Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

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1 Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group over K. Let me recall some basic facts on the Lie algebra g = Lie G of G. (Some material here and below was not covered in my lecture. It is included in response to questions I got in the seminar following the lecture.) By definition g is the tangent space T 1 G of G at the neutral element 1 G. One way of introducing this tangent space is g = (m/m 2 ) where m = { f K[G] f(1) = 0 } is the maximal ideal in K[G] of all regular functions vanishing at 1. convenient to set g = Der K (K[G], K) where the right hand side consists of all K linear maps It will be more X: K[G] K with X(f 1 f 2 ) = X(f 1 ) f 2 (1) + f 1 (1) X(f 2 ) for all f 1, f 2 K[G]. These maps are called derivations from K[G] to K where K is considered as a K[G] module via f a = f(1) a. The equivalence between these definitions is easy: We can regard any ϕ (m/m 2 ) as a linear form on m annihilating m 2. We then extend ϕ to K[G] = K 1 m by mapping 1 to 0 and check that this extension belongs to Der K (K[G], K). Conversely, one checks that any X Der K (K[G], K) annihilates 1 and m 2 and thus comes from an element in (m/m 2 ). Take for example G = GL n (K). In this case g identifies with the space M n (K) of all (n n) matrices over K. Denote by E ij the standard basis element for M n (K) that has (i, j) entry 1 and all other entries 0. We have K[GL n (K)] = K[(T ij ) 1 i,j n, 1 det ] where T ij is the function mapping any matrix to its (i, j) entry. So the T ij are the basis of M n (K) dual to the basis of all E ij. If we now identify any E ij M n (K) = g with an element of Der K (K[GL n (K)], K), then one gets E ij (f) = f T ij (1) for all f. 1

2 More generally, if G is a closed subgroup of GL n (K), then we can identify K[G] with K[GL n (K)]/I G where I G is the ideal of all functions vanishing on G. We have then a natural isomorphism { X Der K (K[GL n (K)], K) X(I G ) = 0 } Der K (K[G], K) mapping X to X with X(f + I G ) = X(f). In this way g = Lie G is identified with a subspace of M n (K). For example, if G = { ( ) 1 a a K } GL (K), then I G is generated by T 11 1, T 21, and T 22 1; we can identify K[G] with the polynomial ring K[T ] such that T corresponds to the restriction of T 12 to G. Then we get g = K E 12, and the basis element acts as derivation via E 12 (f) = df dt (1). So far we have introduced g only as a vector space, not as a Lie algebra. One way of defining that extra structure is as follows (for arbitrary G): For any g G the inner automorphism Int(g): G G with h ghg 1 has a tangent map at 1 that we denote by Ad(g): g g. One checks that Ad: G GL(g) is a homomorphism of algebraic groups and thus has a tangent map ad g End(g). Set now [X, Y ] = ad(x) (Y ). In case G is a closed subgroup of some GL n (K) and thus g a subspace of M n (K), then one checks that [X, Y ] = X Y Y X (matrix multiplication!). This implies that our bracket turns g into a Lie algebra. From G modules to g modules. Any G module V has a natural structure as a module for g = Lie G: Given v V there are f 1, f 2,..., f r K[G] and v 1, v 2,..., v r V such that r g v = f i (g) v i for all g G. i=1 Then set for all X g = Der K (K[G], K) X v = r X(f i ) v i. i=1 If dim V <, then the G module structure on V is given by a homomorphism G GL(V ) of algebraic groups. Its tangent map g End(V ) yields then the g module structure described above. This construction takes obviously G submodules to g submodules. It is compatible with taking factor modules, direct sums, tensor products, and dual modules. 2

3 An example for SL 2. The Lie algebra of G = SL 2 (K) is the space sl 2 (K) of all (2 2) matrices over K with trace 0. It has basis ( ) ( ) ( ) e =, f =, h = Consider for any n N the G module S n (V ) from Lecture I with basis (v i ) 0 i n. Recall the formulae for the G action ( ) 1 a i ( ) ( ) i v 0 1 i = a i j 1 0 n ( ) n i v j and v j a 1 i = a j i v j n j j=0 and for all t K ( ) t 0 0 t 1 v i = t n 2i v i. Using the recipe described above one gets for all i j=i e v i = { 0 if i = 0, i v i 1 if i > 0, f v i = { (n i) vi+1 if i < n, 0 if i = n, and h v i = (n 2i) v i. If char K = 0, then this g module is simple for each n: The basis elements v i are eigenvectors for h with distinct eigenvalues. Any non-zero g submodule contains an eigenvector for h, hence some v i. Applying powers of e and f one gets then that all v j belong to the submodule. In fact, one of the first steps in the structure theory of semi-simple Lie algebras is to prove that the S n (V ) are representatives for the isomorphism classes of finite dimensional simple sl 2 (K) modules. Assume that char K = p > 0. The same argument as above shows that all S n (V ) with 0 n < p are simple g modules. (Actually one has to modify the argument in case p = 2 and n = 1.) We have seen in Lecture I that S p (V ) is not simple as a G module; so it cannot be simple as a g module. However, also the simple G submodule L(p) S p (V ) is not simple for g: We have L(p) = Kv 0 + Kv p and a look at the formulae above yields 0 = e L(p) = f L(p) = h L(p). In other words, L(p) is a two dimensional trivial module and thus not simple. Frobenius twists. The latest example can be explained more generally. Suppose again that char K = p > 0, but with arbitrary G. Consider an arbitrary G module V. Lemma: The Frobenius twist V (1) is trivial as a g module. Proof : Recall: Given v V there are f 1, f 2,..., f r K[G] and v 1, v 2,..., v r V such that g v = r i=1 f i(g) v i for all g G. We get then in V (1) g v = r f i (g) p v i, i=1 3

4 hence for all X g X v = r X(f p i ) v i. i=1 But the derivation property of X implies X(f p ) = p f p 1 (1) X(f) = 0 in characteristic p. Remark: The tensor product formula for L(n) in Lecture I shows now that no L(n) with n p is simple as an sl 2 (K) module. More simple modules for sl 2 (K). Suppose again that char K = p > 0. We want to exhibit more simple finite dimensional sl 2 (K) modules beyond those S n (V ) with 0 n < p from above. For the sake of simplicity assume that p 2. For any a, b K there exists an sl 2 (K) module M a,b of dimension p with basis z 0, z 1,..., z p 1 such that for all i e z i = { 0 if i = 0, i (a + 1 i) z i 1 if i > 0, f z i = { zi+1 if i < p 1, bz 0 if i = p 1, and h z i = (a 2i) z i. In fact, one checks easily that the three linear maps defined by these formulae satisfy the commutator relations for our basis elements of sl 2 (K). Claim 1: If b 0 or if a / F p, then M a,b is simple. Well, the basis vectors z i are eigenvectors for h with distinct eigenvalues. (Here we use that p 2.) Therefore any non-zero submodule has to contain at least one z i. If b 0 then we get all other z j by applying powers of f to z i. In case b = 0 we get thus only the z j with j > i, whereas powers of e yield the z j with j < i if a / F p. On the other hand, if b = 0 and a = n1 with 0 n < p, then one checks that there is a surjective homomorphism of sl 2 (K) modules M n1,0 S n (V ) with z i { (n!/(n i)!) vi if i n, 0 if i > n. This is an isomorphism in case n = p 1; otherwise the kernel turns out to be isomorphic to S p (n+2) (V ). Claim 2: We have e p z = 0 and f p z = bz and (h p h) z = (a p a) z for all z M a,b. We have to check this only when z is one of our basis elements z i, and in that case it is pretty obvious. (Note that (a 2i) p = a p (2i) p = a p 2i since we are in characteristic p.) Claim 3: If a / F p, then M a,b M a,b if and only if a = a and b = b. Assume here that M a,b M a,b. Claim 2 implies that b = b and (a ) p a = a p a 0, hence a / F p. Furthermore a look at the action of e shows in case a / F p that { z M a,b e z = 0 } = Kz 0. 4

5 So a is determined by M a,b as the eigenvalue of h acting on the left hand side. This claim does not generalise to the case where a F p. If a = n1 with 0 n < p 1 and if b 0, then one finds an isomorphism M n1,b M (p n 2)1,b. Twisting by group elements. The claims above show that we have a 2 parameter family of isomorphism classes of simple modules for sl 2 (K) in characteristic p. But there are even more. We can construct them by twisting with group elements. Return to general G and arbitrary K. For any g G the adjoint action Ad(g): g g on the Lie algebra of G is a Lie algebra automorphism. We can therefore define for any g module a new g module g V that is equal to V as a vector space and where any X g acts on g V as Ad(g 1 ) (X) does on V : X v = Ad(g 1 ) (X) v for all v V. We call g V the g module V twisted by g. The use of g 1 in the definition makes sure that g ( g V ) gg V for all g, g G. In the case G = SL 2 (K) and char K > 2, we get now many new simple modules of the type g M a,b. It turns out that each simple g module in that case is isomorphic to some g M a,b or to some S n (V ) with 0 n < p. Lie algebra and invariant derivations. We have seen that the representation theory of sl 2 (K) in prime characteristic is very different from that in characteristic 0. In the latter case the simple modules are parametrised by Z, in the first case there are simple modules parametrised by a positive dimensional variety. This difference arises from an extra structure on any Lie G in prime characteristic, the p th power map. In order to explain it, we first need another description of the Lie algebra. Return again to general G and arbitrary K. Recall that g = Der K (K[G], K) and recall from Lecture I the G module structure on K[G] where (g f) (g ) = f(g 1 g ). For any f K[G] and X g one now defines a new function f X on K[G], the right convolution of f by X, setting (f X) (g) = X (g 1 f) for all g G. We have f X K[G]: Since K[G] is a G module there exist f i, f i K[G] such that g 1 f = r i=1 f i(g) f i ; then f X = r i=1 X(f i ) f i. Theorem: The map X X is an isomorphism of Lie algebras from g to the Lie algebra Der K (K[G]) G of all G invariant derivations of the K algebra K[G], i.e., the space of all K linear maps D: K[G] K[G] satisfying D(ff ) = D(f) f + f D(f ) and D(g f) = g D(f) 5

6 for all f, f K[G] and g G. It is elementary to check that X satisfies these properties and that X X is an injective linear map. And one gets an inverse map sending any D as above to X D : K[G] K, X D (f) := (Df) (1). It is more subtle to check that X X is compatible with the Lie bracket which on Der K (K[G]) G is just the commutator of two endomorphisms. Actually, one can argue that it is more natural to define the Lie algebra structure on g using this isomorphism. In the case G = GL n (K) we have identified g with the space M n (K) of all (n n) matrices over K. We have described before how a standard basis element E ij looks as a derivation K[G] K. I leave it now as an exercise to show for all f K[G] and all i, j that f E ij = n k=1 T ki f T kj. (Note that a derivation of K[G] is determined by its values on all T ij.) Restricted Lie algebras. By definition any derivation of an associative K algebra A satisfies D(ab) = D(a) b + a D(b). This implies by induction for all r > 0 D r (ab) = r i=0 ( ) r D i (a) D r i (b). i In case char K = p > 0 this implies that D p is again a derivation of A. If D commutes in case A = K[G] with the G action, then so does D p. So the theorem above implies (when char K = p > 0): For each X g there exists a unique X [p] g such that One calls the p th power map of g. (X [p] ) = ( X) p. g g, X X [p] If G is a closed subgroup of GL n (K), then we usually identify g with a Lie subalgebra of M n (K). In this case the p th power X p] turns out to coincide with the p th power of X as a (square) matrix. For example, we get in case G = SL 2 (K) that e [p] = 0, f [p] = 0, h [p] = h. Return to general G with char K = p > 0. There are two important properties of the p th power map. The first one: ad(x [p] ) = ad(x) p for all X g. (1) 6

7 Denote by U(g) the universal enveloping algebra of g. If follows from (1) that each X p X [p] with X g belongs to the centre of U(g). [Here X p is the p power of X as an element in the associative algebra U(g).] The second important property can be stated as The map ϕ: g U(g), X X p X [p] is semilinear. (2) This means it satisfies ϕ(x + Y ) = ϕ(x) + ϕ(y ) and ϕ(ax) = a p ϕ(x) for all a K. A restricted Lie algebra over K is defined as a Lie algebra g over K together with a map X X [p] from g to itself such that (1) and (2) hold. So our discussion above says that the Lie algebra of an algebraic group has a natural structure as a restricted Lie algebra. p characters. Let g be a finite dimensional restricted Lie algebra over our algebraically closed field K of characteristic p. Consider a simple g module E. We can regard E as a simple U(g) module. The central elements X p X [p] (X g) in U(g) have to act on E via scalars. (If we assume that dim E <, then this follows from Schur s lemma. And one can actually prove that all simple g modules have finite dimension, which is quite different from the situation in characteristic 0.) So there exist χ E (X) K such that (X p X [p] ) v = χ E (X) p v for all v E. We have taken here a p th root of the eigenvalue of X p X [p] which is possible as any element in K has a unique p th root. The reason for doing so is: Lemma: The map χ E : g K is linear. This follows quickly from (2) above. We call χ E the p character of E. More generally, given χ g we say that a g module M has p character χ if (X p X [p] ) v = χ(x) p v for all v E and X g. For example, our Claim 2 above implies that the sl 2 (K) module M a,b has p character given by e 0, f p b, h p a p a. Restricted modules. We continue to assume that char K = p > 0. A module for a restricted Lie algebra is called restricted if it has p character 0. Suppose now that g = Lie G for an algebraic group over K. Recall that any G module has a natural structure as a g module. It turns out that this g module always is restricted. You may check this for G = SL 2 (K) and the modules S n (V ). Suppose now that G is semi-simple, connected and simply connected. notation collected at the beginning of Lecture II. Curtis has proved: Let us use the Theorem: Each L(λ) with λ X p is also simple as a g module. The L(λ) with λ X p are a system of representatives for the isomorphism classes of simple restricted g modules. Coadjoint action. Let again g = Lie G with char K = p > 0. We have seen that we can twist g modules by elements of G. The adjoint action of G on g preserves the p th power map: We have Ad(g) (X [p] ) = (Ad(g) (X)) [p] for all X g and g G. 7

8 Using this equality a simple calculation shows: Lemma: Let χ g and g G. If M is a g module with p character χ, then g M has p character gχ. Here gχ is the image of χ under the coadjoint action of g on g given by (gχ) (X) = χ(ad(g 1 ) (X)). The lemma says that the category of all g module with p character χ depends up to equivalence only on the coadjoint orbit of χ. Consider the example G = SL 2 (K) where g = sl 2 (K). We have a symmetric bilinear form on g given by (X, Y ) = trace (XY ). Let us assume that p > 2. Then (, ) is non-degenerate and thus defines a isomorphism of vector spaces γ: g g, X (X, ). This is actually an isomorphism of vector spaces since Ad(g) (X) = gxg 1 in this case implies (Ad(g) (X), Ad(g) (Y )) = (X, Y ) for all X, Y g and g G. Therefore γ induces a bijection of G orbits. Any element in g = sl 2 (K) is either conjugate to a diagonal matrix or to the nilpotent matrix e. (We use again that p > 2.) If X g is a non-zero diagonal matrix, then χ = γ(x) satisfies χ(e) = χ(f) = 0 and χ(h) 0. The simple g modules with this p character are then all M a,0 with a p a = χ(h) p. On the other hand, χ 1 = γ(e) satisfies χ 1 (e) = χ 1 (h) = 0 and χ 1 (f) = 1. The simple g modules with p character χ 1 are all M a,1 with a F p. It then follows that any simple g module with non-zero p character is isomorphic to either some g M a,0 with a / F p or to some g M a,1 with a F p. The reductive case. Suppose now that g = Lie G where G is connected and reductive. We extend notations like T, B, B +, U, U +, Φ from the beginning of Lecture II to this more general case. We set and have then the triangular decomposition h = Lie T, n = Lie U, n + = Lie U + g = n h n +. One can show that h has a basis (h i ) 1 i r with h [ p] i = h i for all i. On the other hand n + (resp. n ) has a basis consisting of root vectors e β, β Φ + (resp. β Φ + ); they satisfy e [p] β = 0. It turns out that each coadjoint orbit in g contains a representative χ with χ(n + ) = 0. (This is somewhat dual to the fact that each adjoint orbit intersects b + = h n +.) Using 8

9 twists with group elements it therefore suffices to consider χ with this property. Given such χ, there exist p r linear forms λ h such that λ(h i ) p λ(h i ) = χ(h i ) p for all i, 1 i r. For any such λ one constructs a baby Verma module Z χ (λ) as the quotient of the enveloping algebra U(g) by the left ideal generated by n +, all h λ(h) with h h, and all e p β χ(e β) p with β Φ + : Z χ (λ) = U(g) /( U(g) n + + U(g) (h λ(h)) + ) U(g) (e p β χ(e β) p ). h h β<0 Then Z χ (λ) is a g module with p character χ; it has dimension p N where N = Φ +. In case G = SL 2 (K) one gets Z χ (λ) = M a,b with a = λ(h) and b = χ(f) p. One shows now that every simple g module with p character χ is a homomorphic image of some Z χ (λ) with λ as above. (The proof uses that each simple b + module with p character χ b + has dimension one and is given by some λ as above, extended by 0 on n +.) However, a given Z χ (λ) may have several, non-isomorphic simple quotients, and a simple module may be a simple quotient of several Z χ (λ). (See, e.g., the isomorphism M n1,b M (p 2 n)1,b in the SL 2 case.) In fact, there is no general classification of the simple modules. Exceptions are the groups G = GL n (K) for any p and G = SL n (K) for p not dividing n and a few low rank cases. Regular semi-simple χ. One calls a linear form χ g semi-simple if there exists g G with (gχ) (n n + ) = 0. One calls χ nilpotent if there exists g G with (gχ) (h n + ) = 0. (In many cases there exists an isomorphism of G modules g g. In these cases a linear form is semi-simple or nilpotent if and only if its inverse image has this property.) Under mild restrictions on the characteristic p one can reduce the description of the simple modules to the case where the p character is nilpotent. (For GL n there is no restriction, for SL n one needs that p does not divide n. For the general case consult the references.) More precisely, one reduces to the case of a nilpotent p character for a Levi subalgebra of g. The simplest case is that of a regular semi-simple χ. Here regular means that the stabiliser of χ in G for the coadjoint action is a maximal torus of G. If we assume (as we may) that χ (n n + ) = 0, then this condition amounts to χ([e β, e β ]) 0 for all β Φ. For such χ one gets: Proposition: (a) Each Z χ (λ) is simple. We have Z χ (λ) Z χ (λ ) if and only of λ = λ. (b) Each g module with p character χ is semi-simple. Back to the groups. Non-restricted g modules play an important role in the approach by Andersen, Soergel, and me to Lusztig s conjecture. More precisely, we worh with nonrestricted modules for g A := g K A where A is a suitable K algebra. Consider the symmetric algebra S(h) of the Lie subalgebra h. The kernel of the augmentation homomorphism S(h) mapping h to 0 is a maximal ideal. Let A denote the completion of S(h) with respect to this maximal ideal, and set F equal to the field of fractions of A. 9

10 Then g F is a restricted Lie algebra over F. Simple g F modules need not have p characters since F is not algebraically closed. However, consider the category C F of all finite dimensional g F modules that are annihilated by all e p β, β Φ, and where each of our basis elements h i of h acts diagonalisably with all eigenvalues of the form h i + a, a F p where we regard h i as an element in F via the embeddings h S(h) A F. These g F modules are then actually modules with a p character that takes values in an extension field of F. (One has to adjoin a few p th roots.) Now the proposition above on regular semi-simple p characters generalises and yields that all modules in C F are semi-simple and that the simple modules are baby Verma modules Z F (λ), λ Λ, where Λ is the set of all linear maps λ: h F with λ(h i ) h i + F p F for all i. We define similarly a category C A of g A modules that we require to be free of finite rank over A. We have then an obvious base change functor C A C F, M M F := M A F. There are baby Verma modules Z A (λ), λ Λ, in C A such that Z A (λ) A F Z F (λ). On the other hand, there is a natural homomorphism A K killing the maximal ideal of the local ring A. The corresponding functor M M := M A K takes C A to restricted g modules: Any e p β also annihilates M and any h i acts diagonalisably with all eigenvalues in F p since h i is in the kernel of A K. We have Z A (λ) Z 0 (λ) where λ is related to λ via λ(h i ) = h i + λ(h i ). Lusztig s conjecture can be shown to be equivalent to a conjecture on the projective indecomposable objects in the category of all restricted g modules. These objects lift to the projective indecomposable objects in the category C A. If M is one of these projective indecomposables in the category C A, then M F is a direct sum of baby Verma modules Z F (λ). Lusztig s conjecture is then equivalent to a statement on the multiplicities in these decompositions. (Here I am cheating a bit. One needs some extra structure, a grading of the modules by the weight lattice so that one can talk of formal characters. I am ignoring this for the sake of simplicity.) For any such projective indecomposable module M the multiplicity of any Z F (λ) as a direct summand of M F is equal to the rank of the A module Hom CA (Z A (λ), M). So we really want to understand the family of all these A modules, parametrised by Λ. This is the starting point for the construction of the combinatorial category mentioned at the end of Lecture II. It involves slightly more complicated Hom-spaces that then determine the relevant modules up to isomorphism. References. I have written two more detailed surveys: Representations of Lie algebras in prime characteristic, pp in: A. Broer (ed.), Representation Theories and Algebraic Geometry (Proc. Montreal 1997), NATO ASI Series C 514, Dordrecht 1998 (Kluwer) Representations of Lie algebras in positive characteristic, pp in: T. Shoji et al. (eds.), Representation Theory of Algebraic Groups and Quantum Groups (Proc. Tokyo 2001), Advanced Studies in Pure Math. 40, Tokyo 2004 (Math. Soc. Japan) Both papers can be downloaded from 10