A proof of Casselman's comparison theorem for standard minimal parabolic subalgebra
aa r X i v : . [ m a t h . R T ] F e b A PROOF OF CASSELMAN’S COMPARISONTHEOREM FOR STANDARD MINIMAL PARABOLICSUBALGEBRA
NING LI, GANG LIU, AND JUN YU
Abstract.
Let G be a real linear algebraic group and K be amaximal compact subgroup. Let p be a standard minimal para-bolic subalgebra of the complexified Lie algebra g of G , and n be itsnil-radical. In this paper we show that: for any admissible finitelygenerated moderate growth smooth Fr´echet representation V of G , the map V K ⊂ V induces isomorphisms H i ( n , V K ) ∼ = H i ( n , V )( ∀ i ≥ k ≥ n k V is a closed subspace of V and the inclusion V K → V induces anisomorphism V K / n k V K = V / n k V . This strengthens Casselman’sautomatic continuity theorem ([6],[18]). Contents
1. Introduction 12. Bruhat filtration and degree filtration 23. Casselman-Jacquet modules 84. A cohomological comparison result between tempereddistributions and polynomial multiple volume forms 105. Casselman’s comparison theorem 13References 151.
Introduction
Let G be a real linear algebraic group and K a maximal compact sub-group. Let W be an admissible finitely generated ( g , K )-module and W ∗ be its dual ( g , K )-module. Schmid constructed the minimal glob-alization W min which consists of analytic vectors, and the topologicaldual of ( W ∗ ) min is called the maximal globalization of W ([14]). Cassel-man and Wallach constructed the smooth globalization W ∞ which is amoderate growth smooth Fr´echet representaiton consisting of smooth Mathematics Subject Classification.
Key words and phrases. ( g , K )-module, smooth representation, Casselman-Jacquet modules, n -homology, n -cohomology, tempered E -distribution, Bruhat fil-tration, transversal degree filtration. vectors, and one has W −∞ as the topological dual of ( W ∗ ) ∞ . They fitin a sequence W ⊂ W min ⊂ W ∞ ⊂ W −∞ ⊂ W max . The globalizations have important applications. For example, the smoothglobalization W ∞ is very useful in the theory of automorphic forms,and the maximal globalization W max is used to realize standard de-rived functor modules ([15]).Let p be a standard minimal parabolic subalgebra of g and n beits nil-radical. In this paper we show that: for any admissible finitelygenerated moderate growth smooth Fr´echet representation V of G , themap V K ⊂ V induces isomorphisms H i ( n , V K ) ∼ = H i ( n , V ) ( ∀ i ≥ k ≥ n k V is a closed subspace of V and the inclusion V K → V induces an isomorphism V K / n k V K = V / n k V .This strengthens Casselman’s automatic continuity theorem ([6],[18]).For each of the four globalizations above, there is a comparison con-jecture for n -homology where n is the nilradical of certain parabolicsubalgebras p of g (cf. [17, Conj. 10.3]). Casselman claimed a proofof the comparison theorem when p is any standard parabolic subal-gebra, but the proof remains unpublished. For minimal globalization,the comparison conjecture is shown in [9], [5] and [4]. In [10] theabove comparison theorem for smooth globalization and for p a stan-dard minimal parabolic subalgebra is deduced from the correspondingcomparison theorem for minimal globalization. Schmid claimed thatthe powerful techniques in [11] imply comparison theorem for all fourglobalizations and for a wide range of parabolic subalgebras. Despiteall of these developments, our proof of the comparison theorem is moredirect and more elementary and should be close to Casselman’s originalidea. The closeness of n k V and the isomorphism V K / n k V K = V / n k V ( ∀ k ≥
1) shown in this paper are new.2.
Bruhat filtration and degree filtration
In this section we review some contents of [7]: tempered E -distributionssupported in a subanalytic submanifold and degree filtration; Bruhatfiltration associated to the space of smooth sections of a finite rankequivariant analytic vector bundle on a flag manifold G/P . These cor-respond to § § Tempered E -distributions supported on a subanalytic sub-manifold and degree filtration. Let X be a compact analytic man-ifold and E be an analytic vector bundle on X with finite rank. Let ASSELMAN’S COMPARISON THEOREM 3 C ∞ ( X, E ) denote the space of smooth sections of E . It is a nuclearFr´echet space (NF space for short) with seminorms p D,f : s max x ∈ X | f ( x )(( Ds )( x )) | , where f ∈ C ∞ ( X, E ∗ ) with E ∗ the dual vector bundle of E and D ∈ Diff(
X, E ) is an E -coefficient smooth differential operator on X . Let C ∞ ( X, E ) ′ be the strong dual of C ∞ ( X, E ) ([13]), which is a dualnuclear Fr´echet space (DNF space for short). For an open subset U of X , let C ∞ ( U, E ) be the space of compactly supported E -sections on U , which is again a nuclear Fr´echet space. The strong dual C ∞ ( U, E ) ′ of C ∞ ( U, E ) is called the space of E -distributions on U . There is anatural restriction map res U : C ∞ ( X, E ) ′ → C ∞ ( U, E ) ′ .In the below, when E is the trivial line bundle, we omit the symbol E from notations like C ∞ ( X, E ), C ∞ ( X, E ) ′ , etc. So, for example, C ∞ ( U ) ′ means the space of distributions on an open subset U . Definition 2.1.
The image of res U is called the space of tempered E -distributions on U with respect to X . There is a short exact sequence0 → ker res U → C ∞ ( X, E ) ′ → im res U → . It is clear that ker res U consists of E -distributions on X supported in Z := X − U . By dualizing, there is a dual exact sequence0 → (im res U ) ′ → C ∞ ( X, E ) → (ker res U ) ′ → . Put S ( U, E ) = (im res U ) ′ . Then, S ( U, E ) ′ = im res U by duality. Thefollowing lemma gives a local characterization of the space (im res U ) ′ . Lemma 2.2. ( [7, Lemma 2.2] ) The subspace (im res U ) ′ of C ∞ ( X, E ) consists of all global sections s ∈ C ∞ ( X, E ) vanishing with all of theirderivatives on Z .Proof. Any distribution supported in a point x ∈ Z is a finite linearcombination of derivatives of delta like distributions s α ( s ( x )) ( α ∈ E ∗ x ). Evaluating these distributions at s ∈ (im res U ) ′ , it follows that s vanishes with all derivatives at x . Then, all s ∈ (im res U ) ′ vanishwith all derivatives on Z . The converse follows from [7, Lemma 2.1],which asserts that any order ≤ m E -distribution D ′ on X vanisheson s ∈ C ∞ ( X, E ) such that Ds | supp( T ) = 0 for all order ≤ m smoothdifferential operators D on E . (cid:3) Let Y be a submanifold of X which is also a subanalytic set ([3]) in X (note that being a submanifold implies that Y is an open subset of Y ).An open subset U containing Y is called a subanalytic neighborhood of Y if U is subanalytic in X and Y is a closed subset of U , i.e, Y ∩ U = Y . Definition 2.3.
Let T ( Y, E ) denote the space of tempered E -distributionson U with support in Y for a subanalytic open neighborhood U of Y . NING LI, GANG LIU, AND JUN YU
By the following Lemma 2.4, the space T ( Y, E ) is independent of thechoice of the subanalytic open neighborhood U . Hence, the notation T ( Y, E ) is well suited.
Lemma 2.4. ( [7, Lemma 2.6] ) If U ′ ⊂ U are two subanalytic openneighborhoods of Y , then the restriction map res U,U ′ induces an iso-morphism of the space of tempered E -distributions on U with supportin Y onto the space of tempered E -distributions on U ′ with support in Y .Proof. Injectivity of the restriction map res
U,U ′ is due to supporting setcondition. The inverse of the restriction map res U,U ′ is given by “theextension by zero” map. (cid:3) For a closed subset K of X , let C ∞ ( X, E ) ′ K denote the space of E -distributions on X with support contained in K . Since Y is a closedsubset of U , then Y − Y ⊂ Z . Thus, Y ∪ Z is a closed subset of X .The following statement is clear from the definition. Lemma 2.5.
The space T ( Y, E ) is equal to C ∞ ( E ) ′ Z ∪ Y /C ∞ ( E ) ′ Z . For an integer p , let M p denote the closed space of S ( U, E ) consistingof sections s which vanish with all derivatives of order ≤ p along Y .Put F p T ( Y, E ) = M ⊥ p . Then, F T ( Y, E ) = { F p T ( Y, E ) } ∞ p =0 is an exhaustive increasing filtration of T ( Y, E ). Definition 2.6.
We call F p T ( Y, E ) the space of tempered E -distributionswith support in Y and of transversal degree ≤ p . For each p ∈ Z , let M p denote the subspace of M p consisting ofsections with compact support contained in U . The following lemmagives a useful characterization of F p T ( Y, E ) in practice. The proof ofLemma 2.7 is a bit complicated. We refer interested readers to consultthe original proof in [7].
Lemma 2.7. ( [7, Lemma 2.9] ) For any integer p , one has M ⊥ p = M ⊥ p . The following is [7, Ex. 2.3] plus a bit generalization of [7, Ex. 2.11].
Example 2.8.
Let X = S n be an n -dimensional sphere, p ∈ S n and π : S n − { p } → R n be the stereographical projection of S n − { p } onto R n with respect to the pole p . The closed subspace of C ∞ ( S n ) consistingof smooth functions vanishing with all derivatives at p can be identifiedwith the “classical” Schwartz space S ( R n ) via the map p . Dually, thisidentifies the space of tempered distributions on S n −{ p } with respect to S n with the space S ( R n ) ′ of “classical” tempered distributions on R n .Let S k be a k -dimensional sphere in S n passing through p such thatthe stereographical projection image of Y = S k − { p } is equal to the ASSELMAN’S COMPARISON THEOREM 5 linear subspace R k of R n defined by x k +1 = · · · = x n = 0 . UsingLemma 2.7 one can show that F p T ( Y ) = { T ∈ S ( R n ) ′ : ( x a k +1 k +1 · · · x a n n ) T = 0 , a j ≥ , X k +1 ≤ j ≤ n a j = p + 1 } . In particular, F T ( Y ) is identified with the image of the inclusion of S ( R k ) ′ into S ( R n ) ′ . Moreover, it is well known that ( [16, Ch. III, § ) T ( Y ) = M a j ≥ ∂ a k +1 k +1 · · · ∂ a n n F T ( Y ) and F p T ( Y ) = M a j ≥ , P a j ≤ p ∂ a k +1 k +1 · · · ∂ a n n F T ( Y ) . Bruhat filtration.
Let G be a reductive Lie group with Lie al-gebra g and K be a maximal compact subgroup with Lie algebra k .Let g = k + q be the corresponding Cartan decomposition. Choose amaximal abelian subspace a of q . Let g , k , q , a be the complexifica-tions of g , k , q , a respectively. Write Φ = Φ( g , a ) for the restrictedroot system from the adjoint action of a on g . Choose a positive sys-tem Φ + = Φ + ( g , a ) of Φ. Let n be the subalgebra of g spanned byroot spaces of positive roots in Φ + , and n = n ∩ g . There is anIwasawa decomposition G = KAN where A and N are the closed Liesubgroups corresponding to Lie subalgebras a , n of g respectively.Put M = Z K ( a ) and P = M AN . Then, P is a minimal parabolicsubgroup of G .Let X = G/P be the flag variety associated to the parabolic sub-group P . Let m = dim G/P . Write W = N G ( A ) /Z G ( A ). It is wellknown that the conjugaction action of W on a identifies W with theWeyl group of the restricted root system Φ. For each w ∈ W , choose˙ w ∈ N G ( A ) representing w and write C ( w ) = N ˙ wP/P ⊂ G/P . The set C ( w ) is called Bruhat cell associated to w . The Bruhat decompositionof G relative to P asserts that G/P = G w ∈ W C ( w ) . Let w be the longest element in W , i.e., the unique element in W which maps Φ + to − Φ + . For each w ∈ W , put N w = wN w − . Then, N = ( N ∩ N w ) × ( N ∩ N ww ) as an affine algebraic variety. Thus, C ( w ) ∼ = N/ ( N ∩ N w ) ∼ = N ∩ N ww by sending n to n ˙ wP/P . Moreover, we know that the complexified Liealgebra ¯ n = n w of ¯ N := N w is spanned by root spaces of negativeroots in Φ − = − Φ + . The parabolic subgroup ¯ P := M A ¯ N is called theopposite parabolic subgroup of P . Let ( σ, V σ ) be a finite-dimensional NING LI, GANG LIU, AND JUN YU complex linear representation of P with trivial N action, i.e., it factorsthrough a representation of M A via the projection P → P/N = M A . Definition 2.9.
Write E ( σ ) := G × P V σ for a smooth equivariantvector bundle with fibre space V σ at P/P ∈ G/P .Write I ( σ ) = Ind GP ( σ ) for un-normalized smooth parabolic inductionfrom the representation σ of P . Then, C ∞ ( E ( σ )) = I ( σ ) and C ∞ ( E ( σ )) ′ = I ( σ ) ′ . Definition 2.10.
The Bruhat filtration of I ( σ ) is a finite decreasingfiltration of U ( g ) -submodules: I ( σ ) = C ∞ ( σ ) ⊃ C ∞ ( σ ) ⊃ · · · ⊃ C ∞ m ( σ ) ⊃ C ∞ m +1 ( σ ) = { } , where C ∞ p ( σ ) is the subspace of all smooth sections of E ( σ ) which van-ish with all derivatives along Z p := ⊔ dim C ( w )
Let T ( C ( w ) , E ( σ )) be the space of tempered E ( σ ) -distributions on V w with support in C ( w ) , and J ( w, σ ) be the strongdual of T ( C ( w ) , E ( σ )) . By Lemma 2.4, T ( C ( w ) , E ( σ )) depends only on the cell C ( w ). Lemma 2.12. ( [7, Lemma 3.3] ) The space of tempered E ( σ ) -distributionssupported in Z p +1 − Z p is equal to ⊕ dim C ( w )= p T ( C ( w ) , E ( σ )) .Proof. Put Y = Z p +1 − Z p and U = X − Z p . Let O , . . . , O q be all p -dimensional N -orbits in X . For each i (1 ≤ i ≤ q ), put U i = X − ( Z p ∪ [ j = i O j ) . Then, each U i is an open subanalytic neighborhood of O i , U i ∩ O j = ∅ whenever i = j , and X − Z p = S ≤ i ≤ q U i . Let φ i : C ∞ ( U, E ) ′ Y → C ∞ ( U i , E ) ′ O i be the natural restriction map. Let φ := P ≤ i ≤ q φ i : C ∞ ( U, E ) ′ Y → L ≤ i ≤ q C ∞ ( U i , E ) ′ O i . Then, φ is clearly an injectivemap. Choose a smooth partition { ϕ i : 1 ≤ i ≤ q } of { U i } in U . Foreach tuple T i ∈ C ∞ ( U i , E ) ′ O i (1 ≤ i ≤ q ), define T ( s ) = X ≤ i ≤ q T i ( ϕ i s | U i ) , ∀ s ∈ C ∞ ( U, E ) . Then, T ∈ C ∞ ( U, E ) ′ Y and φ ( T ) = ( T , . . . , T q ). Thus, φ is sur-jective. Identifying T ( Y, E ( σ )) (resp. T ( O i , E ( σ ))) with C ∞ ( U, E ) ′ Y (resp. C ∞ ( U i , E ) ′ O i ), we then get an isomorphism T ( Y, E ( σ )) ∼ = M ≤ i ≤ q T ( O i , E ( σ )) . (cid:3) ASSELMAN’S COMPARISON THEOREM 7
By dualizing, the following theorem follows from Lemma 2.12 imme-diately.
Theorem 2.13. ( [7, Theorem 4.1] ) According to the Bruhat filtration,the graded module of I ( σ ) is the U ( g ) -module: gr I ( σ ) = M p ∈ Z + ( M dim C ( w )= p J ( w, σ )) . Moreover, we have a short exact sequence −→ C ∞ p +1 ( σ ) −→ C ∞ p ( σ ) −→ M dim C ( w )= p J ( w, σ ) −→ . for each p ∈ Z . As in the last subsection, let M p be the closed subspace of s ∈S ( V w , E ( σ )) which vanishes with all derivatives of order ≤ p along C ( w ). Then, F p T ( C ( w ) , E ) = M ⊥ p , ( ∀ p ∈ Z )and F T ( C ( w ) , E ) = S ( C ( w ) , E | C ( w ) ) ′ . For each p ∈ Z , putGr p T ( C ( w ) , E ( σ )) = F p +1 T ( C ( w ) , E ( σ )) /F p T ( C ( w ) , E ( σ )) . Lemma 2.14. ( [7, Lemma 3.5] ) For any p ∈ Z , the subspace F p T ( C ( w ) , E ) of T ( C ( w ) , E ) is U ( p ) -invariant.Proof. Let M ′ p be the space of smooth sections of E ( σ ) vanishing withderivatives of order ≤ p along the orbit C ( w ). By Lemma 2.7, M ′ ⊥ p = M ⊥ p = F p T ( C ( w ) , E ). Since C ( w ) and M ′ p are clearly P -stable, then F p T ( C ( w ) , E ) is U ( p )-invariant. (cid:3) For an affine space Y , write R ( Y ) for the ring of complex polynomialson Y . Let J be the ideal of polynomials in R ( V w ) vanishing along C ( w ).From the multiplication map J ⊗ R ( V w ) F p T ( C ( w ) , E ) → F p − T ( C ( w ) , E ) , we get J p /J p +1 ⊗ R ( C ( w ))) Gr p T ( C ( w ) , E ) → F T ( C ( w ) , E ) . Put L p ( w ) := Hom R ( C ( w )) ( J p /J p +1 , R ( C ( w ))). Since J p /J p +1 is a free R ( C ( w ))-module of finite rank, so is L p ( w ). Then, we get a U ( p )-module homomorphism α p : Gr p T ( C ( w ) , E ) → L p ( w ) ⊗ R ( C ( w )) S ( C ( w ) , E ) ′ , which is clearly an isomorphism. NING LI, GANG LIU, AND JUN YU
Lemma 2.15. ( [7, Lemma 4.3] ) As a U ( p ) module, Gr p T ( C ( w ) , E ) ∼ = L p ( w ) ⊗ R ( C ( w )) S ( C ( w ) , E ) ′ . In particular, Gr p T ( C ( w ) , E ) is isomorphic to a direct sum of finitelymany copies of S ( C ( w ) , E ) ′ . Lemma 2.16. ( [7, Lemma 4.4] ) The R ( C ( w )) module L p ( w ) possessesa finite increasing filtration such that each graded piece is the directsum of finite many copies of R ( C ( w )) as an N -equivariant R ( C ( w )) module.Proof. The R ( C ( w )) module L p ( w ) is the the space of global regularsections of an N -equivariant finite rank locally free sheaf F p ( w ) on C ( w ). By the nilpotence of N and the finite rank condition, F p ( w )has a finite increasing filtration such that each graded piece is an N -equivariant finite rank free sheaf. Since C ( w ) is affine, then the globalsection functor is exact. Therefore, L p ( w ) has a finite increasing fil-tration with each graded piece the direct sum of finite many copies of R ( C ( w )) as an N -equivariant R ( C ( w )) module. (cid:3) For convenience, we choose a different filtration of I ( σ ) ′ . Write W = { w , . . . , w r } so that dim C ( w i ) ≤ dim C ( w i +1 ) (1 ≤ i ≤ r − I k for the space of E ( σ )-distributions on X with support contained in S ≤ i ≤ k C ( w i ) (0 ≤ k ≤ r ). In particular, I = 0 and I m = I ( σ ) ′ . Then, I k /I k − = T ( C ( w k ) , E ( σ ))by the same reason as in the proof of Lemma 2.12.3. Casselman-Jacquet modules
Definition 3.1.
For a U ( p ) -module V , define its Casselman-Jacquetmodule V [ n ] := { v ∈ V : ∃ k ∈ Z > , n k · v = 0 } . Then, V [ n ] is still a U ( p )-module, and n acts on it locally nilpotentlyin the sense that for any v ∈ V [ n ] , U ( n ) · v is a finite-dimensional space.Moreover, if V is a U ( g )-module, then so is V [ n ] . Our proof of thecomparison theorem is based on several lemmas concerning Casselman-Jacquet modules of the Bruhat filtration of I ( σ ) ′ and the transversaldegree filtration of its graded pieces T ( C ( w ) , E ( σ )) ( w ∈ W ).Let U be a Zariski closed subgroup of N . In [7, p. 169] it appears adefinition of polynomials on the affine space N/U : a smooth function f on N/U is called a polynomial if its annihilator in U ( n ) contains n k forsome k ≥
1. Actually, the meaning of polynomial and regular functioncoincide in this case, as the following lemma shows. Lemma 3.2 is anadjustment of [1, Prop. A.4]. The same proof as the proof in [1] works.
ASSELMAN’S COMPARISON THEOREM 9
Write R l ( N/U ) for the space of regular functions on
N/U of degree ≤ l . Define a distribution δ N/U on N/U by δ N/U ( f ) = Z N/U f ( x ) d x where d x is a fixed N invariant measure on N/U . Lemma 3.2.
For any k ≥ , there exists l ≥ such that if a distribu-tion T on N/U satisfies n k T = 0 , then T ∈ R l ( N/U ) δ N/U . Conversely,for any l ≥ , there exists k ≥ such that n k ( R l ( N/U ) δ N/U ) = 0 . Note that each C ( w ) = N/ ( N ∩ N w ) ∼ = N ∩ N ww is geometricallya Euclidean space. Thus, there is the space S ( C ( w )) ′ of tempereddistributions on C ( w ). The following lemma follows from Lemma 3.2. Lemma 3.3.
For any w ∈ W , S ( C ( w )) ′ [ n ] = R ( C ( w )) · δ C ( w ) . For X ∈ g and ϕ ∈ C ∞ ( X, E ( σ )), define( R X ϕ )( g ) = ddt ϕ (exp( − tX ) g ) | t =0 . Define R ( X · · · X r ) ϕ = R X ( · · · ( R X r ϕ ) · · · ). This defines an actionof U ( g ) on C ∞ ( X, E ( σ )). For X ∈ U ( g ), f ∈ R ( C ( w )) and u ′ ∈ V ′ σ ,define < δ w ( X, f, u ′ ) , ϕ > = Z N ∩ N ww f ( n ) u ′ (( R ( X ) ϕ )( n ˙ w )) d n ( ∀ φ ∈ C ∞ ( V w , E ( σ ))). Set I ′ w = { l X i =1 δ w ( X i , f i , u ′ i ) : X i ∈ U ( n ww ∩ n w ) , f i ∈ R ( C ( w )) , u ′ i ∈ V ′ σ } . For each p , set I ′ p,w = { l X i =1 δ w ( X i , f i , u ′ i ) : X i ∈ U p ( n ww ∩ n w ) , f i ∈ R ( C ( w )) , u ′ i ∈ V ′ σ } . Lemma 3.4.
Let T ∈ T ( C ( w ) , E ) . If there exists k ≥ such that X k T = 0 for all X ∈ n , then T ∈ I ′ w .Proof. This is a special case of [1, Lemma 2.9]. (cid:3)
Lemma 3.5.
For any T ∈ I ′ w , there exists k ≥ such that n k T = 0 .Proof. By [1, Remark 2.2], this follows from [1, Lemma 3.1]. (cid:3)
The following is a direct consequence of Lemmas 3.4 and 3.5.
Lemma 3.6.
One has T ( C ( w ) , E ( σ )) [ n ] = I ′ w . Lemma 3.7.
For any w ∈ W and each p ∈ Z , there is an exactsequence → F p T ( C ( w ) ,E ( σ )) [ n ] → F p +1 T ( C ( w ) ,E ( σ )) [ n ] → Gr p T ( C ( w ) ,E ( σ )) [ n ] → . Proof.
Similar to Lemma 3.6, one has F p ( C ( w ) , E ( σ )) [ n ] = I ′ p,w . Also, there is a description for Gr p T ( C ( w ) ,E ( σ )) [ n ] using U p +1 ( n ww ∩ n w ) / U p ( n ww ∩ n w )instead. This could be shown either along the same line as that forLemma 3.6, or by combining Lemma 2.15 and Lemma 3.3. Then, theexactness of the sequence in the conclusion follows. (cid:3) Lemma 3.8.
For any k , there is an exact sequence → I [ n ] k − → I [ n ] k → ( I k /I k − ) [ n ] → . Proof.
One only needs to show that I [ n ] k ։ ( I k /I k − ) [ n ] . Since( I k /I k − ) [ n ] = T ( C ( w k ) , E ( σ )) [ n ] by Lemma 3.4, it suffices to find a distribution in I k with restriction δ w k ( X, f, u ′ ) on V w k for any X ∈ U ( g ), f ∈ R ( C ( w k )) and u ′ ∈ V ′ σ .This is shown in the proof of [1, Lemma 4.6]. (cid:3) A cohomological comparison result between tempereddistributions and polynomial multiple volume forms
Proposition 4.1.
Let U be a closed connected subgroup of a unipotentreal algebraic group N . Then the inclusion R ( N/U ) δ N/U ⊂ S ( N/U ) ′ induces isomorphisms H i ( n , R ( N/U ) δ N/U ) = H i ( n , S ( N/U ) ′ ) for all i ≥ .Proof. The proof is based on ideas in Casselman-Hecht-Milicic [7, § U = N , the proposition is trivial. Assume that U = N below.Suppose that C is a one-dimensional central connected subgroup of N which is not contained in U . Let V be the connected Lie subgroupwith Lie algebra v = c + u , and let p : N/U −→ N/V be the naturalprojection. Then,
N/V is the quotient of
N/U by the right actionof C . The pullback map p ∗ gives a homomorphism from the algebraof smooth functions on N/V into the algebra of smooth C -invariantsmooth functions on N/U . There is a natural exact sequence:0 / / S ( N/U ) ξ / / S ( N/U ) π / / S ( N/V ) / / . ASSELMAN’S COMPARISON THEOREM 11 where ξ is an N invariant differential operator on N/U associated to anonzero element ξ ∈ z and the map π is defined by π ( h )( p ( x )) = Z C h ( xz ) d z ( ∀ h ∈ S ( N/U )) , ∀ x ∈ N/U.
Then, it induces the following exact sequences:0 / / S ( N/V ) ′ π ∗ / / S ( N/U ) ′ ξ ∗ / / S ( N/U ) ′ / / . and0 / / R ( N/V ) δ N/V π ∗ / / R ( N/U ) δ N/U ξ ∗ / / R ( N/U ) δ N/U / / . Note that for any f ∈ R ( N/V ) and any h ∈ S ( N/U ), one has h π ∗ ( f δ N/V ) , h i = h f δ N/V , π ( h ) i = Z N/V f ( y ) π ( h )( y ) d y = Z N/V f ( y ) Z V/U h ( yz ) d z d y = Z N/U ˜ f ( x ) h ( x ) d x = h ˜ f δ N/U , h i where ˜ f ∈ R ( N/U ) is given by ˜ f ( x ) = f ( p ( x )) ( ∀ x ∈ N/U ). Thus, π ∗ ( f δ N/V ) = ˜ f δ
N/U . Consequently,(4.1) H i ( z , S ( N/U ) ′ ) = (cid:26) S ( N/V ) ′ for i = 00 for i = 0and(4.2) H i ( z , R ( N/U ) δ N/U ) = (cid:26) R ( N/V ) δ N/V for i = 00 for i = 0 . Case I: N is abelian. In this case, N = U × D for some comple-mentary abelian Lie subgroup D . The subgroup U acts on S ( N/U ) ′ and R ( N/U ) δ N/U trivially. Then,(4.3) H j ( u , S ( N/U ) ′ ) = j ^ u ∗ ⊗ S ( N/U ) ′ , j ∈ Z . and(4.4) H j ( u , R ( N/U ) δ N/U ) = j ^ u ∗ ⊗ R ( N/U ) δ N/U , j ∈ Z . Claim 4.2.
One has (4.5) H j ( d , S ( D ) ′ ) = (cid:26) C δ D for j = 00 for j = 0 . and (4.6) H j ( d , R ( D ) δ D ) = (cid:26) C δ D for j = 00 for j = 0 . Assume this claim first. Then, by the Hochschild-Serre spectral se-quence H p ( n / u , H q ( u , · )) ⇒ H p + q ( n , · )the equalities (4.3), (4.4), (4.5) and (4.6) imply that the inclusion R ( N/U ) δ N/U ⊂ S ( N/U ) ′ induces isomorphisms H i ( n , R ( N/U ) δ N/U ) = i ^ u ∗ = H i ( n , S ( N/U ) ′ )( ∀ i ≥ Proof of Claim 4.2.
We show (4.6). The equality (4.5) can be provedin the same way. Alternatively, it is a consequence of [7, Lemma 5.7].Prove by induction on dim D . When dim D = 0, it is trivial. Suppose(4.6) holds whenever dim D < k . Now let dim D = k ≥
1. Choose aone-dimensional connected subgroup Z of D . By (4.2) the Hochschild-Serre spectral sequence H p ( d / z , H q ( z , R ( D ) δ D )) = ⇒ H p + q ( d , R ( D ) δ D )degenerates and H p ( d , R ( D ) δ D ) = H p ( d / z , R ( D/Z ) δ D/Z ). Then, Claim4.2 follows from induction hypothesis. (cid:3)
Case II: N is non-abelian. We prove the proposition by inductionon dim N . Choose a one dimensional subalgebra c of n and let C be the corresponding connected Lie subgroup of N . We may assume c ⊂ [ n , n ].First, assume that C ⊂ U . Put N ′ = N/C and U ′ = U/C . Then,
N/U = N ′ /U ′ . Note that the action of C on R ( N/U ) δ N/U is trivial.Hence,(4.7) H i ( c , S ( N/U ) ′ ) = (cid:26) S ( N/U ) ′ i = 0 , i = 0 , . and(4.8) H i ( c , R ( N/U ) δ N/U ) = (cid:26) R ( N/U ) δ N/U for i = 0 ,
10 for i = 0 , . Then, the Hochschild-Serre spectral sequences H p ( n ′ , H q ( c , R ( N/U ) δ N/U )) = ⇒ H p + q ( n , R ( N/U ) δ N/U )and H p ( n ′ , H q ( c , S ( N/U ) ′ )) = ⇒ H p + q ( n , S ( N/U ) ′ ) ASSELMAN’S COMPARISON THEOREM 13 degenerate into short exact sequences. Consider the following commu-tative diagram:0 (cid:15) (cid:15) / / H i ( n ′ ,H ( c ,R ( N/U ) δ N/U )) (cid:15) (cid:15) / / H i ( n ,R ( N/U ) δ N/U ) (cid:15) (cid:15) / / H i − ( n ′ ,H ( c ,R ( N/U ) δ N/U )) (cid:15) (cid:15) / / (cid:15) (cid:15) / / H i ( n ′ ,H ( c , S ( N/U ) ′ )) / / H i ( n , S ( N/U ) ′ ) / / H i − ( n ′ ,H ( c , S ( N/U ) ′ )) / /
0. By (4.7), (4.8) and the induction hypothesis, each vertical arrow in theabove commutative diagram is an isomorphism. By the five lemma, weget the isomorphisms H i ( n , R ( N/U ) δ N/U ) = H i ( n , S ( N/U ) ′ ) ( ∀ i ≥ C * U . Put N ′ = N/C and U ′ = N C/C . Let v = u + c and V be the corresponding connected Lie subgroup of N .By (4.1) and (4.2), the Hochschild-Serre spectral sequence H p ( n / c , H q ( c , ∗ )) ⇒ H p + q ( n , ∗ )for ∗ = S ( N/U ) ′ or R ( N/U ) δ N/U degenerates and we have H i ( n , S ( N/U ) ′ ) = H i ( n ′ , S ( N ′ /U ′ ) ′ )and H i ( n , R ( N/U ) δ N/U ) = H i ( n ′ , R ( N ′ /U ′ ) δ N ′ /U ′ )( ∀ i ≥ N ′ = dim N −
1, the conclusion of the propositionfollows from induction hypothesis. (cid:3)
Remark 4.3.
In the above proof, while we run the Hochschild-Serrespectral sequence, at every step it is a comparison of the form H i (˜ d , R ( ˜ N / ˜ U ) δ ˜ N/ ˜ U ) = H i (˜ d , S ( ˜ N / ˜ U ) ′ ) for ˜ N a quotient group of N and ˜ D, ˜ U connected subgroups of ˜ N . Forthis reason, we can use the induction hypothesis whenever dim ˜ N < dim N and ˜ D = ˜ N .When we run the induction argument for every comparison resultshown below, similarly we always keep the form of the comparison sothat the induction hypothesis is valid while in use. Casselman’s comparison theorem
A comparison theorem for principal series.Theorem 5.1.
Let ( σ, V σ ) be a finite-dimensional representation of P with trivial N -action. Then the inclusion I ( σ ) ′ [ n ] → I ( σ ) ′ inducesisomorphisms H i ( n , I ( σ ) ′ [ n ] ) = H i ( n , I ( σ ) ′ ) for all i ∈ Z . Proof.
Since I ( σ ) = C ∞ ( E ( σ )), then I ( σ ) ′ = C ∞ ( E ( σ )) ′ and I ( σ ) ′ [ n ] = C ∞ ( E ( σ )) ′ [ n ] . Take the Bruhat filtration of C ∞ ( E ( σ )) ′ . By Lemma 3.8,it suffices to show that ( I k /I k − ) [ n ] ⊂ I k /I k − induces isomorphisms forcohomology on all degrees. Since I k /I k − = T ( C ( w k ) , E ( σ )), it suf-fices to show so for the inclusions T ( C ( w ) , E ( σ )) [ n ] ⊂ T ( C ( w ) , E ( σ ))( w ∈ W ). Consider the filtration through transversal degree. ByLemma 3.7, taking induction on the degree p it suffices to show that theinclusion (Gr p T ( C ( w ) , E )) [ n ] ⊂ Gr p T ( C ( w ) , E ) induces isomorphismsfor cohomology on all degrees. By Lemma 2.16, it suffices to show sofor the inclusions S ( C ( w ) , E ( σ )) ′ [ n ] ⊂ S ( C ( w ) , E ( σ )) ′ ( w ∈ W ). Then,the conclusion follows from Lemma 3.3 and Prop. 4.1. (cid:3) A comparison theorem for admissible representations.Theorem 5.2.
Let V be an admissible finitely generated moderategrowth smooth Fr´echet representation of G . Then the inclusion V K ⊂ V induces isomorphisms H i ( n , V K ) = H i ( n , V ) for all i ∈ Z , which are allfinite-dimensional spaces. Equivalently, the inclusion V ′ → V ∗ K inducesisomorphisms H i ( n , V ′ ) = H i ( n , V ∗ K ) for all i ∈ Z , which are all finite-dimensional spaces.Proof. We first show the homological version and the cohomologicalversion are equivalent. Suppose we have the cohomological comparisonstatement. Then, H i ( n , V ) are all finite-dimensional as H i ( n , V K ) areso. Then, by duality (cf. [7, Lemma 5.11]) we get the homologicalcomparison statement. The proof for the converse direction is the same.We show the cohomological comparison statement below.By [8, Lemma 2.37] one has H i ( n , V ∗ K ) = H i ( n , ( V ∗ K ) [ n ] ). By Cas-selman’s automatic continuity theorem (cf. [6, p. 416], [2, Theorem11.4] or [18, p.77]), one has V ′ [ n ] = ( V ∗ K ) [ n ] . Then, it reduces to show H m ( n , V ′ [ n ] ) = H m ( n , V ′ ). First, when V = I ( σ ) is a principal se-ries with σ an irreducible finite-dimensional representation of P , thisis shown in Theorem 5.1. Second, by induction on dim σ and usingshort exact sequence of the form 0 → σ ′ → σ → σ/σ ′ → σ ′ is a proper and nonzero P -subrepresentation of σ when σ is a non-irreducible finite-dimensional representation of P , one shows the con-clusion for V = Ind GP ( σ ) by cohomological long exact sequence and fivelemma. Third, by the dual form of Casselman’s representation theo-rem there exists a finite-dimensional representation σ of P such that I ( σ ) ։ V . Let U be the kernel. By the cohomological long exactsequences associated to 0 → V ′ → I ( σ ) ′ → U ′ → → V ∗ K → I ( σ ) ∗ K → U ∗ K →
0, one sees that the map H ( n , V ′ ) → H ( n , V ∗ K ) isinjective. For the same reason, H ( n , U ′ ) → H ( n , U ∗ K ) is injective.Then, by the above two cohomological long exact sequences and fivelemma, one sees that the map H ( n , V ′ ) → H ( n , V ∗ K ) is an isomor-phism. Fourth, we finish the proof by induction. For each i ≥
1, after
ASSELMAN’S COMPARISON THEOREM 15 showing that H j ( n , V ′ ) → H j ( n , V ∗ K ) (0 ≤ j ≤ i −
1) are isomorphisms,one first shows H i ( n , V ′ ) ֒ → H i ( n , V ∗ K ) from the above two cohomologi-cal long exact sequences. For the same reason, H i ( n , U ′ ) ֒ → H i ( n , U ∗ K ).Then, by the five lemma one shows that H i ( n , V ′ ) ֒ → H i ( n , V ∗ K ) is anisomorphism. (cid:3) Theorem 5.2 has the following immediate consequence.
Corollary 5.3.
Let V be an admissible finitely generated moderategrowth smooth Fr´echet representation V of G . For the Koszul complexassociated to V : / / ∧ d n ⊗ C V ∂ d / / · · · ∂ / / n ⊗ C V ∂ / / V / / where d = dim n , each Im( ∂ i ) is an NF space and it is a closed subspaceof ∧ i − n ⊗ C V ( ≤ i ≤ d ).Proof. By Theorem 5.2, Im( ∂ i ) is a finite co-dimensional subspace ofker( ∂ i − ). Since ∂ i − is a linear continuous map, then ker( ∂ i − ) is aclosed subspace of the NF space ∧ i − n ⊗ C V . Thus, ker( ∂ i − ) is also anNF space. Since the linear continuous map ∂ i : ∧ i n ⊗ C V → ker( ∂ i − )has finite co-dimensional image, by [7, Lemma A.1] it follows thatIm( ∂ i ) is an NF space. Hence, it is a closed subspace of ∧ i − n ⊗ C V . (cid:3) Closeness and higher level comparison.Theorem 5.4.
Let V be an admissible finitely generated moderategrowth smooth Fr´echet representation of G . Then for every k ≥ , n k V is a closed subspace of V and the inclusion V K ⊂ V induces anisomorphism V K / n k V K = V / n k V .Proof. By the homological comparison theorem in Theorem 5.2, n V isa finite co-dimensional subspace of V . Thus, we can choose a finite-dimension subspace U of V such that V = n V + U . Then, for each k ≥ V = n k V + ( X ≤ j ≤ k − n j U ) . Thus, each n k V is a finite co-dimensional subspace of V . Since n k ⊗ V → V is a linear continuous map having finite co-dimensional image, by[7, Lemma A.1] it follows that n k V is a closed subspace of V . ByCasselman’s automatic continuity theorem: V K / n k V K = V / n k V ( ∀ k ∈ Z ≥ ). Then, it follows that V K / n k V K = V / n k V for all k ∈ Z ≥ . (cid:3) References [1] N. Abe,
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Ning Li, Beijing International Center for Mathematical Research,Peking University, No. 5 Yiheyuan Road, Beijing 100871, China
Email address : [email protected] Gang Liu, Institut Elie Cartan de Lorraine, CNRS-UMR 7502, Uni-versit´e de Lorraine, 3 rue Augustin Fresnel, 57045 Metz, France
Email address : [email protected] Jun Yu, Beijing International Center for Mathematical Research,Peking University, No. 5 Yiheyuan Road, Beijing 100871, China
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