denote the usual conjugation of quaternions. The invariants of K acting on 2 H aregenerated by ( p, q ) (¯ pp, ¯ qq, ¯ qp ) where the first two invariants lie in R and the last in H . Let α and β denote the first two invariants and let γ be the real part of ¯ qp . Let δ , ǫ and ζ be theinvariants which are the i , j and k components of ¯ qp , respectively. Then there are certainlypoints in 2 H where δ , ǫ and ζ vanish and where α = β = γ is any positive real number. Let x be a coordinate on the copy of R in W and let X be the subset of W defined by δ = ǫ = ζ = 0, α = β = γ and ( α − + x = 1 /
2. Then α never vanishes on X which implies that the isotropygroup at the corresponding point of W is trivial, so we have that X = X pr . The quotient X/K is a smooth curve, hence X is smooth of dimension 4. The complexification Y of X also hasdimension four and contains some of the points ( s, t, ± p − /
4) where ( s, t ) lies in the null coneof 2 H ⊗ R C ≃ C for the action of K C ≃ SL ( C ). But this null cone has dimension 5. Hence Y is not G -saturated, let alone G -reduced, and Y = Y apr . Moreover, X is neither K -reduced noralmost K -reduced.Now we recover the theorems of the introduction. Theorem 1.2 is just Corollary 5.2. Theorem1.3 is a consequence of Proposition 5.1 and Theorem 1.4 follows from Theorems 3.6 and 3.11. Proof of Theorem 1.1.
Suppose that X is K -reduced. Then Proposition 5.1 shows that X pr isdense in X . Conversely, if ( W, K ) is cofree (equivalently, (
V, G ) is cofree) and X pr is dense in X , then Y pr is dense in Y by Proposition 5.1 and Y is G -reduced by Corollary 3.9. Hence X is K -reduced. The proof in the almost K -reduced case is similar. (cid:3) References [Kra84] Hanspeter Kraft,
Geometrische Methoden in der Invariantentheorie , Aspects of Mathematics, D1,Friedr. Vieweg & Sohn, Braunschweig, 1984.[KS11] Hanspeter Kraft and Gerald W. Schwarz,
Reduced null cones , to appear.[Lun72] Domingo Luna,
Sur les orbites ferm´ees des groupes alg´ebriques r´eductifs , Invent. Math. (1972), 1–5.[Mat80] Hideyuki Matsumura, Commutative algebra , second ed., Mathematics Lecture Note Series, vol. 56,Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.
EDUCED INVARIANT SETS 7 [Sch80] Gerald W. Schwarz,
Lifting smooth homotopies of orbit spaces , Inst. Hautes ´Etudes Sci. Publ. Math.(1980), no. 51, 37–135.[Sch01] ,
Algebraic quotients of compact group actions , J. Algebra (2001), no. 2, 365–378.
Department of Mathematics, Brandeis University, Waltham, MA 02454-9110
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