Representations of Algebraic Groups and Their Lie Algebras Jens Carsten Jantzen Lecture I Set-Up. Let K Be an Algebraically Clos

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

Set-up. Let K be an algebraically closed field. By convention all our algebraic groups will be linear algebraic groups over K. For such a group G let K[G] denote the algebra of regular functions on G. For the background on algebraic groups I refer to the books Linear Algebraic Groups by J. E. Humphreys, T. A. Springer, and A. Borel. Modules. Let G be an algebraic group. A G–module is by definition a module V over the group algebra KG of G as an abstract group such that V satisfies two additional conditions: • V is locally finite: each element is contained in a finite dimensional KG–submodule. ∗ • For each v ∈ V and ϕ ∈ V the matrix coefficient cϕ,v given by cϕ,v(g) = ϕ(g v) for all g ∈ G is a regular function: cϕ,v ∈ K[G]. For example, any finite dmensional vector space V is a module under the natural action for the general linear group GL(V ) as well as for any closed subgroup of GL(V ). For any algebraic group G the algebra of regular functions K[G] is a G–module under the (left) regular representation given by (gf)(g0) = f(g−1g0). It is the easy to chech that the category of all G–modules is closed under: • submodules • factor modules • direct sums • tensor products • symmetric and exterior powers. If a G–module V is finite dimensional, then also its dual space V ∗ is a G–module. Frobenius twist. Assume char(K) = p > 0. In this case we have another method to construct new G–modules from known ones. If V is a vector space over K we denote by V (1) the vector space over K that coincides with V as an additive group and where the scalar multiplication is given by √ a • v = p a v for all a ∈ K, v ∈ V where the left hand side is the new multiplication and the right hand side the old one. (Since K is algebraically closed, the Frobenius morphism a 7→ ap is a bijection K → K, so any element in K has a unique p–th root.)

1 If V is a G–module, then V (1) is a G–module using the given action of any g ∈ G on the additive group V (1) = V . The action is again K–linear. A finite dimensional G– stable subspace of V is also a finite dimensional G–stable subspace of V (1) (of the same dimension); so also V (1) is locally finite. For any ϕ ∈ V ∗ the map

ϕ(1): V (1) → K, v 7→ ϕ(v)p

is K–linear. One checks that (V (1))∗ = { ϕ(1) | ϕ ∈ V ∗ } and gets ﬁnally for all g ∈ G

(1) p p cϕ(1),v(g) = ϕ (gv) = ϕ(gv) = cϕ,v(g)

p hence cϕ(1),v = cϕ,v ∈ K[G]. One deﬁnes higher Frobenius twists inductively: Set V (r+1) = (V (r))(1). Exercise: Consider K[G] as a G–module under the (left) regular representation. Show that f 7→ f p is a homomorphism K[G](1) → K[G] of G–modules. Induction. Let H be a closed subgroup of an algebraic group G. Any G–module V is by restriction an H–module since the restriction to H of any function in K[G] belongs to K[H]. This restriction of modules is clearly a functor from G–modules to H–modules. G We can deﬁne an induction functor indH in the opposite direction, from H–modules to G–modules. For any H–module M set

G −1 indH M = { f: G → M regular | f(gh) = h f(g) for all g ∈ G, h ∈ H }.

Here a function f: G → M is called regular if there exists m1, m2, . . . , mr ∈ M and Pr f1, f2, . . . , fr ∈ K[G] such that f(g) = i=1 fi(g) mi for all g ∈ G. All regular functions G G → M form a vector space over K and indH M is then a subspace. We get a G–action on this subspace by setting (gf)(g0) = f(g−1g0). In order to show that this KG–module is a G–module one uses the fact that K[G] is a G–module. G A trivial example for an induced module is ind{1} = K[G]. We’ll soon see more interesting examples for G = SL2(K). G The functor indH is right adjoint to the restriction functor from G–modules to H–modules. This means that we have functorial isomorphisms

G ∼ HomG(V, indH M) −→ HomH (V,M)

G Here any G–homomorphism ϕ: V → indH M is sent to ϕ: V → M with ϕ(v) = ϕ(v) (1). G Conversely any H–homomorphism ψ: V → M goes to ψe: V → indH M such that ψe(v)(g) = ψ(g−1v). 2 Example: SL2(K). Consider G = SL2(K) acting on its natural module V = K with its ∗ ∗ standard basis e1, e2. Then G acts also on V and on the symmetric algebra S(V ) that we can identify with the algebra K[X1,X2] of polynomial functions on V . Here X1,X2 is ∗ the basis of V dual to e1, e2.

2 The action of G on S(V ∗) is given by (gf)(v) = f(g−1v). Each symmetric power Sn(V ∗) i i n−i is a submodule. We choose as basis all vi = (−1) X1X2 , 0 ≤ i ≤ n, and get (e.g.) for all a ∈ K

i n 1 a X i 1 0 X n − i v = ai−jv and v = aj−iv (1) 0 1 i j j a 1 i n − j j j=0 j=i and for all t ∈ K× t 0 v = tn−2i v . (2) 0 t−1 i i Consider the closed subgroups in G

1 0 t 0 U = { | a ∈ K } and B = { | a ∈ K, t ∈ K× }. a 1 a t−1

Let us ﬁrst observe that we have an isomorphism of G–modules

∗ ∼ G S(V ) −→ indU K where K is the trivial U–module such that

G indU K = { f ∈ K[G] | f(gu) = f(g) for all g ∈ G, u ∈ U }.

∗ G Well, any ψ ∈ S(V ) defines ψb: G → K setting ψb(g) = ψ(ge2). One checks that ψb ∈ indU K G and that ψ 7→ ψb is a morphism of G–modules. On the other hand, any f ∈ indU K defines f: V \{0} → K: For any v ∈ V , v 6= 0 there exists g ∈ G with v = ge2; we then set 0 0 0 f(v) = f(g). This is well defined as ge2 = g e2 implies g ∈ gU, hence f(g) = f(g ) since G f ∈ indU K. One checks that f is a regular function on V \{0}. Algebraic geometry tells you that f has a (unique) extension to a regular function on V . So there exists ψ ∈ S(V ∗) with f(v) = ψ(v) for all v 6= 0, hence with f(g) = ψ(ge2) for all g ∈ G, i.e., with f = ψb.

For each n ∈ Z let Kn denote K considered as a B–module such that

t 0 v = tn v for all a, t, v ∈ K, t 6= 0. a t−1

G G Since any u ∈ U acts trivially on any Kn, it is clear that indB Kn ⊂ indU K. The inverse G ∗ G image of indB Kn under the isomorphism S(V ) ' indU K from above consists of all ψ ∈ S(V ∗) with ψ(tv) = tnψ(v) for all t ∈ K× and v ∈ V . This shows that we get isomorphisms n n ∗ indG K −→∼ S (V ) if n ≥ 0, B n 0 if n < 0.

Simple modules for SL2(K). Consider again G = SL2(K). If char K = 0, then one can show that all Sn(V ∗) with n ∈ N are simple G–modules. [Using (2) above one checks that

3 each non-zero submodule contains some vi. Then (1) shows that the submodule contains all vj. Here one uses that all binomial coeﬃcients are non-zero in K.] Furthermore, each simple G–module is isomorphic to Sn(V ∗) for a unique n. If char K = p > 0, then the same argument shows that all Sn(V ∗) with 0 ≤ n < p are simple G–modules. But one can quickly see that Sp(V ∗) is not simple: The map

ϕ: V ∗ −→ Sp(V ∗), f 7→ f p

is semi-linear (i.e., satisfies ϕ(f + f 0) = ϕ(f) + ϕ(f 0) and ϕ(af) = apϕ(f) for all a ∈ K) and G–equivariant. So its image is a G–submodule of Sp(V ∗) that is non-zero and not the whole module. Going back to the definition of a Frobenius twist one checks that ϕ is an isomorphism of G–modules ∗ (1) ∼ p (V ) −→ KGX2 p ∗ p onto the submodule of S (V ) generated by X2 . For general n the characteristic 0 argument shows first that each non-zero submodule n ∗ of S (V ) contains some vi. Then the first formula in (1) implies that the submodule n contains v0 = X2 . This shows that the submodule

n L(n) := KG v0 = KGX2

n ∗ n n ∗ of S (V ) generated by X2 is simple, in fact, it is the only simple submodule of S (V ). It then turns out that each simple G–module is isomorphic to L(n) for exactly one n. One can describe L(n) alternatively as follows. Write n in p–adic expansion:

2 r n = n0 + n1p + n2p + ··· + nrp with 0 ≤ ni < p for all i.

Then the map p p2 pr f0 ⊗ f1 ⊗ f2 ⊗ · · · ⊗ fr 7→ f0 f1 f2 . . . fr is an isomorphism of G–modules

∼ Sn0 (V ∗) ⊗ Sn1 (V ∗)(1) ⊗ Sn2 (V ∗)(2) ⊗ · · · ⊗ Snr (V ∗)(r) −→ L(n).

Notations. Let G be a connected reductive group over K. Choose a maximal torus T in G. We want to generalise the SL2–example. This involves some notation: × • X = Homalg.gr(T,K ), the character group of T ∨ × • X = Homalg.gr(K ,T ), the cocharacter group of T • h , i, the pairing X × X∨ → Z such that λ(ϕ(t)) = thλ,ϕi for all t ∈ K× • Φ ⊂ X, the root system of G with respect to T • Φ∨ = {α∨ | α ∈ Φ} ⊂ X∨ the corresponding dual roots • Φ+ a chosen system of positive roots

4 • B and B+ the Borel subgroups of G containing T that correspond to −Φ+ and Φ+ respectively • U and U + the unipotent radicals of these Borel subgroups. So we have semi-direct decompositions B = TU and B+ = TU + as algebraic groups. Each λ ∈ X can be extended to a character λ: B → K× such that λ(tu) = λ(t) for all t ∈ T and u ∈ U. Denote by Kλ the corresponding one dimensional B–module. The general theory of connected soluble algebraic groups yields:

Proposition: Each simple B–module is isomorphic to Kλ for a unique λ ∈ X. A corresponding statement holds for B+ instead of B. Simple G–modules. Any G–module is by definition locally finite, i.e., a union of finite dimensional submodules. Therefore any simple G–module V is finite dimensional. Consid- ered as a B–module it then has a composition series. This implies in particular that there is a surjective homomorphism of B–modules from V onto a simple B–module, hence by the proposition above onto some Kλ with λ ∈ X. The basic property of induction implies then G HomG(V, indB Kλ) ' HomB(V,Kλ) 6= 0. Using the simplicity of V we get now: G Claim 1: Each simple G–module is isomorphic to a simple submodule of some indB Kλ with λ ∈ X.

In the case of SL2(K) this implies that any simple module is (as claimed above) isomorphic to some L(n). The next step towards a classiﬁcation of the simple modules requires more subtle arguments that I cannot discuss here. They yield: G Claim 2: indB Kλ 6= 0 ⇐⇒ λ dominant where the set X+ of dominant characters is deﬁned by

∨ + X+ = { λ ∈ X | hλ, α i ≥ 0 for all α ∈ Φ }.

G As a third step we want to see that each indB Kλ with λ dominant contains a unique simple submodule. This is done by looking at ﬁxed points for the subgroup U +. For any non-zero U +–module M the space of ﬁxed points

+ M U = { x ∈ M | u x = x for all u ∈ U + } is non-zero as U + is connected and unipotent. G G U + The deﬁnition of indB Kλ and of the G–action on this space shows for any f ∈ (indB Kλ) that −1 + f(u1tu2) = λ(t) f(1) for all u1 ∈ U , t ∈ T , u2 ∈ U.

5 So the restriction of f to the subset U +TU of G is determined by f(1). As U +TU is dense in G, it follows that f itself is determined by f(1). This implies that

G U + (indB Kλ) = K fλ

where fλ ∈ K[G] is the unique regular function on G with −1 + fλ(u1tu2) = λ(t) for all u1 ∈ U , t ∈ T , u2 ∈ U.

G U + Now every non-zero G–submodule M of indB Kλ satisﬁes M 6= 0, hence fλ ∈ M and KG fλ ⊂ M. It follows that L(λ) := KG fλ G is the unique simple submodule of indB Kλ.

Theorem: Each simple G–module is isomorphic to L(λ) for a unique λ ∈ X+. U + In order to get uniqueness one notes that L(λ) = Kfλ and that T acts on this subspace via λ. G In case char K = 0 one can show that L(λ) = indB Kλ for all λ ∈ X+. The examples above show that this fails in prime characteristic already for G = SL2(K). Steinberg’s tensor product theorem. In order to simplify notation assume now that G is semi-simple and simply connected. Our choice of positive roots Φ+ determines a system of simple roots. The corresponding dual roots form under our assumption a basis for the free abelian group X∨ and X has then a dual basis consisting of fundamental weight. For any integer m > 0 the set

∨ Xm = { λ ∈ X | 0 ≤ hλ, α i < m for all simple roots α } ⊂ X+ consists of those characters whose coeﬃcients with repect to the basis of fundamental weights belong to the interval from 0 to m − 1.

Suppose now that char K = p > 0. Any λ ∈ X+ has a p–adic expansion 2 r λ = λ0 + pλ1 + p λ2 + ··· + p λr with all λi ∈ Xp.

Now Steinberg’s tensor product theorem says (generalising the result for SL2) that we have an isomorphism (1) (2) (r) ∼ L(λ0) ⊗ L(λ1) ⊗ L(λ2) ⊗ · · · ⊗ L(λr) −→ L(λ). G Note that each L(λi) ⊂ indB Kλi ⊂ K[G] is a submodule of K[G]; similarly for L(λ). The isomorphism is then simply given by

p p2 pr f0 ⊗ f1 ⊗ f2 ⊗ · · · ⊗ fr 7→ f0 f1 f2 . . . fr

Application to finite groups of Lie type. Assume again that G is semi-simple and simply connected. Suppose that char K = p > 0 and that G is defined over a finite subfield n Fq of K. (So q = p for some n > 0.) The group G(Fq) of points of G over Fq is then a finite group of Lie type. We can describe G(Fq) as the fixed points of a suitable Frobenius map on G.

6 In this situation a theorem due to Curtis for q = p and to Steinberg in general says:

Theorem: Each L(λ) with λ ∈ Xq is still simple when considered as a KG(Fq)–module. Any simple KG(Fq)–module is isomorphic to L(λ) for a unique λ ∈ Xq. Note that the Ree and Suzuki groups are finite groups of Lie type, which are not of the form G(Fq). However, a modification of this theorem yields also a classification of their irreducible representations over an algebraically closed field of their defining characteristic. References. For a more detailed survey of the material in Lectures I and II see my article Modular representations of reductive groups in the proceedings of a conference in Beijing published under the title Group Theory, Beijing 1984 as volume 1185 of the Springer Lecture Notes in Mathematics.