18.737 Algebraic groups (G.Lusztig)
Mondays, Wednesdays 2-3:30 in 2-142.
An algebraic group is an algebraic variety which is also a group. The theory of algebraic groups has much in common with that of Lie groups. But it uses the tools of algebraic geometry instead those of differential geometry. Also it includes the case when the ground field has positive characteristic which is not accessible by the methods of differential geometry. In this course we study affine algebraic groups over an algebraically closed field K. The topics that we hope to cover are:
1.Elementary theory of algebraic groups.
2.Semisimple, unipotent elements, Jordan decomposition.
3.Structure of connected solvable/nilpotent groups.
4.Borel subgroups, flag manifolds. Maximal tori. Weyl group.
5.Structure of connected reductive groups.
6.Reductive groups over a finite field. Use of l-adic cohomology to construct their representations.
The main topic of the course is No.6. The topics 1-5 are preparation for it. Some material will be given without proof (but with reference to literature.) Some knowledge of elementary algebraic geometry is desirable. For topic No.6 it will be helpful if the student is familiar with ordinary cohomology theory from algebraic topology.
Homework 1.
Let k be an algebraically closed field. Let V be a k-vector space of finite dimension 2n (at least 2). Let q:V\to k be a quadratic form, that is a function such that (x,y)=q(x+y)-q(x)-q(y) is a bilinear form V\times V\to k and q(cx)=c^2q(x) for x in V, c in k. We assume that q is nondegenerate, that is if (x,y)=0 for all y in V then x=0. Let O(V)={T\in GL(V); q(Tx)=q(x) for all x\in V}.
1) Show that O(V) is a closed subgroup of GL(V).
2) Show that there exists a basis e_1,e_2,...,e_{2n} of V such that q(a_1e_1+...+a_{2n}e_{2n})=a_1a_{2n}+a_2a_{2n-1}+...+a_na_{n+1} for all a_1,..,a_{2n} in k. A subspace V' of V is said to be isotropic if the restriction of q to V' is 0. Show that if V' is isotropic then dim V' is at most n. A subspace of V is said to be Lagrangian if it is isotropic and has dimension n. Let L(V) be the set of Lagrangian subspaces of V.
3) Show that L(V) is nonempty.
Define a relation \sim on L(V): we say that V'_1\sim V'_2 if the dimension of the intersection of V'_1 and V'_2 is congruent to n mod 2.
4) Show that \sim is an equivalence relation on L(V). (The only nontrivial part is the transitivity of this relation).
Homework 2.
Let V be a three dimensional k-vector space. Let G=SL(V)=\{g\in GL(V);det(g)=1\}. Let V^* be the dual vector space
For a natural number m let P_m(V) be the vector space of polynomials V\to k which are homogeneous of degree m. Show that dim P_m(V)=(m+1)(m+2)/2.
Show that G acts naturally on P_m(V) and this representation of G is irreducible if char(k)=0 or if char(k) is a prime p and m is strictly less than p. Show that it is not irreducible if char k is a prime p and m=p.
Let m,m' be natural numbers and let F_{m,m'} be the vector space of polynomial functions f:V\times V^*\to k such that x\to f(x,y) is in P_m(V) for any y in V^* and y\to f(x,y) is in P_{m'}(V^*) for any x in V. Show that dim F_{m,m'}=(m+1)(m+2)(m'+1)(m'+2)/4.
If m,m' are nonzero define j:F_{m-1,m'-1}\to F_{m,m'} by f\to ff_0 where f_0 \in F_{1,1} is f_0(x,y)=y(x)\in k. Show that j is an injective linear map map compatible with the actions of G. Define L_{m,m'}=F_{m,m'}/j(F_{m-1,m'-1}. If m=0 we set L_{m,m'}=F_{0,m'}=P_{m'}(V^*). If m'=0 we set L_{m,m'}=F_{m,0}=P_m(V). Show that dim L_{m,m'}=(m+1)(m'+1)(m+m'+2)/2 for any m,m'.
Note that L_{m,m'} is naturally a representation of G for any m,m'.
Show that L_{1,1) is irreducible if char k=0.
Give an example of a prime number p such that L_{1,1) is not irreducible
with char k=p.