Variations on a Casselman-Osborne theme

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1 Variations on a Casselman-Osborne theme Dragan Miličić Introduction This paper is inspired by two classical results in homological algebra o modules over an enveloping algebra lemmas o Casselman-Osborne and Wigner. They have a common theme: they are statements about derived unctors. While the statements or the unctors themselves are obvious, the statements or derived unctors are not and the published proos were completely dierent rom each other. In the irst section we give simple, pedestrian arguments or both results based on the same principle. They suggest a common generalization which is the topic o this paper. In the second section we discuss some straightorward properties o centers o abelian categories and their derived categories. In the third section, we consider a class o unctors and prove a simple result about their derived unctors which generalizes the irst two results. The original arguments were considerably more complicated and based on dierent ideas [1], [3] and [5]. 1 Classical approach 1.1 Wigner s lemma Let g be a complex Lie algebra, U (g) its enveloping algebra and Z (g) the center o U (g). Denote by M (U (g)) the category o U (g)-modules. Dragan Miličić Department o Mathematics, University o Utah, Salt Lake City, UT 84112, USA, milicic@math.utah.edu 1

2 2 Dragan Miličić Let χ : Z (g) C be an algebra morphism o Z (g) into the ield o complex numbers. We say that a module V in M (g) has an ininitesimal character χ i z v = χ(z)v or any z Z (g) and any v V. Theorem 1.1 Let U and V be two objects in M (U (g)) with ininitesimal characters χ U and χ V. Then χ U χ V implies Ext p U (g) (U,V ) = 0 or all p Z +. Proo. Clearly, the center Z (g) o U (g) acts naturally on Hom U (g) (U,V ) or any two U (g)-modules U and V, by z(t ) = z T = T z or any z Z (g) and any T Hom U (g) (U,V ), i.e., we can view it as a biunctor rom the category o U (g)-modules into the category o Z (g)-modules. Hence, its derived unctors Ext U (g) are biunctors rom the category o U (g)-modules into the category o Z (g)-modules. Fix now a U (g)-module U with ininitesimal character χ U. Consider the unctor F = Hom U (g) (U, ) rom the category M (U (g)) into the category o Z (g)- modules. Since the ininitesimal character o U is χ U, any element o z Z (g) acts on F(V ) = Hom U (g) (U,V ) as multiplication by χ U (z) or any object V in M (U (g)). Fix now a U (g)-module V with ininitesimal character χ V. Let 0 V I 0 I 1... I n... be an injective resolution o V. Let z ker χ V. Then we have the commutative diagram 0 I 0 I 1... I n... z z z. 0 I 0 I 1... I n... We can interpret this as a morphism φ : I I o complexes. Clearly, since H 0 (I ) = V, we have H 0 (φ ) = 0. Thereore, φ is homotopic to 0. By applying the unctor F to this diagram we get 0 F(I 0 ) F(I 1 )... F(I n )... F(z) F(z) F(z), 0 F(I 0 ) F(I 1 )... F(I n )... i.e., a morphism F(φ ) : F(I ) F(I ) o complexes. Since φ is homotopic to 0, F(φ ) is also homotopic to 0. This implies that all H p (φ ) : H p (I ) H p (I ), p Z, are equal to 0. Since H p (I ) = R p F(V ) = Ext p (U,V ), we see that U (g) Ext p (U,V ) are annihilated by z. U (g)

3 Variations on a Casselman-Osborne theme 3 On the other hand, by the irst remark in the proo, z must act on Ext p U (g) (U,V ) as multiplication by χ U (z). Since χ U χ V, there exists z ker χ V such that χ U (z) 0. This implies that Ext p U (g) (U,V ) must be zero or all p Z Casselman-Osborne lemma Now we assume that g is a complex semisimple Lie algebra. Let h be a Cartan subalgebra o g, R the root system o (g,h) and R + a set o positive roots. Let n be the nilpotent Lie algebra spanned by root subspaces o positive roots. Let γ : Z (g) U (h) be the Harish-Chandra homomorphism, i.e., the algebra morphisms such that z γ(z) nu (g) [2, Ch. VIII, 6, no. 4]. Let V be a U (g)-module. Since h normalizes n, the quotient V /nv = C U (n) V has a natural structure o U (h)-module. Also, Z (g) acts naturally on V /nv, and this action is given by the composition o γ and the U (h)-action. We can consider F(V ) = V /nv as a right exact unctor F rom the category o U (g)-modules into the category o U (h)-modules. Let For g denote the orgetul unctor rom the category o U (g)-modules into the category o U (n)-modules. Let For h denote the orgetul unctor rom the category o U (h)-modules into the category o linear spaces. Then we have the ollowing commutative diagram F M (U (g)) M (U (h)) For g For h M (U (n)) H 0 (n, ) M (C). By the Poincaré-Birkho-Witt theorem, a ree U (g)-module is a ree U (n)- module, hence we can use ree let resolutions in M (U (g)) to calculate Lie algebra homology H p (n, ) o U (g)-modules, i.e., we get the commutative diagram L p F M (U (g)) M (U (h)) For g For h M (U (n)) H p (n, ) M (C), or any p Z +. Thereore, Lie algebra homology groups H p (n, ) o U (g)- modules have the structure o U (h)-modules. Theorem 1.2 Let V be an object in M (U (g)). Let z Z (g) be an element which annihilates V. Then γ(z) annihilates H p (n,v ), p Z +. Proo. Let z Z (g). Let

4 4 Dragan Miličić... P n... P 1 P 0 V 0 be a projective resolution o V in M (U (g)). Multiplication by z gives the ollowing commutative diagram:... P n... P 1 P 0 0 z z z... P n... P 1 P 0 0 We can interpret this diagram as a morphism ψ : P P o complexes o U (h)- modules. Applying the unctor F we get the diagram... F(P n )... F(P 1 ) F(P 0 ) 0 F(z) F(z) F(z)... F(P n )... F(P 1 ) F(P 0 ) 0 representing F(ψ ), where F(z) is the multiplication by γ(z). Now, assume that z Z (g) annihilates V. Then we have H 0 (ψ ) = 0. It ollows that ψ is homotopic to 0. This in turn implies that F(ψ ) is homotopic to 0. Hence, the multiplication by γ(z) on F(P ) is homotopic to zero. Thereore, the multiplication by γ(z) annihilates the cohomology groups o the complex F(P ), i.e., γ(z) H p (n,v ) = 0 or p Z +. 2 Centers o derived categories 2.1 Center o an additive category Let A be an additive category. This implies that or any object V in A, all its endomorphisms orm a ring End(V ) with identity id V. An endomorphism z o the identity unctor on A is an assignment to each object U in A o an endomorphism z U o U such that or any two objects U and V in A and any morphism ϕ : U V we have z V ϕ = ϕ z U. Lemma 2.1 Let z be an endomorphism o the identity unctor on A and V an object in A. Then z V is in the center o the ring End(V ). Proo. Let e : V V be an endomorphism o V. Then, z V e = e z V, i.e., z V commutes with e. This implies that z V is in the center o End(V ). All endomorphisms o the identity unctor on A orm a commutative ring with identity which is called the center Z(A ) o A. Let B be the ull additive subcategory o A. Then, by restriction, any element o the center o A determines an element o the center o B. Clearly, the induced map

5 Variations on a Casselman-Osborne theme 5 r : Z(A ) Z(B) is a ring homomorphism. I the inclusion unctor B A is an equivalence o categories, the morphism o centers is an isomorphism. Let U and V be two objects in A. Then the center Z(A ) acts naturally on Hom(U,V ) by z(ϕ) = z V ϕ = ϕ z U or z Z(A ). Thereore, Hom(U,V ) has a natural structure o a Z(A )-module. Clearly, in this way Hom(, ) becomes a biunctor rom A A into the category o Z(A )-modules. 1 Assume that C is a triangulated category and T its translation unctor. Let z be an element o the center o C. Let U and V be two objects in C and ϕ : U V a morphism. Then T 1 (ϕ) : T 1 (U) T 1 (V ) is a morphism and we have By applying T to this equality we get z T 1 (V ) T 1 (ϕ) = T 1 (ϕ) z T 1 (U). T (z T 1 (V ) ) ϕ = ϕ T (z T 1 (U) ). Since ϕ : U V is arbitrary, we conclude that the assignment U T (z T 1 (U) ) is an element o the center o A, which we denote by T (z). It ollows that T induces an automorphism o the center Z(C ) o C. The elements o the center Z(C ) ixed by this automorphisms orm a subring with identity which we call the t-center o C and denote by Z 0 (C ). Let W U h [1] g V be a distinguished triangle in C and z an element o the t-center Z 0 (C ) o C. Clearly, since z is in the center, we have the commutative diagram U V g W z U z V z W. U V g Moreover, since z is in the t-center o C, we have T (z U ) = z T (U) and the diagram W 1 A is the category opposite to A.

6 6 Dragan Miličić W h T (U) z W T (z U ) W h T (U) commutes. Thereore, U V g W h T (U) z U z V z W T (z U ) U V g W h T (U) is an endomorphism o the above distinguished triangle. It ollows that the elements o the t-center induce endomorphisms o distinguished triangles in C. Let X be another object o C. The above remark implies that the distinguished triangle determines long exact sequences and Hom(X,U) Hom(X,V ) Hom(X,W) Hom(X,T (U))... Hom(T (U),X) Hom(W,X) Hom(V,X) Hom(U,X)... o Z 0 (C )-modules. 2.2 Center o a derived category Let C (A ) (where is b, +, or nothing, respectively) be the category o (bounded, bounded rom below, bounded rom above or unbounded) complexes o objects o A. Then C (A ) is also an additive category. Let z be an element o the center o A. I... V 0 V 1... V n... is an object in C (A ), we get the commutative diagram... V 0 V 1... V n... z V 0 z V 1 z V n... V 0 V 1... V n... which we can interpret as an endomorphism z V o V.

7 Variations on a Casselman-Osborne theme 7 Let ϕ : U V be a morphism in C (A ). Then z V p ϕ p = ϕ p z U p or any p Z, i.e., z V ϕ = ϕ z U. Thereore, the assignment V z V deines an element C (z) o the center o C (A ). Moreover, we have the ollowing trivial observation. Lemma 2.2 The map z C (z) deines a homomorphism o the center Z(A ) o A into the center Z(C (A )) o C (A ). Let K (A ) be the corresponding homotopic category o complexes. Let [z V ] be the homotopy class o endomorphism z V o V in C (A ). Then it deines an endomorphism o V in K (A ). Clearly, the assignment V [z V ] is an endomorphism K (z) o the identity unctor in K (A ). Moreover, the category K (A ) is triangulated and the translation unctor is given by T (U ) p = U p+1 or any p Z or any object U in K (A ). I a morphism ϕ : U V is the homotopy class o a morphism o complexes : U V, the morphism T (ϕ) : T (U ) T (V ) is the homotopy class o the morphism o complexes given by p+1 : T (U ) p T (V ) p or p Z. This immediately implies that T ([z U ]) = [z T (U )] or any element z o the center o A. It ollows that K (z) is in the t-center Z 0 (K (A )) o K (A ). Thereore, we have the ollowing observation. Lemma 2.3 The map z K (z) deines a homomorphism o the center Z(A ) o A into the t-center Z 0 (K (A )) o K (A ). Finally, assume that A is an abelian category and let D (A ) be the corresponding derived category o A, i.e., the localization o K (A ) with respect to all quasiisomorphisms. Clearly, or any z Z(A ), [z V ] determines an endomorphism [[z V ]] o V in D (A ). Let U and V be two complexes in D (A ) and ϕ : U V a morphism o U into V in D (A ). We can represent ϕ by a roo (see, or example [4]): s U V where s : U is a quasiisomorphism and : V is a morphism in K (A ). On the other hand, [[z U ]] and [[z V ]] are represented by roos id U U [z U ] U U and V id U [z V ] V V.

8 8 Dragan Miličić To calculate the composition [[z V ]] ϕ we consider the composition diagram s id W id V V [z V ] U V V which obviously commutes. This implies that the composition is represented by the roo s [z V ] U V. Analogously, to calculate ϕ [[z U ]] we consider the composition diagram id U U s [z U ] s [z W ] U U V which commutes since K (z) is in the center o K (A ). This implies that the composition is represented by the roo s [z W ] U V. Since [z ] = [z V ], these two roos are identical and [[z V ]] ϕ = ϕ [[z V ]]. Hence, the assignment V [[z V ]] deines an element o the t-center Z 0 (D (A )) o D (A ) which we denote by D (z). Moreover, we have the ollowing result. Lemma 2.4 The map z D (z) deines an injective morphism o the center Z(A ) o A into the t-center Z 0 (D (A )) o D (A ). For any z Z(A ), we have H p ([[z V ]]) = z H p (V ) or any V in D (A ) and any p Z. Proo. The second statement ollows immediately rom the construction.

9 Variations on a Casselman-Osborne theme 9 To prove injectivity, assume that D (z) = 0 or some z Z(A ). For an object V in A, denote by D(V ) the complex such that D(V ) 0 = V and D(V ) p = 0 or p 0. By our assumption, we have [[z D(V ) ]] = 0. This implies that z V = H 0 ([[z D(V ) ]]) = 0. Thereore, z V = 0 or any V in A, i.e., z = 0. Let z be an element o the t-center Z 0 (D (A )) o D (A ). Then H p+1 (z U ) = H p (T (z U )) = H p (z T (U )) or any object U in D (A ) and p Z. Thereore, H 0 (z U ) = 0 or all objects U in D (A ) is equivalent to H p (z U ) = 0 or all objects U in D (A ) and p Z. In particular, I 0 (D (A )) = {z Z 0 (D (A )) H p (z U ) = 0 or all U in D (A ) and p Z} = {z Z 0 (D (A )) H 0 (z U ) = 0 or all U in D (A )} is an ideal in Z 0 (D (A )). On the other hand, let D : A D (A ) be the unctor which attaches to each object V in A the complex D(V ), such that D(V ) 0 = V and D(V ) p = 0 or all p 0. This unctor is an isomorphism o A onto the ull additive subcategory o D (A ) consisting o all complexes U such that U p = 0 or all p 0 [4]. Thereore, we have a natural homomorphism r o Z(D (A )) into Z(A ) which attaches to an element z o the center o D (A ) the element o the center o A given by V H 0 (z D(V ) ) or any V in A. In particular, we have a natural homomorphism r : Z 0 (D (A )) Z(A ). From 2.4, we see that r(d (z)) V = H 0 (D (z) D(V ) ) = H 0 ([[z D(V ) ]]) = z V or any z in the center o A and any V in A. Thereore, we have the ollowing result. Proposition 2.5 The natural homomorphism r : Z 0 (D (A )) Z(A ) is a let inverse o the homomorphism D : Z(A ) Z 0 (D (A )). In particular, it is surjective. Its kernel is the ideal I 0 (D (A )). The situation is particularly nice or bounded derived categories. 2 Proposition 2.6 The natural homomorphism r : Z 0 (D b (A )) Z(A ) is an isomorphism. Proo. We have to prove that I 0 (D b (A )) = 0. Let z be an element o I 0 (D b (A )). Clearly, or any object V in A, we have z D(V ) = 0. Moreover, since z is in the t-center, z T p (D(V )) = 0 or any p Z. 2 I do not know any example where this result ails in unbounded case.

10 10 Dragan Miličić For any object U in D b (A ) we put l(u ) = Card{p Z H p (U ) 0}, and call l(u ) the cohomological length o U. Now we want to prove that z U = 0 or all U in D b (A ). The proo is by induction in the cohomological length l(u ). I l(u ) = 0, U = 0 and z U = 0. I l(u ) = 1, there exists p Z such that H q (U ) = 0 or all q p. In this case, U is isomorphic to the complex which is zero in all degrees q p and in degree p is equal to H p (U ), i.e., to T p (D(H p (U ))). Hence, by the above remark, z U = 0. Assume now that l(u ) > 1. Let τ p and τ p be the usual truncation unctors [4]. Then, or any p Z, we have the truncation distinguished triangle τ p+1 (U ) [1] τ p (U ) U and by choosing a right p Z, we have l(τ p (U )) < l(u ) and l(τ p+1 (U )) < l(u ). Thereore, by the induction assumption, there exists p Z such that z τ p (U ) = 0 and z τ p+1 (U ) = 0. As we remarked beore, this distinguished triangle leads to the long exact sequence Hom(U,τ p (U )) Hom(U,U ) Hom(U,τ p+1 (U ))... o Z 0 (D b (A ))-modules. By our construction, z annihilates the irst and third module. Thereore, it must annihilate Hom(U,U ) too. This implies that 0 = z(id U ) = id U z U = z U. 3 Centers and derived unctors 3.1 Homogeneous unctors Let A and B be two abelian categories. Let R be a commutative ring with identity and α : R Z(A ) and β : R Z(B) ring morphisms o rings with identity. By 2.4, α and β deine ring morphisms α = D α : R Z 0 (D (A )) and β = D β : R Z 0 (D (B)) Let F : A B be an additive unctor. We say that F is R-homogeneous i or any r R we have

11 Variations on a Casselman-Osborne theme 11 β(r) F(V ) = F(α(r) V ) or any object V in A. Assume now that F is let exact. Assume that there exists a subcategory R o A right adapted to F [4, ch. III, 6, no. 3] 3. Then F has the right derived unctor RF : D + (A ) D + (B). Theorem 3.1 The unctor RF : D + (A ) D + (B) is R-homogeneous. Proo. Let V be a complex in D + (A ). Since R is right adapted to F, there exists a bounded rom below complex R consisting o objects in R and a quasiisomorphism q : V R. Let z be an element o the center o A. Then we have the commutative diagram V q R [[z V ]] [[z R ]]. V R q By applying the unctor RF to it, we get the diagram RF([[z V ]]) RF(q) RF(V ) F(R ) [[F(z) F(R ) ]] RF(V ) RF(q) F(R ). I r R, α(r) is in the center o A and the above diagram implies that RF([[α(r) V ]]) RF(V ) RF(q) F(R ) [[β(r) F(R ) ]] RF(V ) RF(q) F(R ) is commutative. Moreover, β(r) is in the center o B, hence we also have [[β(r) RF(V )]] RF(q) RF(V ) F(R ) [[β(r) F(R ) ]] RF(V ) RF(q) F(R ). Hence, we conclude that RF([[α(r) V ]]) = [[β(r) RF(V )]], i.e., RF is R-homogeneous. 3 I would preer a proo o the next theorem which doesn t use the construction o the derived unctor, but its universal property. Unortunately, I do not know such argument.

12 12 Dragan Miličić Let V be an object in A. Then By taking cohomology, we get β(r) RF(D(V )) = RF(α(r) D(V ) ) or any r R. β(r) R p F(V ) = H p (β(r) RF(D(V )) ) = R p F(α(r) V ) or any r R and p Z +. Thereore, we have the ollowing consequence. Corollary 3.2 The unctors R p F : A B are R-homogeneous. We leave to the reader the ormulation and proos o the analogous results or a right exact unctor F and its let derived unctor LF : D (A ) D (B). 3.2 Special cases Now we are going to illustrate how 1.1 and 1.2 ollow rom the above discussion. First, we prove a well-known result about the center o the category o modules. This is not necessary or our applications, but puts the constructions in a proper perspective. Let A be a ring with identity and Z its center. Let M (A) be the category o A- modules. Any element z in Z determines an endomorphism z U o an A-module U. Clearly, the assignment U z U deines an element o the center Z(M (A)) o M (A). Thereore, we have a natural homomorphism i : Z Z(M (A)) o rings. Lemma 3.3 The morphism i : Z Z(M (A)) is an isomorphism. Proo. I we consider A as an A-module or the let multiplication, we see that i(z) A is the multiplication by z or any z Z. Thereore, i(z) A (1) = z and i : Z Z(M (A)) is injective. Let ζ be an element o the center o A. Then ζ A is an endomorphism o A considered as A-module or let multiplication. Let z = ζ A (1). Then ζ A (a) = aζ A (1) = az or any a A. Moreover, any b A deines an endomorphism ϕ b o A given by ϕ b (a) = ab or all a A. Since we must have ζ A ϕ b = ϕ b ζ A, it ollows that bz = (ζ A ϕ b )(1) = (ϕ b ζ A )(1) = zb. Since b A is arbitrary, z must be in the center Z o A. Let M be an arbitrary A-module and m M. Then m determines a module morphism ψ m : A M given by ψ m (a) = am or any a A. Thereore, ζ M (m) = (ζ M ψ m )(1) = (ψ m ζ A )(1) = zm = i(z) M m.

13 Variations on a Casselman-Osborne theme 13 Hence ζ = i(z), and i is surjective. Now we return to the notation rom the irst section. By 3.3, the center o the category M (U (g)) is isomorphic to Z (g). First we discuss 1.1. The unctor F = Hom U (g) (U, ) is a unctor rom the category U (g) into the category o Z (g)-modules. I we deine α as the natural morphism o Z (g) into the center o M (U (g)) and β as multiplication by χ U (z), F is clearly Z (g)-homogeneous. This implies that the unctors R p F are Z (g)- homogeneous. Hence, or any V in M (U (g)) we have R p F(z V ) = χ U (z) or all p Z +. In, particular, i z ker χ V we have 0 = R p F(0) = R p F(z V ) = χ U (z). This clearly contradicts χ U χ V i Ext p (U,V ) 0 or some p Z. U (g) No we discuss 1.2. The unctor F = H 0 (n, ) is a unctor rom the category U (g) into the category o Z (g)-modules. I we deine α as the natural morphism o Z (g) into the center o M (U (g)) and β as the composition o the Harish-Chandra homomorphism with the natural morphism o U (h) into the center o M (U (h)), F is clearly Z (g)-homogeneous. This implies that the unctors L p F are Z (g)- homogeneous. Hence or any V in M (U (g)), we have L p F(z V ) = γ(z) Lp F(V ) or all p Z +. In, particular, i z annihilates V, γ(z) annihilates L p F(V ) = H p (n,v ) or all p Z. Reerences 1. A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations o reductive groups, second ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, Nicolas Bourbaki, Groupes et algèbres de Lie, Hermann, William Casselman and M. Scott Osborne, The n-cohomology o representations with an ininitesimal character, Compositio Math. 31 (1975), no. 2, Sergei I. Geland and Yuri I. Manin, Methods o homological algebra, Springer-Verlag, Berlin, Anthony W. Knapp and David A. Vogan, Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995.