Knop-Luna slices

Let A be an affine C-algebra and ξ ∈ issn A a point in the quotient space corresponding to an n-dimensional semi-simple representation Mξ of A. In the next chapter we will present a method to study the ´etale local structure ofissn A nearξ and the ´etale localGLn-structure of the representation varietyrep_n Anear the closed orbitO(Mξ) =GLn.Mξ. First, we will outline the main idea in the setting of differential geometry.

Let M be a compact C^∞-manifold on which a compact Lie groupG acts differentially. By a usual averaging process we can define aG-invariant Riemannian metric onM. For a pointm∈M we define

• TheG-orbitO(m) =G.mofminM,

• the stabilizer subgroupH =StabG(m) ={g∈G | g.m=m}and

• the normal spaceNm defined to be the orthogonal complement to the tangent space inmto the orbit in the tangent space toM. That is, we have a decomposition ofH-vectorspaces

TmM =TmO(m)⊕Nm

The normal spaces Nx when x varies over the points of the orbit O(m) define a vectorbundle N -^p- O(m) over the orbit. We can identify the bundle with the associated fiber bundle

N 'G×^HNm

3.8. Knop-Luna slices 157

Any pointn∈ N in the normal bundle determines a geodesic γn:R - M defined by

(

γn(0) =p(n)

dγn

dt (0) =n

Using this geodesic we can define aG-equivariant exponential map from the normal bundle N to the manifoldM via

N ^exp- M where exp(n) =γn(1)

•

YY222

222

n

x γn

O(m)

Nx M

Now, takeε >0 and define theC^∞ sliceSε to be

Sε={n∈Nm | knk< ε}

thenG×^HSεis aG-stable neighborhood of the zero section in the normal bundleN =G×^HNm. But then we have aG-equivariant exponential

G×^HSε exp

- M

which for small enoughε gives a diffeomorphism with aG-stable tubular neighborhoodU of the orbit O(m) in M as in figure 3.10 If we assume moreover that the action of G on M and the action of H onNm are such that the orbit-spaces are manifolds M/GandNm/H, then we have the situation

G×^HSε exp

' - U ^⊂ - M

Sε/H

??

'- U/G

??

⊂ - M/G

??

giving a local diffeomorphism between a neighborhood of 0 inNm/H and a neighborhood of the pointminM/Gcorresponding to the orbitO(m).

Nm

•0 ε

−ε

G/H exp

-m•

O(m)

U

Nx M

Figure 3.10: Tubular neighborhood of the orbit.

Returning to the setting of the orbitO(Mξ) inrep_nAwe would equally like to define aGLn -equivariant morphism from an associated fiber bundle

GLn×^GL(α)Nξ e

- rep_nA

whereGL(ξ) is the stabilizer subgroup ofMξandNξis a normal space to the orbitO(Mξ). Because we do not have an exponential-map in the setting of algebraic geometry, the mapewill have to be an ´etale map. Such a map does exist and is usually called a Luna slicein case of a smooth point onrep_n A. Later, F. Knop extended this result to allow singular points, or even points in which the scheme is not reduced.

Although the result holds for any reductive algebraic groupG, we will apply them only in the caseG=GLnorGL(α) =GLa₁×. . .×GLa_k, so restrict to the case ofGLn. We fix the setting :

3.8. Knop-Luna slices 159

XandYare (not necessarily reduced) affineGLn-varieties,ψ is aGLn-equivariant map x=ψ(y) ` X

ψ Y a y

X/GLn π_X

??

Y/GLn π_Y

?? and we assume the following restrictions :

• ψ is ´etale iny,

• theGLn-orbitsO(y) inYandO(x) inXare closed. For example, in representation varieties, we restrict to semi-simple representations,

• the stabilizer subgroups are equalStab(x) =Stab(y). In the case of representation varieties, for a semi-simplen-dimensional representation with decomposition

M =S₁^⊕e1⊕. . .⊕S^⊕e_k ^k into distinct simple components, this stabilizer subgroup is

GL(α) = 2 6 4

GLe1(C⊗rr

d₁) . ..

GLe_k(C⊗rr

d_k) 3 7 5

⊂ - GLn

wheredi=dim Si. In particular, the stabilizer subgroup is again reductive.

In algebraic terms : consider the coordinate ringsR=C[X] andS=C[Y] and the dual morphism R ^ψ

∗

- S. LetI / Rbe the ideal describingO(x) andJ / S the ideal describingO(y). WithRbwe will denote theI-adic completionlim

← R

Iⁿ ofRand withSbtheJ-adic completion ofS.

Lemma 3.8 The morphismψ^∗induces for allnan isomorphism R

Iⁿ

ψ^∗

- ^S Jⁿ In particular,Rb'Sb.

Proof. LetZ be the closed GLn-stable subvariety ofYwhere ψ is not ´etale. By the separation property, there is an invariant functionf∈S^GLn vanishing onZ such thatf(y) = 1 because the two closedGLn-subschemesZ andO(y) are disjoint. ReplacingS bySf we may assume thatψ^∗ is an ´etale morphism. BecauseO(x) is smooth,ψ⁻¹ O(x) is the disjoint union of its irreducible components and restricting Y if necessary we may assume that ψ⁻¹ O(x) = O(y). But then J=ψ^∗(I)S and asO(y) ^'- O(x) we have ^R_I '^S_J so the result holds forn= 1. and the result follows from induction onnand the commuting diagram

0 - ^I

For an irreducible GLn-representation s and a locally finite GLn-moduleX we denote its s-isotypical component byX(s).

Lemma 3.9 Letsbe an irreducibleGLn-representation. There are natural numbersm≥1 (inde-pendent ofs) and n≥0such that for all k∈Nwe have

I^mk+n∩R(s)^⊂ - (I^GLn)^kR(s) ^⊂ - I^k∩R(s)

Proof. Consider A=⊕^∞_i=0Iⁿt^n⊂ - R[t], thenA^GLn is affine so certainly finitely generated as R^GLn-algebra say by

{r1t^m1, . . . , rzt^mz} withri∈Randmi≥1.

Further,A(s) is a finitely generatedA^GLn-module, say generated by {s1tⁿ¹, . . . , syt^ny} withsi∈R(s)andni≥0.

3.8. Knop-Luna slices 161

Takem=max miandn=max ni andr∈I^mk+n∩R(s), thenrt^mk+n∈A(s) and rt^mk+n=X

j

pj(r1t^m1, . . . , rzt^mz)sjt^nj

withpj a homogeneous polynomial oft-degreemk+n−nj≥mk. But then each monomial inpj

occurs at least with ordinary degree ^mk_m =kand therefore is contained in (I^GLn)^kR(s)t^mk+n.

LetR\^GLnbe theI^GLn-adic completion of the invariant ringR^GLnand letS\^GLnbe theJ^GLn-adic completion ofS^GLn.

Lemma 3.10 The morphismψ^∗induces an isomorphism

R⊗_RGLnR\^{GLn '}- S⊗S^GLnS\^GLn

Proof. Let s be an irreducible GLn-module, then the I^GLn-adic completion of R(s) is equal to Rd(s)=R(s)⊗_RGLnR\^GLn. Moreover,

Rb(s)=lim

←(R

I^k)(s)=lim

←

R(s)

(I^k∩R(s))

which is theI-adic completion ofR_(s). By the foregoing lemma both topologies coincide onR_(s) and therefore

Rd(s)=Rb(s) and similarly Sd(s)=Sb(s)

BecauseRb'Sbit follows thatRb_(s) 'Sb_(s)from which the result follows as the foregoing holds for

alls.

Theorem 3.13 Consider aGLn-equivariant map Y ^ψ- X, y ∈ Y, x= ψ(y) and ψ ´etale in y.

Assume that the orbitsO(x)and O(y)are closed and that ψ is injective onO(y). Then, there is an affine open subsetU ^⊂ - Ycontaining ysuch that

1. U =π⁻¹_Y (πY(U))andπY(U) =U/GLn. 2. ψ is ´etale onU with affine image.

3. The induced morphismU/GLn ψ

- X/GLn is ´etale.

4. The diagram below is commutative

U ^ψ - X

U/GLn π_U

?? _ψ

- X/GLn π_X

??

Proof. By the foregoing lemma we haveR\^GLn'S\^GLn which means thatψ is ´etale inπY(y). As

´etaleness is an open condition, there is an open affine neighborhoodV ofπY(y) on whichψis ´etale.

IfR=R⊗_RGLnS^GLn then the above lemma implies that R⊗_SGLnS\^GLn'S⊗_SGLnS\^GLn

LetS^GL_locⁿbe the local ring ofS^GLninJ^GLn, then as the morphismS^GL_locⁿ - S^\GLnis faithfully flat we deduce that

R⊗_SGLnS^GL_locⁿ'S⊗_SGLnS_loc^GLn

but then there is anf∈S^GLn−J^GLn such thatRf 'Sf. Now, intersectV with the open affine subset wheref6= 0 and letU⁰be the inverse image underπY of this set. Remains to prove that the image ofψis affine. AsU^{0 ψ}- X is ´etale, its image is open andGLn-stable. By the separation property we can find an invarianth∈R^GLn such thathis zero on the complement of the image andh(x) = 1. But then we takeU to be the subset ofU⁰of pointsusuch thath(u)6= 0.

Theorem 3.14 (Slice theorem) Let X be an affine GLn-variety with quotient map X ^π-- X/GLn. Let x ∈ X be such that its orbit O(x) is closed and its stabilizer subgroup Stab(x) =H is reductive. Then, there is a locally closed affine subscheme S^⊂ - Xcontaining x with the following properties

1. Sis an affineH-variety,

2. the action mapGLn×S - Xinduces an ´etaleGLn-equivariant morphismGLn×^HS ^ψ- X with affine image,

3. the induced quotient mapψ/GLn is ´etale

(GLn×^HS)/GLn'S/H ^ψ/GL-ⁿ X/GLn

and the right hand side of figure 3.11 is commutative.

3.8. Knop-Luna slices 163

GLn×^HNx

GL_n×^Hφ

GLn×^HS ^ψ - X

Nx/H

??

φ/H S/H

?? _ψ/GL

n - X/GLn π

?? Figure 3.11: Etale slice diagram

If we assume moreover that X is smooth in x, then we can choose the slice S such that also the following properties are satisfied

1. Sis smooth,

2. there is anH-equivariant morphismS ^φ- Tx S=Nx withφ(x) = 0having an affine image, 3. the induced morphism is ´etale

S/H ^φ/H- Nx/H and the left hand side of figure 3.11 is commutative.

Proof. Choose a finite dimensionalGLn-subrepresentationV ofC[X] that generates the coordinate ring as algebra. This gives aGLn-equivariant embedding

X^⊂i- W =V^∗

Choose in the vectorspace W an H-stable complement S0 of gl_n.i(x) = T_i(x) O(x) and denote S1=i(x) +S0 andS2 =i⁻¹(S1).Then, the diagram below is commutative

GLn×^HS2 ^⊂ - GLn×^HS1

X

ψ

?⊂ i

- W

ψ0

?

By construction we have thatψ0 induces an isomorphism between the tangent spaces in (1, i(x))∈ GLn×^HS0 and i(x) ∈ W which means that ψ0 is ´etale in i(x), whenceψ is ´etale in (1, x) ∈ GLn×^HS2. By the fundamental lemma we get an affine neighborhoodU which must be of the formU=GLn×^HSgiving a sliceSwith the required properties.

Assume thatXis smooth inx, thenS1 is transversal toXini(x) as Ti(x) i(X) +S0 =W

Therefore,Sis smooth inx. Again using the separation property we can find an invariantf∈C[S]^H such thatf is zero on the singularities ofS(which is aH-stable closed subscheme) andf(x) = 1.

Then replaceS with its affine reduced subvariety of pointss such that f(s) 6= 0. Letm be the maximal ideal ofC[S] inx, then we have an exact sequence ofH-modules

0 - m² - m ^α- Nx^∗ - 0

Choose aH-equivariant sectionφ^∗:N_x^∗ - m^⊂ - _C[S] ofαthen this gives anH-equivariant morphismS ^φ- Nx which is ´etale in x. Applying again the fundamental lemma to this setting

finishes the proof.

References.

More details on ´etale cohomology can be found in the textbook of J.S. Milne [64] . The material of Tsen and Tate fields is based on the lecture notes of S. Shatz [77]. For more details on the coniveau spectral sequence we refer to the paper [18]. The description of the Brauer group of the functionfield of a surface is due to M. Artin and D. Mumford [6] . The ´etale slices are due to D.

Luna [63] and in the form presented here to F. Knop [45] . For more details we refer to the lecture notes of P. Slodowy [79].

Im Dokument noncommutative geometry@n, volume 1 : the tools (Seite 164-173)