a mackey imprimitivity theory for algebraic...

Math. Z. 182, 447-471 (1983) Mathematische Zeitschrift

�9 Springer-Verlag 1983

A Mackey Imprimitivity Theory for Algebraic Groups*

Ed Cline 1 , , , Brian Parshall 2, and Leonard Scott 3

1 Department of Mathematics, Clark University, Worchester, Massachusetts 01601, USA 2, 3 Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903, USA

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question arises, given a rational G-module M, how can one recognize M as an induced module VIG? For a finite group G the answer is the Mackey imprimitivity theorem: the module M is induced if and only if it is a direct sum of subspaces permuted transitively by G (with H the stabilizer of one of these subspaces, called V). One uses this result, for example, in proving the famous Mackey decomposition theorem which describes the restriction of any induced module to a second subgroup L as a direct sum of suitable induced modules; given the imprimitivity theorem, the proof is just a matter of grouping the summands permuted by G into their orbits under L.

In the case of algebraic groups the situation is quite different. For some subgroups H, all G-modules are induced. This occurs, for example, if G is connected and k [G/H] ,= k, e.g., if H is parabolic. Also, if G is a connected unipotent group, then a rational G-module M is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule E, for the conjugation action of G, with E___ k. 1 and such that k. l is precisely the subspace annihilated by the action of the Lie algebra of G on E [27] (cf. also (5.5) below). The latter is an application of a general criterion in case G/H is affine: a rational G-module M is induced if and only if there is an action of the coordinate ring A of G/H compatible with the action of G on both A and M. Note that this generalizes the imprimitivity theorem in the case of finite groups, since then A has a k-basis of [G:H] orthogonal idempotents permuted transitively by G.

* Research supported by the National Science Foundation ** The first author thanks the University of Virginia for its hospitality during the writing of this paper

448 E. Cline et al.

Mackey's original results [25] in the generality of locally compact groups were formulated similarly in terms of idempotents (projection operators associated with Borel sets) and were reformulated by Blattner [2] in terms of the action of an algebra of continuous functions. Other imprimitivity theorems appear in [-2, 11, 23, 29, 35], and a version of the decomposition theorem for finite group schemes appears in [36].

In this paper we present a general imprimitivity theory, encompassing and considerably extending the criterion in the affine case. Our results do not require G or H to be reduced, and they are meaningful even in the apparently trivial case mentioned above where k [G/H] =k. One striking consequence of the machinery we develop is the following induction theorem (4.5) for a semisimple group G: Let Ps and PK be standard parabolic subgroups associated with a Borel subgroup B and subsets J , K of the set F/ of simple roots. If K u J* =II, where J* denotes the image of J under the opposition involution, and if V is a rational Pj-module, then

Vale~, ~ VW~

Here w o is the long word in the Weyl group and V w~ is the module V for pj~o, acting through conjugation by w o. The condition K•J* =17 guarantees that the complement in G of the subvariety PS~ is sufficiently small, though it is by no means empty, as in the analogous results for discrete groups.

The general imprimitivity theorems (2.7), (3.5) may be regarded as a sheafif- ication of the affine result [-27] discussed above: Each induced module may be regarded as the global sections of a suitable sheaf on G/H and the main problem is to characterize these sheaves. Our task is considerably facilitated, both conceptually and technically, by the adoption of the Grothendieck-De- mazure-Gabriel philosophy of faisceaux [12]. From this point of view, one regards the category of schemes as embedded in suitable functor categories, giving increased flexibility for dealing with, for example, quotient structures. In our case, this procedure turns out to be enormously effective, with the general imprimitivity theorems ultimately derived from a generality due to Grothen- dieck and Verdier [34], in which a seemingly innocuous property of group actions on sets is translated pro forma into a result on faisceaux (see Theorem (1.2)).

This paper is organized as follows. Section 1 developes the machinery of group actions on fiber bundles which is used in the proof of the imprimitivity theorem in Sect. 2 and its generalization in (3.5). The remainder of Sect. 2 as well as Sect. 3 collect together a large number of examples and first applications. In Sect. 4, we prove some analogues of Mackey's decomposition theorem and give several consequences, such as the result on parabolic induction mentioned above. In Sect. 5, we use our local results to give a global imprimitivity theorem generalizing that given in [27] to the case where G/H is quasi- affine. The paper ends with three appendices. The first, Sect. 6, gives some further applications of the results of Sect. 4 to the case of infinitesimal thicken- ings of a Borel subgroup (see (4.2) for definitions). The second appendix, Sect. 7, uses these results to treat the Andersen-Haboush tensor identity [1, 18]

A Mackey Imprimitivity Theory for Algebraic Groups 449

which implies the Kempf vanishing theorem [22]. The third, Sect. 8, treats briefly a form of the Maekey decomposition theorem observed by Voigt [36] for finite group schemes.

We would like to thank Andy Magid for several helpful conversations, and for making available to us the results of [14], all of which contributed at least indirectly to this work. We also mention that Theorem (2.7) had been treated earlier by Demazure [11] for the case G/H complete, and under similar hypotheses (locally free sheaves), though without proof, by Haboush [35].

w 1. Group Actions on Fiber Bundles

Let k be an algebraically closed field, k-Alg the category of (commutative) k- algebras, and Ens be category of sets. (These and similar categories should be suitably restricted to avoid set-theoretic difficulties, as explained in [12].) A k- functor is a functor X:k -Alg~Ens . Every scheme X over k in the usual geometric sense gives rise to a k-functor hx: R ~HOmk(Spec R, X); in this way the category of schemes is embedded in the category of k-functors. We will usually write X for h x when no confusion can arise.

(1.1) Faisceaux. We now recall what it means for a k-functor X to be a faisceau. Given a commutative k-algebra A, a family (Bi)~ I of commutative A- algebras is called a covering of A provided that I is finite and the product a lgebra /7 B~ is a finitely presented, faithfully flat A-algebra. (This is the "fppf topology" of Grothendieck.) Then X is a faisceau [12 ; I I I ,w if for every A and every covering (B~)~ of A we have an exact sequence

X(A)~]-~X(Bi) ~ ~ X(BI| A Bj) i (i, j )

of sets, where the map u has X(A)~X(B~) as its ith projection, the (i,j)th projection of v is obtained from X(Bi)~X(Bi| and that of w from X(Bj)~X(Bi| Recall that "exactness" requires that u be the kernel for the pair (v, w). Every scheme is a faisceau.

Given a k-functor X, there exists a faisceau )( and a morphism ~ : X ~ ) ? having the universal property that any morphism of X into a faisceau factors uniquely through ~. The functor X~--,Yf commutes with finite projective limits.

A k-group functor is a functor H:k-Alg~Gr, where Gr denotes the "sub"category of groups in Ens. If the k-functor H is also a faisceau, we say that H is a k-group faisceau. A k-group H acts on a k-functor X if H(R) acts on X(R) for each k-algebra R in the usual sense, with the action functorial in R. If H acts on X, then, following [12], we let X/H denote the k-functor R~--+X(R)/H(R), and we let XTH denote the associated faisceau. It is of course XTH which is the correct notion of quotient when X and H are faisceaux (e.g., schemes). In w 2 when we are dealing with faisceaux only, we will drop this notation and just write X/H for XTH.

We remark that X~Y=XTH is an epimorphism in the category of faisceaux when X is a faisceau, and that all epimorphisms in this category are

450 E. Cline et al.

effective (that is, X x r X _ ~ X ~ Y is exact). Consequently, given any epimorphism X ~ Y and commutative diagram

W I--~Z

\ / Y

of faisceaux, if f x r X : W x r X ~ Z x r X is an isomorphism, then so is f . See [12, III ,w 1, 2.6] for more details.

(1.2) Induced Fiber Bundles. If H is a subgroup functor (in the obvious sense) of a k-group functor G, we write X xHG for (X x G)/H where H(R) acts on (X x G) (R) =X(R) x 6(R) via (x, g) h=(sh, h- 1 g) for x~X(R), g~G(R), hell(R), R a

k-algebra. Also, we write X v ~ G for (X x G)~H. Finally, we prefer the quotients G/H and GTH to be taken with respect to the (right) action of H in which the result of applying h~H(R) to g~G(R) is defined as h - l g . The natural right action of G on G by multiplication commutes with this right action of H, and hence induces a right action of G on G/H and GTH.

Theorem ([34; IV,w Let G be a k-group functor (resp. faisceau) Z and suppose we have a G-equivariant map of k-functors Z ~ G / H (resp. Z--,G[H) where H is a subgroup functor of G. 1 Let X be the fiber Z XG/l~e (resp. Z x GT~ e) where e is the subfunctor H/H of G/H or 6[H, and let H act on X in the obvious way. Then Z-~X xUG (resp. Z~-XvHG) via the natural G-equivariant map X x I~ G ~ Z induced by multiplication.

Proof. These results for k-functors and k-group functors are straightforward consequences of the corresponding (trivial!) facts for groups and sets (where X is just the inverse image of e), using the natural map Xx~G---,Z. In the faiceau case, note that X = Z x G ) I e is also ZOXG/He, where Zo=ZxG;HG/H. Applying the result for Z 0 and G/H we have Zo~-XxUG. Thus, Z_-__Z xGT~G~H~(ZxG)tG/H)~=2o_~(Xx~G)'=XvX~G, giving the desired

isomorphism. Q.E.D.

(1.3) Induced Sheaves. Again let G be a k-group functor and H a subgroup functor acting on a k-functor X. We end this section with a description of the set F(U, X xt/G) of sections over a subfunctor U of G/H, and we give as well a similar result when everything is a faisceau and U ~_ G~H.

Let rc: G ~ G/H be the quotient map. If G is a faisceau, let ~ be the quotient map G~G}H. Define LZ~x(U ) (resp. LTex(U), in case G is a faisceau) to be the set of all morphisms f : W ~ X , where W=rc I(U) (resp. W=~- I (U) ) , which are H-equivariant in the sense that f (hw)=f(w)h -z for all hell(R), weW(R), R a k-algebra. 2 Then with this notation we have

Note that H need not be a faiceau in either case 2 In w 2 where we deal only with faisceaux (and mostly with schemes) we will write s for 2x(V)


Theorem. There is a natural bijection

el'v: 5~x(U)-~+ F(U, X x~G).

Similarly, when G and X are faisceaux and U is a subfaisceau of GTH we have a natural bijection

~u: ~ x ( u ) - ~ r ( v , x v~O).

Proof. Given feG, qx(U ) we have a map q~v(f):W--,XxnG which sends weW(R), R a k-algcbra, to Ef(w), w], the image in (X xUG)(R) of (f(w), w)e(W xG)(R). Clearly, ~bv(f) factors through ~r :W~W/H=U to give a map q~v(f): U~'X x~ G.

Next, we describe a candidate 5r v for an inverse of q~v. Let # : (XxUG) x a/u W--*X x W be the isomorphism defined by

~([x, w'7, w)=(xw' w -1, w)

for xeX(R), w, w'eW(R). If H acts on the left of (X xnG) xGm W by acting on the W factor, and on X x W by h(x,w)=(xh-l , hw), for hell(R), xeX(R), weW(R), then the map /~ is H-equivariant. Thus, the composite of/~ with the projection Px: X x W---*X is H-equivariant. Now suppose we are given a section s: U---,X xnG. This gives a map sTr: W-~X x G, and we set

Tv(s) =Px ~(s ~ x ~/~ lw).

Since Px# is H-equivariant, so is ~Pv(s), and thus T v ( s ) ~ x ( U ). Clearly, 7~vq)v=l~(v). To prove the other composite is the identity, note

that the map P : ( X x ~ G ) xGm W ~ X x W ~ X xH G

is the same thing as projection on the first factor. Given s~F(U, X xUG), then by definition 4'v Tv(s) is the composite Po(sn xGmlw), hence is just sn, and so ~b v 7Jv(S)=S, as desired.

This argument goes through essentially as given, in the faisceau case, though it is necessary to note some adjustments: First observe

(1.3.1) W~H ~ U

if U is a subfaisceau of G~H and W=~-~(U). Indeed, we have that WTH ~_ (G/H x ~TH U) ~~ G~H x 6?u U ~ U.

For the rest of the proof, let Or(f) be defined for f e~x (U ) exactly as above, and let ~v(f) be the induced map W--,X vHG. The factorization of (~v through ~ gives ~5 v. Next define # exactly as above and let /2:(XvHG) x G-;n W---,X x W be the induced map of faisceaux. For s~F(U, X v R G) define

~ (s) = px ~(s ~ x ~/~ 1 w).

If P above is replaced by its induced map /~, it is still projection on the first factor of (X v ~ G)x am W and ~j ~s(S)=/5(s~ x ~mlw). Now the proof is com- pleted as above. Q.E.D.

452 E. Cline et al.

Observe that Sx(U ) can be defined for any morphism U---,X, and the above result goes through, mu~atis mutandis.

w 2. An Imprimitivity Theorem

In this section G is an affine algebraic group scheme over the algebraically closed field k, and H is a closed subgroup scheme. (Thus, G is an affine algebraic group in the classical sense, except that its coordinate ring may contain nilpotent elements.) If H acts on the right on a scheme (or faisceau) F we write F/H for the quotient faisceau which was denoted FTH in w 1. The faisceau G/H is a scheme [12, III, w 3, 5.4] and most quotients we deal with will in fact usually be schemes, though we may not always know this d priori. (2.1) For a scheme F upon which H acts, we construct a sheaf 5~ on G/H by defining ~F(U), for U an open subscheme of G/H, by analogy with the construction of ~cfv(U ) given in w 1. If U is identified with the subfunctor of ha/~ it defines, this Yr(U) in fact agrees with the set ~v(U) defined there.

The morphism 7:: G~G/H is faithfully flat, and also G x G/H G ~ G x H (an easy argument using faisceaux). In particular, G--*G/H is an affine morphism 1-12; III ,w 4, 1.9].

If V is a vector space over k which is a rational left H-module (i.e., a comodule for the coordinate ring of H, or equivalently and more in the spirit of the present exposition: H(R) acts linearly and functorially on V(R)=R| k V for each k-algebra R), we define a right action of H on V by setting v h = h- 1 v, for v~V(R), h~H(R), and R a k-algebra. (Recall, we use a similar convention for the right action of H on G.) Then we have a sheaf 5~ as discussed above, with Lfv(U ) the set of left H-equivariant maps from the preimage in G of U to V, for U open in G/H. Alternately, we can write Sv(U)=((gG(Tz-I(U))| for a suitable diagonal action on the tensor product, from which it is easily seen that Yv is a quasi-coherent (gGm-module, cf. the argument in [7; (4.1)]. Anoth- er obvious consequence of this description of ~ v is that we have a natural inclusion S v _ 7z, ((9 G| V). We observe:

(2.1.1) The corresponding map 7r* ~ev---,(gG| k V is an isomorphism.

To prove this, let Uc_G/H be open affine. Then W=Tz-I(U) is open affine as noted above, and so

(~* ~v) ( w) ~- (9 ~( w) |162 2e~( u) ,~ (g G(W)|162 ((9 G(W)| V)"

((g G(W)| (g G(W)| V/r

(using flatness of G ~ G/H)

-~ ((gH(H) | k OG(W ) | V)"

(since G x G/H G ~ G x H)

(9,(w)| v.


We leave it to the reader to check that this isomorphism is indeed the map in question. For a similar argument see [5; w 4].

(2.2) Theorem. Let V be a rational H-module and V*=Hom(V,k) its dual. Then over G/H we have

V* v R G ~Spec S(Sev),

where S(2'v) denotes the symmetric algebra of •v over (gG/H. In particular, V* v ~I G is a scheme [-12; III,w 4, 1.9].

Proof for V Finite Dimensional. The natural inclusion A~~174 V) of g0G/H-modules gives rise to a G/H-morphism V* x G~SpecS(~v) which obviously factors to give a G/H-morphism t: V*v~lG~SpecS(~v) . Now it is straightforward to check that t x~/HG becomes an isomorphism of faisceaux (actually schemes), noting that (V*v~IG) x~/HG~-V*xG and SpecS(Yv) xG/r~G~Spec(S(V)| V* x G (cf. [13; I, w 9]). Thus t is an isomorphism,

since G-?G/H is an effective epimorphism. The proof in the infinite dimensional case will be given below in (2.6).

(2.3) We now define the notion of a quasi-coherent X. G-module, where X is a k-scheme upon which G acts. Let g be a quasi-coherent (gx-module and form the vector bundle ~r(g)=SpecS(E) with structural morphism p:~g(C)~X [13;I,w Observe that there is a natural action G a x V ( E ) ~ r ( g ) , where G u ~ S p e c k [ t ] is the one-dimensional unipotent group. Writing ~a xV(g) =SpecS(g)[t] and V(g)=SpecS(8), the action is obtained from the (9 x- algebra homomorphism S(g)---,S(g) It] gotten by sending a section f of g to ft . (This agrees with the natural R-module action on ~g(g)(R), for R a k- algebra, as defined in [13, I, w The following definition is in the spirit of Kempf [20].

Definition. A quasi-coherent X.G-module is a quasi-coherent (gx-module equipped with a right action of G on u commuting with the natural action of 113a defined above.

Observe that any action V(g)x G--,~r(g) arises from an (gx-algebra homomorphism S(g)~p.(S(8)| where p: X x G ~ X defines the action of G on X. The fact that this action is to commute with the G u action is easily seen to be the equivalent to the assertion that this algebra homomorphism carries into P,(g| (9~). The definition of a quasi-coherent X-G-modules could now be formulated in terms of the corresponding map p * ( C ) ~ g | k (gG; the result is essentially Mumford's notion 1-26] of a "G-linearized sheaf" [20, p. 314]. For functorial definitions see (3.2) and (3.5) below. These definitions and the present one are in the spirit of G-vector bundles for a Lie group [31; 2.4.2].

As a consequence of (the proof of) Theorem (2.2), we have the

(2.4) Corollary. Let H be a closed subgroup scheme of G and let V be a rational H-module. Then ~ v is an X. G-module for X = G/H.

Proof. If V is finite dimensional, then Theorem (2.2) gives a natural isomorphism V*v~lG=SpecS(~v), and one easily checks that the ll~u-action on Spec S(L,~v) comes from the vector space structure of V*. Hence if we let G act

454 E. Cline et al.

on u162 S(~v) by acting in the obvious way on V* v u G, we have the required commutativity with the Ga-action.

In the infinite dimensional case we can represent Ae v as a direct limit of Yw'S with W finite dimensional. The map 2#v~p,(~V| mentioned above is now just obtained as a limit of the corresponding maps for the 5r and the required properties can be checked. Q.E.D.

(2.5) Lemma. Let X = G/H where H is a closed subgroup scheme of G. Then for V a rational H-module we have an isomorphism of rational H-modules

(2.5.1) k| ~ (~Q'Q~V)e "--> V.

Proof. Let U be an open affine subset in X containing e. Then we have a map

given by

v(F) = G v

where e: (gG(u-l(U))~k is the comorphism associated with the inclusion Speck =e~rc - l (U) . One verifies as in [5; (4.2)] that (gG(u-l(U))| is sur-

jective. Thus, #v is surjective by the faithful flatness of re-~(U) over U. When V is finite dimensional, the isomorphism

(~ G(7~- l ( u ) ) @ ( o . / . ( u ) ~ v ( U ) ~ > (~G(7g- l(U))@k W

shows that 5~ v is locally free of rank equal to dim V, and so (2.5.1) must be an isomorphism. Taking limits we get an isomorphism for general V. Q.E.D.

(2.6) We now complete the proof of Theorem (2.2). By the above corollary (which used the finite dimensional version of Theorem (2.2) only), we now know that X~ v is an X.G-module. Hence, by Theorem (1.2) we have that Y(Yv)=Y(Lfv)e vHG. Observe by the above lemma that vf(~fV)e~Spec(S(SYv) |162 S(V)= V*. Q.E.D.

We now state our first imprimitivity theorem. See also (3.5) and (5.1) below.

(2.7) Theorem. Let H be a closed subgroup of an affine algebraic k-group scheme G. Write X = G / H and let e~X be the point H/H. Then the correspon- dences

V~--~fv and #F--*k|162

define an equivalence between the category of rational H-modules and that of quasi-coherent X .G modules. In particular, every quasi-coherent X.G-module is induced.

Proof. We have from (2.5.1) above that the composition

V~Sev ~--,(Yv)e | ~ k


is the identity. Next suppose we are given an X.G-module & Since H stabilizes V(d~)e=V(g~| ek) and commutes with the action of ~3,, it follows that V=g~| is a k [H]-comodule - that is, a rational H-module. By Theorem (1.2), V(g )~V* v ~ G ~ V ( ~ v ) , and one checks easily these isomorphisms are 113a-linear, whence ~ ~ ~ v . Q.E.D.

(2.8) Corollary. The functor V~-*Sf v is an exact functor from the category of rational H-modules to the category of quasi-coherent (gx-modules.

(2.9) Corollary. Let i, denote the induction functor I ~ from the category of rational H-modules to that of rational G-modules. Then the right derived functors of i, are given by

R"i,(V)~-H"(G/H,~v) , n>O,

for each rational H-module V.

Proof. Since i, V~-H~ ~v), it is enough by (2.8) to show

(2.9.1) The functor V~--~s v takes rationally injective H-modules to F(G/H,-)- acyclic sheaves.

The conclusion will then follow by, say, the Grothendieck spectral sequence [17]. Clearly it suffices to consider the case of the H-module k[H], since any rationally injective H-module is a direct summand of a sum of copies of k [HI, cf. [5]. But for U open in G/H, we have an isomorphism

/r ~(oA~- l(u))| k [H]) ~

= e~(~- l(V))

of (PC/H-modules (since VIn~V for any rational H-module V). Thus, ~k~m~n,(PG as (gG/H-modnles. Since n is an affine morphism, an elementary application of the Leray spectral sequence [19; III, Ex. 8.2] shows that

H"(G/H, .LFv)~-H"(G/H , n, Gv)~- H"(G, (gG)=0

for each n > 0 by Serre's vanishing theorem [19; III, Theorem 3.7]. Q.E.D.

We remark as in [7; (4.8)] that R"i,(V) has an alternate description in terms of rational group cohomology as H=(H, k[G]| as is clear from the construction of induced modules.

Finally, we record here the following result obtained in the course of the proof, cf. (2.5).

(2.10) Corollary. For V a finite dimensional rational H-module, the (gx-module S v is locally free of finite rank.

w 3. Examples and Farther Properties of X.G-Modules

(3.1) Throughout this subsection G is an affine algebraic group scheme over k. All schemes are taken over k.

456 E. Cline et al.

(3.1.1) If G acts on a scheme X, the structure sheaf (9 x is always an X.G- module, since ~g((_gx) ~ A 1 x X.

(3.12) Suppose G acts on schemes X and Y and f : X ~ Y is a G-equivariant morphism. Then if ~- is a quasi-coherent Y.G-module, f * ~ is a quasi-coherent X.G-module. (Observe that X x y V ( @ ) = V ( f * ~ ) . ) If f is affine and N is a quasi-coherent X.G-module, then f . N is a quasi-coherent Y.G-module.

(3.1.3) If G acts on a scheme X and ~ , N are quasi-coherent X.G-modules, then ~ - | is naturally a quasi-coherent X.G-module. If, in addition, ~- is locally free of finite rank, H o ~ = ( ~ , , (4) is also a quasi-coherent X.G-module. This has been essentially noted by Mumford [26], and can also be easily verified using the functorial definition in (3.2) below. Similarly if G acts on schemes X and Y and if ~, N are quasi-coherent X.G-modules, Y.G-modules, respectively, then ~-| N is a quasi-coherent (X x Y).G-module. (Recall that ~- |174 where Px, PY are the projections onto X,Y, respectively.)

(3.1.4) Let G act on a scheme X, let o~ be a quasi-coherent X.G-module and V a rational G-module. Since V is clearly a Spec k.G-module, it follows from (3.1.3) that ~ | V is a quasi-coherent X.G-module. If X=G/H (where H is a closed subgroup scheme of G), we have by Theorem (2.7) that ~ =5~ for some rational H-module W and Sw|174 ). This generalizes the tensor identities of [5] and [10; 1.3.1]. A similar argument shows that 5Yv| w =Sly| w for rational H-modules V and W. Also, Y * = ~ v * when V is finite dimensional (recall Yv is locally free of finite rank). The same holds for V infinite dimensional (cf. (2.2) and (1.3)) if V* is considered as a non-rational G- module.

(3.1.5) Let ~- be a quasi-coherent X.G-module for some scheme X. If h: H ~ G is a morphism of affine k-group schemes, ~- is an X.H-module. If U is an H-stable open subscheme of X, then ~ ( U ) is a rational H-module. See also (3.4).

(3.1.6) For a given action of G on a scheme X, the category of quasi-coherent X.G-modules is abelian - in particular and more precisely, if f : ~ , ~ is a G- equivariant morphism of X.G-modules (in the sense that ~g(f): V(N)--,~r(~) is G-equivariant) then the kernel and cokernel of f are also quasi-coherent X.G- modules.

(3.1.7) If G acts on a scheme X, the sheaf ~x/k of Kihler differentials is an X.G-module. Recall that f2x/k=A*(I/IZ), where A: X--+XxX is the diagonal embedding and I is the sheaf of ideals defining A (X) (assume for simplicity that X is separated). In particular, if X = G/H, then ~?x/k is by Theorem (2.7) an induced sheaf 5~v where V= (g/t?)* and g and b denote the Lie algebras of G and H respectively.

(3.1.8) Suppose G acts rationally as a group of k-algebra automorphisms of a commutative k-algebra R. Given an R-module M, we say that it is an R.G- module if it is a rational G-module in which the R and G actions are related


by requiring that g(rm)=(gr)(gm) for geG(S), reS| , m~S| for any commutative k-algebra S. Setting X = Spec R and ~ =M, we clearly have that Y is a quasi-coherent X.G-module. In this context (and assuming that G is reduced), X.G-modules have been studied in [-14] and [-27].

(3.2) A First Functorial Definition of an X.G-Module. Recall that there is a natural way to define the action of an abstract group F on a sheaf over a topological space X upon which F acts. Namely, for each g~F we must have a morphism 0 g : ~ g , ~ such that 01=id and Ogh=OhgOOh for each g, hsG, where 0~: h , y - - - , g , h, j~ is the morphism defined by setting 0, h v=Og vh-i for U open in X. We will usually just write ga for Og(a)eo~(Ug 2"1) whenever geF and a ~ ( U ) for some open set U; thus, the above condition on Og h becomes (g h) (a) = g(h a).

Whenever F acts on X and X is a ringed space with structure sheaf (9 x, then by definition F acts on (9 x. If ~" is an (gx-module, we say that F acts X- linearly on ~ if it acts on the sheaf ~ as above and if the maps 0g are (9 x- module maps (thus g(am)=(ga)(gm) if g~F, ae(gx(U ) for some open U in X, and mso~(U)).

Next suppose that G is an affine group scheme over k as in w 2, X is a scheme over k upon which G acts and ~ is a quasi-coherent X.G-module. For any commutative k-algebra R, let X a = X x Spec R, regarded as a scheme over R. In terms of functors there is a natural isomorphism between X R and the restriction of X to the category of commutative R-algebras. Let YR denote p* ~ where p: X x Spec R ~ X is the projection morphism. Since V(~R) ~$r(~, ~ ) x S p e e R = I / ( J ) R as noted in (3.1.2), we have an action of G(R) on u over

X R which is functorial in R. It follows that the abstract group G(R) acts X R- linearly on the sheaf YR in the sense of the previous paragraph. Conversely, if Y is a quasi-coherent (gx-module such that for each commutative k-algebra R, G(R) acts XR-linearly on ~R and such that this action is functorial in R, then it follows that Y is a quasi-coherent X.G-module. This motivates the

Definition. Let G be a k-group functor acting on a scheme X over k. A quasi- coherent (gx-module ~- is called an X.G-module provided that, for each commutative k-algebra R, there is an XR-linear action of the group G(R) on ~R, functorial in R.

The generalization beyond schemes to k-group functors is quite useful for our treatment of infinitesimal actions below. Also, the general format of the definition makes it very easy to check, as we now demonstrate. For an even more general notion of an X.G-module see (3.5) below.

(3.3) Local Cohomology. Suppose G is a k-group functor acting on a Noe- therian scheme X over k. Let U_~X be a G-stable open subset and put X = Z - U . We will show that if @ is a quasi-coherent X.G-module, then the local cohomology sheaves ~16] ~ ( o ~ ) are also X.G-modules. This result is essentially due to Kempf [20] with slightly weaker hypotheses.

Let R be a commutative k-algebra and let p : X x S p e c R ~ X be the projection morphism. We first note that if V is an injective quasi-coherent (9 x- module, then p*~/" is ~~ The question is local, so we may assume

458 E. Cl ine et al.

that X = S p e c A with A Noetherian. We can suppose that ~=/~7/ where M is an injective A-module. Then if Z =SpecA/I , we have

~4~z~ (P* M ) ~ lira Ext"A|174 R, M | ~ 1

lira p* Ext"A(A/I ~, M) ~ l

= p* ~ ( ~ ) = 0

(cf. [-16, 2, pp. 29-35]). Next observe that we have an isomorphism

~L(P* g)--p* ~e~(g).

This follows by observing that if y-- ,qU, is a resolution of ~- by injective quasi-coherent (gx-modules, then p * ~ - ~ p * ~ , is a resolution of p * ~ by J r~ R- acyclic modules. Because p* is exact, however, the cohomology of this complex is just p* J r* (~-).

Now let gsG(R). The morphism Og: o ~ g , g given by the X.G-module structure on ~ clearly induces corresponding morphisms Og:p* J t ~ ( ~ ) ~ g , p* ~f~}(~) for each n. We must show that these are functorial in R. If z: R - * S is a k-algebra homomorphism, we have a commutative diagram

X s gs -~ X s

X R g ~ X R

where t: X s ~ X a is the morphism expressing the base change. The required functoriality follows from the isomorphisms t* g, Jf~R(p* W) ~ gs, t* p* J f ~ ( ~ ) ~gs,~(f}~((pt)* ~) . It follows from (3.2) that .Xg~(~) is an X.G-module ,

It is interesting to observe that the above proof simplifies considerably if we are content with a weaker notion of X.G-module than that of (3.2), cf. (3.5) below.

(3.4) Infinitesimal Actions. Recall that an affine algebraic groups scheme is called infinitesimal provided that G~a=e, the trivial group. In practice, infinitesimal groups arise as kernels of Frobenius morphisms in characteristic p>0. See [-7], [12; II, w 7.1] for more details.

Now let X be a scheme over k upon which an infinitesimal group G acts. For U an open subscheme of X we have a pull-back diagram

( U x G ) x x U , U

U x G - - ~ X


with j an open immersion. Since Gre d = e, J~ea is an isomorphism, so j is also an isomorphism. Thus, U is G-stable, and the results of (3.15), (3.2) apply. We remark that the notion of X.G-module can be defined more simply for infinitesimal groups by exploiting the stability of all the open subsets; namely, an (gx-module ~- is an X.G-module provided that for each open U, ~-(U) is a rational G-module compatible with the restriction maps, cf. [20].

Now let G be an affine algebraic groups scheme over k. Let A be the coordinate ring of G and I the augmentation ideal of A (i.e., the kernel of the count of A). Then we define a k-group functor G by taking G(R) to be the group of all k-algebra homomorphisms annihilating some suitably large power I" of I, R a commutative k-algebra. Noting that G = U G(n), where G(n) is the

n_>0 subfunctor represented by A/I n, we obtain as above that G stabilizes each open subscheme U of any scheme upon which G acts. Thus, the above remarks about infinitesimal groups are applicable to d also. A rational module for a k- group functor H may be defined as a k-vector space V, possibly infinite dimensional, equipped with an R-linear and functorial action of H(R) on R | for each commutative k-algebra R. This agrees with the usual notion when H is affine [12; II, w 2.1]. It is not hard to see that in characteristic 0 the category of rational G-modules is just the category of modules for the Lie algebra of G. In arbitrary characteristic, the category of rational G-modules is the category of modules for the hyperalgebra hy(G)= L) HOmk(A/In, k) of G, cf. [7]. ,_,o

(3.5) The General X.G-Module. First suppose X is a scheme over k and ~- is a quasi-coherent Ox-module. Then we can use ~ to define a faisceau F over X is a natural way: For any commutative k-algebra R, define F(R) to be set of all pairs (p, f) where p: SpecR-~X is a morphism and fe(p*~)(SpecR). Write Fo(R)={fI(f p)~F(R)} for a given psX(R). Thus, if p factors through an open affine U of X, we have F(R)=R| ). It is not hard to check that F is a faisceau, and the map sending (p,f)eF(R) to p~X(R) makes F into a faisceau over X.

N o w suppose in addition that G is a k-group functor acting on X and ~ is a quasi-coherent X.G-module. If geG(R) we get a map 0 g : ~ R ~ g . ~ R as in (3.2), and hence a m a p g * Y R ~ R. Pulling back with p*, we get a map 7g : (p - g ) *~ R ~ p*~R. This gives an action of G(R) on F(R), defined by g" (P' g, f ) = (P , 7g(f)) for (p. g , f )eF(R) . Obviously this action is compatible with the action of G(R) on ~-R, and it is R-linear in the sense that Fp.g(R) is an R-module which is mapped R-linearly by g~G(R) to Fp(R). This leads to the following definition of an X.G-module.

First suppose F is a faisceau over a faisceau X. We say F is a k-vector bundle over X if for each commutative k-algebra R and peX(R) the fiber Fo(R ) of F(R) over p is equipped with an R-module structure, functorial in R in the usual sense.

Definition. Let G be a k-group functor acting on a faisceau X. An X.G-module is a k-vector bundle F over X equipped with a compatible G-action such that gsG(R) maps Fo.g(R ) R-linearly to Fp(R) for p~X(R), R a commutative k- algebra.

460 E. Cline et al.

When X is a scheme over k, and X .G-module F in the above sense arises f rom a quasi -coherent X .G-module ~ in the sense of (3.2) precisely when Fo(R ) =R| ) whenever S=(gx(U ) for U open affine in X and p: S p e c R ~ X factors through the inclusion 1: U ~ X : As expected one just puts ~,~(U)=F,(S). Observe that the fiber at (1, p) of the mapV(~-R)--- ,X R is represented by SpecS(Fp(R)); thus, the act ion of G on F leads to an act ion of G(R) on

I_[ V ( ~ ) p - - V ( ~ ' R ) and consequent ly to an act ion of G(R) on ~R, functorial p~X(R)

in R. Hence the definition of (3.2) is satisfied. With the above definition of X.G-module , we can formulate a very general

version of the imprimit ivi ty theorem. Suppose H is a subgroup functor of a k- g roup functor G, and that H acts on a faisceau X. Let F be an X.H-module . Then F v ~ G is easily seen to be an (XvHG).G-module, and we define F] ~ = F v nG. Arguing as in (1.2) we obta in the

Theorem. The map F ~ F I ~ is an equivalence from the category of X.H-modules to the category of (X vHG).G-modules. The inverse is given by E ~ E x xv~GX , for E an (X v H G).G-module.

Suppose now that we have inclusions H ~ _ K _ G of k-group functors and that H acts on a faisceau X.

(3.5.1). Suppose F is an X.H-module . Then

FI~---F[~[ G.

The p roof is immedia te f rom the theorem; cf. also [34, IV, w 5.1].

w 4. Toward a Mackey Decomposition Theorem

Throughou t this section, G is an affine algebraic group scheme over k. I t will be assumed that G is reduced after (4.3).

P robab ly the simplest case of a Mackey decompos i t ion theorem is the following result. A more sophist icated analogue is given in (4.4).

(4.1) Theorem. Let H be a closed subgroup scheme of G, and L an affine algebraic group scheme over k. Suppose f: L ~ G is a morphism of group schemes such that the morphism H x L ~ G induced by the composition of 1 x f with multiplication is an ePimorphism of faisceau. (In particular, this holds if G is reduced and H(k)f(L(k))= G(k).) Then for every rational H-module V we have an isomorphism

Hn(G/H, s x aL), ~Vln ~oL), n_>0,

of rational L-modules, letting IL denote restriction through f 3

3 If L is a subgroup scheme of G and f is the inclusion morphism, then H x a L is the so-called scheme-theoretic intersection of H and L, and satisfies (H x GL)(R)=H(R)c~L(R) for every commutative k-algebra R. We will sometimes denote H x oL by Hc~L in this case


In particular for n= O, we have an isomorphism

Vf G [L ~- V l . • G L I L

of rational L-modules.

Proof First, if G is reduced and H(k)f(L(k)) = G(k), then it follows from [12; II, w 3.1J that the morphism H x L ~ G is faithfully flat, hence an epimorphism of faisceaux.

Next, suppose that H x L ~ G is an epimorphism of faisceaux. Then the natural morphism J~ L ~ G / H is an epimorphism, since f xn /aG: L xGmG=L xH--+G is one, and G~G/H is faithfully flat. Also, L / H x ~ L ~ G / H is a

monomorphism. (This is clear for quotient faisceaux by the argument of [12; III, w 2.21.) Since all epimorphisms of faisceaux are strict [12; III, w 2.23 we have that L/HxGL~-G/H. Thus, the induced sheaf s on G/H is a quasi- coherent (L/HxoL).HxoL-module. By Theorem (2.7) we have an isomorphism 5fv~-Sfvl~oL of X.L-modules, X = G / H ~ L / H x G L , and the result follows. Q.E.D.

(4.2) Examples. (a) Let G be a reductive algebraic group over k. Let B be a Borel subgroup and P>B a parabolic subgroup. If L is a Levi factor of P, then P = L . B and (4.1) is applicable: V]V]L'~VIB,~L] L for any rational B-module V [9].

(b) Consider the standard embedding of L=Sp2n into G=SLz,. Let P be the parabolic subgroup of G which is the stabilizer of the line ke~ for e~ =(1, 0 . . . . ,0) t, in the standard G-module V of 2n x 1-column vectors. Since L acts transitively on V-{0}, we have that G(k)=P(k)L(k), so (4.1) applies.

(c) Let G be an affine algebraic group scheme defined over ko=GF(p ). Let ~: G ~ G be the Frobenius endomorphism, and for each positive integer r, let Gr be the kernel in the sense of group schemes of c:. Then G~ is an infinitesimal subgroup of G in the sense of (3.4), called the r ~h infinitesimal subgroup of G, cf. [7]. For a closed subgroup scheme H of G defined over k0, we define a new closed subgroup scheme HG~ by means of the pull-back diagram

HQ > H

G ~G. ~r

We shall call HG~ the r th infinitesimal thickening of H. We now show that the product morphism

p: H x G ---, HG r

is an epimorphism of faisceaux, as the notation suggests. In fact, let g~(GHr)(R ) for some commutative k-algebra R. Then ~rr(g)~H(R), so using [12, III, w 1, 2.8] there is a covering R-algebra S such that there exists h~H(S) with o-r(h)=~:(gs). Thus, (h,h-ags)~(HxGr)(S) maps onto gse(HQ)(S). Again by

462 E. Cline et al.

[12; III, w 2.8], p is an epimorphism. Hence, (4.1) applies. This example will play an important role in our treatment of the Andersen-Haboush proof of Kempfs vanishing theorem which we give in the appendix w 7. We will also use the following corollary of (4.1) which just asserts that pulling back a module via a surjective group scheme homomorphism commutes with induction (and all its derived functors).

Let 1 > N - - - - - ~ G ~ > H -+1

1 ~ N - - , K ---~L ~ 1 d

be a commutative diagram of affine group schemes over k in which the rows are exact and the columns are inclusions. For a morphism f : X --> Y of affine group schemes over k, let f * (resp. f . ) denote restriction (resp. induction) or rational modules through f [7; w 13.

(4.3) Corollary. Under the above hypothesis, there is a natural isomorphism of functors

(R i a,) o d* ~ c* o (R ~ b,), i >__ O.

Proof. Since K = G x H L this is clear from (4.1) together with (2.9). Q.E.D.

For the rest of this section G will be a reduced algebraic group over k. Let H, L be two closed subgroup schemes of G. Suppose that L has an open orbit f2 ~ G/H. Choose a point fEf2(k) and let xEG(k) be such that n (x )=s Let L~ be the stabilizer of x in L; thus, L ~ = H ~ x GL, where H~=x * Hx. For a rational H-module V, we denote by V ~ the rational H~-module with V~(R), R a commutative k-algebra, defined as all formal symbols v ~ for veV(R)= V| with R-module structure isomorphic to that of V(R), and with action gv ~ =((x n gx~ 1) v)~ for g~HX(R) and veV(R). Let X = G/H.

(4.4) Theorem. Let d = c o d i m ( X - f 2 ) . Then for 0 _ < i < d - 1 we have that

In particular, for d >_ 2,

Hi(G/H, ~ i s~)IL = H (L/Lx, S~=).

VIGIr~_ = c v IL=I.

Proof One argues as in the proof of (4.1) that the map L / L x ~ P is an isomorphism. Observe that k | 1 6 2 x as an L,-module. (Recall G(k) acts on S v in the sense of (3.2). We have x - l ~ V , e = S V , e x= S v , s . Thus, for geLx(R ) and w~Sfv,~| we have g(x -1 w)=x- l ( (xRgx;~l )w) and moreover ~CPv, z| fV, e@R , and the isomorphism follows.) Thus by Theorem (2.7), we have that S v l e ' ~ S v = as an f2.Lx-module.

Write Z = X - f 2 . For x~Z, we have by the smoothness of X that

depth (gx, = = Krull dim. Ox, z -> d.


Now assume that V is a finite dimensional rational H-module. Then &% is a locally free (gx-module by (2.10), so

depth 2'v, z > d

for all zeZ. Since 5% is coherent, we have that the local cohomology sheaves gf~(&ev) vanish for i<d [16; Theorem (3.8)]. The spectral sequence of local cohomology [16; Proposition (1.4)]

EP'q-HP(X2 - , , ~(Yv))q ~ HqP+q(X,~v)

gives, therefore, that Hiz(X, Yv)=0 for i<d. The long exact sequence of local cohomology [16; Corollary (1.9)]

(4.4.1) O-+ H~ L~'v)---+ H~ 5Yv)-+ H~ :~vIn)-+ H~(X, ~v)-+...

now implies that

HZ(G/H, ~C'q~ Z HI(L/Lx, ~ w )

for i < d - 1 . The result for V infinite dimensional now follows by taking direct limits (using (2.9) and [7; p. 118, fn. (3)] for example. Q.E.D.

(4.5) Examples. Let G be a connected semisimple algebraic group over k. Fix a Borel subgroup B and corresponding set /7 of simple roots. For proper subsets J, K of H, let P=Pj, Q =Pr be the corresponding parabolic subgroups containing B. Obviously Q has an open orbit f2 in G/P, namely the orbit of woP , where w o is the long word in the Weyl group W. We ask for conditions which guarantee that codim(G/P-O)>2, so that the results of (4.4) apply. Considering the dimensions of B, B double cosets, we see that this occurs if and only if QGwoP~_QwoP for each simple root c~. This happens if and only if s~EWKwoVfjWo=WKVfj,, where J* is the image of J under the opposition involution. Thus, codim(G/P-f2)>_2 if and only if KuJ*=FI. Therefore, if KuJ* ~H,

Vl G G, ~ VW~ Wol rK

for every rational Pj-module V. When the codimension condition fails, it is easy to give examples where the

above induction property fails. For instance, take V=k, P=Q--B, and G of type A 1. Here VW~ ~ is infinite dimensional, while V[a is the trivial module.

(4.6) The A, example does, however, illustrate the fact that for any group scheme G, subgroup scheme H, and open subscheme U of G/H we always have an injection V]G-+:LPv(U). This is immediate from (2.10) or (4.4.1). Together with (2.8) this give the following very weak general "decomposition" theorem.

(4.6.1) Let G be an affine algebraic group scheme over k and let H, L be subgroup schemes. Let U be an open L-stable subscheme of G/H, and Y'_~ G(k) a set such that the images of e=H/H under the element of 5~ form a set of

464 E. Cline et al.

representatives for the orbits of L(k) in U(k). Then we have an injection

vL~k -+ ]7 V*L~x~LL L x ~

of L-modules. The intersection H*c~L is, of course, taken in sense of schemes.

w Quasi-Affine Quotients

We consider the situation in which G/H is a quasi-affine scheme. For simplicity we assume, in addition, that G is a reduced affine algebraic group and H is a closed, reduced subgroup. Let A=F(X,(gx) and assume, also for simplicity, that A is a finitely generated k-algebra. This occurs, for example, when G/H is affine or when H is a maximal unipotent subgroup of G [ 1 5 ] / I t is easily checked that j: X ~ J ~ = S p e c A is a G-equivariant open immersion. Let M be an A.G-module in the sense of (3.1.8). Then one can ask for conditions under which M is induced, in the sense of [5], from a rational H-module. We assert first that this occurs if and only if M=F(X , j* f/I). Indeed, 2~/is an Jf.G- module, so j*2~ is a quasi-coherent X.G-module by (3.1.2), and so is an induced sheaf by Theorem (2.7). Thus, if M=F(X,j*f / I) , then M is induced as a module. Conversely, if M is induced from a rational H-module V, then M =F(X, ~v). Since J , ~ v is quasi-coherent, we must have the j , Y v = f / l , and now we get F (X, j* M)= F (X, j* j , ~v ) = F (X, (j, Sv) lx)] = M.

Now we let Z = X - X and let I be the ideal in A defining the closed (reduced) subscheme Z. Then we have an exact sequence of local cohomology groups [16]

0-+ H~ ~/)-+ H~ A))~/4~ ~) + H~(~, ~ / )+ 0

since Hi(Jr, ~ / )=0 by Serre's vanishing theorem [19; III, Theorem (3.7)]. Thus, M is induced if and only if H~(J(, f / ) = 0 for i=1, 0. If M is finitely generated over A, this is the case if and only if the /-depth of M is at least 2 [19; III, Ex. 3.3, 3.4]. Hence we have proved

(5.1) Theorem. Assume that G/H is quasi-affine and that A=k[G] u is a finitely generated k-algebra. Then an A.G-module M, finitely generated as an A-module, is induced from a rational H-module if and only if

depth, (M) > 2,

where I is the ideal defining Spec A - X.

Note in particular that, if G/H is affine, then I =A and so depth~M = o% and we recover the main result of [27].

We now mention the following perhaps more useful criterion in

4 If k has characteristic zero, A is finitely generated in case U is the unipotent radical of a parabolic subgroup [331. It appears to be unknown if this remains true in positive chracteristic for general parabolic subgroups


(5.2) Theorem. Assume that G/H is quasi-affine and that A = k [ G ] H is a finitely generated k-algebra. Suppose M is an A.G-module which is f lat as an A- module. Then M is induced from a rational H-module.

Proof. For f~A, let D ( f ) = S p e c A ! , a basic open subscheme of J f=SpecA. We can choose f l , . . . , f , ~ A so that X = D ( f l ) w . . . u D ( f ~ ) . Since A=F(X,(gx) , we have an exact diagram of A-modules given by

A ~ T T A ~ - ~ ITA L.~ f i ~ I I f l f j

i (i, j )

where u, v, w are as defined in (1.1). If M is a flat A-module, we can tensor this diagram with M to get an exact diagram:

M -* L[i M f , -~ [ I M f f . ( i , j ) ~ j

It is immediate from this that M = F ( X , j * M ) . Therefore, as in the proof of (5.1), we see that M is induced from a rational H-module. Q.E.D.

(5.3) Examples. 5 Let G=SL, , n > l , and let V be the standard n-dimensional module consisting of n x 1-column vectors. If H is the stabilizer in G of the unit vector e1=(1,0 , . . . ,0) t, one has that G/H~-A' -{O} and that A = k [ G T ~ - k [ X 1 . . . . ,X,] . Here G acts on the polynomial ring A by linear substitution. In the notation of Theorem (5.10), I=(X1 , . . . ,X , ) . To give an example of an A.G-module M which is not induced we take M = I . Then one easily finds that d e p t h r M = l , so, by Theorem (5.1), M is not induced from a rational H-module.

Now suppose N is an A. G-module which is finitely generated and flat as an A-module. Since A is Noetherian, N is thus projective, and hence free, by Quillen [28]. The quotient homomorphism A ~ A / t = k is split as a homomorphism of rational G-modules by the map 0: k--~A defining the k-algebra structure on A. Thus, M ~ M / I M is also split as a homomorphism of rational G- modules by ON=N| A O. Setting W = ON(N/IN ) it is c learf rom the freeness of N that the map

A| W ~ N

given by

a|

is an isomorphism of A.G-modules. Here G acts diagonally on A| W. Thus, we have shown that every A.G-module N which is finitely generated flat over A has the form

N ~ W f

for some rational G-module W. Not all finitely generated A. G-modules are flat, however. In fact, whenever

we have an isomorphism N 1 -~N~ of induced A.G-modules, say N~ = W ~ f and

s Similar remarks apply to G=SP2 .

466 E. Cline et al.

N 2 = W2[ a, then W 1 = W 2 as H-modules, because of (2.7) and the isomorphism J 'N1 =J*/q2. (This holds in general in the quasi-affine case). So if W 2 is not a G-module, then the A. G-module N 2 cannot be fiat in the situation of the above paragraph.

(5.4) Suppose as above that G/H is quasi-affine and A =k [G/H] is a finitely generated k-algebra. Again let I be the ideal in Spec A defining S p e c A - G/H. Then the invariance of I under the action of G leads to an interesting rational G-module ~K(W) defined for each rational H-module W, namely, set :##(W) =(W)=(WIG)/I(Wf). In the spirit of [30], we call ~K(W) the WaIlach module of W. Note in the situation of (5.3) that if W is finite dimensional, so is its Wallach module.

Finally we mention the following extension of [27; (4.7)]. Its proof is a straightforward generalization of the one given there, using the results of this paper.

(5.5) Theorem. Let U be a connected (reduced) unipotent group and V a rational U-module. Then a necessary and sufficient condition that V be induced from a proper subgroup scheme H is that Homk(V, V) contain a finite dimensional subspace F, stable under the conjugation action of H, which properly contains the scalar transformations k. 1 and which satisfies k. 1 =F v.

For many further applications of the imprimitivity theorem in the case when G/H is affine see [27].

w 6. Appendix I: Thickening the Borel Subgroup

Suppose that G is a connected semisimple simply connected algebraic group defined over k o =GF(p). Using (4.2c) we will illustrate some of the consequence of (4.1). Let B be a Borel subgroup defined over k o and let T be a maximal k 0- torus of B. Let peX*(B)=X*(T) be the dominant weight which is the sum of the fundamental dominant weights.

For any rational B module V we have by (4.2c) that

(6.1) BGr ~ Gr ViB IG~=VIBr[

In particular, taking V to be the one-dimensional B-module defined by the weight - (ff - 1) p, we have

(6.2) _ ( i f _ ,~r ~ r 1)Pin [G.=St(p )IG.

where St(if) denotes the irreducible (Steinberg) module for G of high weight (p~ - 1) p. To see this, note that there is a Gr-homomorphism

by the universal mapping property of induction [7]. This is an injection since St(p r) is an irreducible G;module [8]. It is not difficult to directly argue that

A Mackey lmprimitivity Theory for Algebraic Groups 467

St(p ~) is a free module of rank one for the hyperalgebra of U,-, where U- is the unipotent radicat of the opposite Borel subgroup B-. (The fact that the hyperalgebra by(U,-) of U,- has an integral [24] easily gives that the vectors

a~] "'" aN!- v§

O<a~<p", form a basis for St(p~). Here v + is the high weight vector of St(p ~)

and the X ~ X ~N -"~' form the standard basis of hy(U~-) as in [-7; (5.1) and al ! "'" aN !

(6.6)].) Hence, dim St(pO=p ~, where N = d i m U. But

- ( p ~ - l ) ~ pl,rl~- ~ - ( p ~ - 1) plB,.~ v; I U;

~k[y ~ ~ k [U,-3

by (4.1) again since the product morphism B~x U~----~Gr is an epimorphism of faisceaux. In fact, we have a pull-back diagram

B xU,- P -~O,

l 1; B x U - -~G

while the inclusion Grog factors through B x U-, thereby proving p is an epimorphism (actually an isomorphism). Since dim k[Ur- ] =prN it follows 1 is an isomorphism as desired.

If 2 is any dominant weight satisfying (2, e")<p~ for each simple root e, then it is easy to see that the natural map

of BG,-modules is an injection. Indeed, if M is the irreducible G-module of low weight -2 , then universal mapping gives a commutative diagram of BG,- modules

- - ~.[GI~G ~ - - ~ l BGr

\ / M

where the arrows from M are nonzero, hence injections, since M is an irreducible Gr-module [-8]. It follows that h is injective on the unique B-stable line in --2IG[BG,.; hence h is injective.

If we apply this argument with 2=(p ~- 1)p, it follows immediately that St(pg~ _(pr_ 1)pl a. This fact is, of course, well-known.

468 E. Cline et al.

w 7. Appendix II: The Andersen-Haboush Identity

In this appendix we use the results of w 6 and the machinery of induction to simplify the Andersen-Haboush proof [1, 18] of Kempfs vanishing theorem [22]. Actually, we only prove their key tensor identity from which the vanishing theorem is easily deduced. This proof is a reworking of one given in [4] by the first author.

Let G be a connected semisimple simply connected algebraic group defined over/Co, etc. as in w 6. Then we have the following diagram

1 ~G~ ~ G - ~" >G ~1

1 >G~ - - - > B G ~ B >1

as in (4.3). If E is a rational B-module, the conclusion of (4.3) is that

(7.1) (R / b , (E)) (m ~- R i a , (E (pr))

for each i >__ 0, where E (pr) denotes the r th Frobenius twist of E [7; w 3]. Let f : B ~ B G r be the inclusion morphism. Then since BGr/B is affine (in

fact isomorphic to U,-), the spectral sequence of induction [7; (4.7)] collapses to give

R i b , ( E ( p ~ ) | a E(m , ,(( | -1) p)WD.

We now apply the tensor identity [7; (2.6)], first for induction to BG,, then, using the results of w 6, for induction from BG, to G to obtain

R i b, (E(Pr)| - (pr _ 1) p) _~ e / a , (E (m | S t(pr))

"~ R / a , (E(pr))| t(p~).

In view of (7.1) we obtain an isomorphism

R ~ b,(E)(P~)|174 - (p~ - 1) p)

of rational G-modules. Translating into sheaf cohomology by (2.9) we obtain the Andersen-Haboush identity

H (G/B, SE(pr~ | _ (p~_ 1)p) = H (G/B, SE) (p~) @ S t(p ~)

for i_>0. The Kempf vanishing theorem [22] states that any line bundle Sa,

),~X*(B), having a nonzero global section has vanishing cohomology groups Hi(G/B, ~ ) , i>0. This follows easily from the above identity using the ample- hess of ~'~ p_~ I l l , [18].


w 8. Appendix 1II: Groups Over a Scheme

The results of w167 1-3 can be easily reformulated in the more general context where the ground ring (thus far an algebraically closed field k) is replaced by a scheme S. Using this set-up we can obtain a very general, though rather formal, kind of Mackey decomposition theorem.

Let S be a scheme over k. An S-functor X is a functor from the category of S-algebras (that is, k-algebras B equipped with a map Spec B ~ S of k-schemes) to the category Ens of sets. If X satisfies in addition the condition analogous to (1.1) for S-functors, then X is called an S-faisceau. Every S-functor X defines by restriction a k-functor kX equipped with a canonical map k X ~ S of k- functors (and conversely) 1-12; I, w 1.6.4]. Moreover, X is an S-faisceau if and only if k X is a faiceau in the sense of (1.1).

Thus, the theory of S-faisceaux reduces formally to that of faisceaux over S. There is, though, one difference. The notion of an S-group is defined as a group-object in the category of S-functors. However, if G is an S-group, then k G is not in general a k-group. Nevertheless, the proof of (1.2), for example, carries over to this more general setting. We leave the details to the interested reader.

Let G be an affine k-group with coordinate ring A. If X is a faisceau, let X G be the G-faisceau (taking S=G in the discussion above) defined by the projection X x G ~ G . Note that if X is a k-group, then X G is a G-group. Let j~G~(A) be the element defined by the product map A| (a| Then j defines by conjugation an automorphism of G a. If T is a G-subgroup of Ga, let T j denote the image of T under this automorphism.

The following result may be viewed as a generalization of 1-36; w 8.2, p. 307].

Theorem. Let G be an affine k-group as above and let H, L be k-subgroups. Let V be an H-module over k (that is, V is a faisceau such that, for each commutative k-algebra R, there is an RH(R)-module structure, functorial in R, on V(R)). Then using the notation of (3.5) we have

Via xn..~/L G..~ j ]La

where I-I\G/L is the quotient-faisceau for the obvious action of I t x L on G, and Vd is the H~-module obtained by letting H~ act on V a through the automorphism j-~

We will only sketch the proof. First observe that there is a pull-back diagram

(H~c~La)\L a , (H\G)~

l G (H\G/L)G

470 E. Cline et al.

as is easi ly checked f r o m the se t - theore t i c case (in w h i c h case j c an be r e g a r d e d

as the d i a g o n a l e l e m e n t in GG=G• ). L e t W=VIG• • G a n d n o t e tha t the re is a n a t u r a l m a p W ~ G • ~n-,G/L)~ (H\G)G = ( H ~ ~LG)\L ~. N o w app ly the a n a l o g u e o f (1.2).

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Received May 6, 1982

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