Math 261A: Lie Groups, Fall 2008Problems, Set 51. (a) Show that SL(2, R) is topologically the product of a circle and two copies of R,hence it is not simply connected.(b) Let S be the simply connected cover of SL(2, R). Show that its finite-dimensionalcomplex representations, i.e., real Lie group homomorphisms S → GL(n, C), are determinedby corresponding complex representations of the Lie algebra Lie(S)C= sl(2, C), and hencefactor through SL(2, R). Thus S is a simply connected real Lie group with no faithfulfinite-dimensional representation.2. (a) Let U be the group of 3 × 3 upper-unitriangular complex matrices. Let Γ ⊆ U bethe cyclic subgroup of matrices1 0 m0 1 00 0 1,where m ∈ Z. Show that G = U/Γ is a (non-simply-connected) complex Lie group that hasno faithful finite-dimensional representation.(b) Adapt the solution to Set 4, Problem 2(b) to construct a faithful, irreducible infinite-dimensional linear representation V of G.3. Following the outline below, prove that if h ⊆ gl(n, C) is a real Lie subalgebra with theproperty that every X ∈ h is diagonalizable and has purely imaginary eigenvalues, then thecorresponding connected Lie subgroup H ∈ GL(n, C) has compact closure (this completesthe solution to Set 1, Problem 7).(a) Show that ad X is diagonalizable with imaginary eigenvalues for every X ∈ h.(b) Show that the Killing form of h is negative semidefinite and its radical is the centerof h. Deduce that h is reductive and the Killing form of its semi-simple part is negativedefinite. Hence the Lie subgroup corresponding to the semi-simple part is compact.(c) Show that the Lie subgroup corresponding to the center of h is a dense subgroup ofa compact torus. Deduce that the closure of H is compact.(d) Show that H is compact—that is, closed—if and only if it further holds that thecenter of h is spanned by matrices whose eigenvalues are rational multiples of i.4. Let Vn= Sn(C2) be the (n + 1)-dimensional irreducible representation of sl(2, C).(a) Show that for m ≤ n, Vm⊗ Vn∼=Vn−m⊕ Vn−m+2⊕ · · · ⊕ Vn+m, and deduce that thedecomposition into irreducibles is unique.(b) Show that in any decomposition of V⊗n1into irreducibles, the multiplicity of Vnisequal to 1, the multiplicity of Vn−2kis equal tonk−nk−1for k = 1, . . . , bn/2c, and all otherirreducibles Vmhave multiplicity zero.5. Let A be a symmetric Cartan matrix, i.e., A is symmetric with diagonal entries 2 andoff-diagonal entries 0 or −1. Let Γ be a subgroup of the automorphism group of the Dynkindiagram D of A, such that every edge of D has its endpoints in distinct Γ orbits. Define thefolding D0of D to be the diagram with a node for every Γ orbit I of nodes in D, with edgeweight k from I to J if each node of I is adjacent in D to k nodes of J. Denote by A0theCartan matrix with diagram D0(a) Show that A0is symmetrizable and that every symmetrizable generalized Cartanmatrix (not assumed to be of finite type) can be obtained by folding from a symmetric one.(b) Show that every folding of a finite type symmetric Cartan matrix is of finite type.(c) Verify that every non-symmetric finite type Cartan matrix is obtained by folding froma unique symmetric finite type Cartan matrix.6. An indecomposable symmetrizable Cartan matrix A is said to be of affine type ifdet(A) = 0 and all the proper principal minors of A are positive.(a) Classify the affine Cartan matrices.(b) Show that every non-symmetric affine Cartan matrix is a folding, as in the previousproblem, of a symmetric one.(c) Let h be a vector space, αi∈ h∗and α∨i∈ h vectors such that A is the matrixhαj, α∨ii. Assume that this realization is non-degenerate in the sense that the vectors αiarelinearly independent. Define the affine Weyl group W to be generated by the reflections sαi,as usual. Show that W is isomorphic to the semidirect product W0n Q where Q and W0arethe root lattice and Weyl group of a unique finite root system, and that every such W0n Qoccurs as an affine Weyl group.(d) Show that the affine and finite root systems related as in (c) have the property thatthe affine Dynkin diagram is obtained by adding a node to the finite one, in a unique way ifthe finite Cartan matrix is symmetric.7. Work out the root systems of the orthogonal Lie algebras so(m, C) explicitly, therebyverifying that they correspond to the Dynkin diagrams Bnif m = 2n + 1, or Dnif m = 2n.Deduce the isomorphisms so(4, C)∼=sl(2, C) × sl(2, C) so(5, C)∼=sp(4, C) and so(6, C)∼=sl(4, C).8. Show that the Weyl group of type Bnor Cn(they are the same because these two rootsystems are dual to each other) is the group Snn (Z/2Z)nof signed permutations, and thatthe Weyl group of type Dnis its subgroup of index two consisting of signed permutationswith an even number of sign changes, i.e., the semidirect factor (Z/2Z)nis replaced by thekernel of Sn-invariant summation homomorphism (Z/2Z)n→ Z/2Z.9. Let (h, R, R∨) be a finite root system, ∆ = {α1, . . . , αn} the set of simple roots withrespect to a choice of positive roots R+, si= sαithe corresponding generators of the Weylgroup W . Given w ∈ W , let l(w) denote the minimum length of an expression for w as aproduct of the generators si.(a) If w = si1· · · sirand w(αj) ∈ R−, show that for some k we have αik= sik+1· · · sir(αj),and hence siksik+1· · · sir= sik+1· · · sirsj. Deduce that l(wsj) = l(w) − 1 if w(αj) ∈ R−.(b) Using the fact that the conclusion of (a) also holds for v = wsj, deduce that l(wsj) =l(w) + 1 if w(αj) 6∈ R−.(c) Conclude that l(w) = |w(R+) ∪ R−| for all w ∈ W . Characterize l(w) in more explicitterms in the case of the Weyl groups of type A and B/C.(d) Assuming that h is over R, show that the dominant cone X = {λ ∈ h :hλ, α∨ii ≥ 0 for all i} is a fundamental domain for W , i.e., every vector in h has a uniqueelement of X in its W orbit.(e) Deduce that |W| is equal to the number of connected regions into which h is separatedby the removal of all the root hyperplanes hλ, α∨i, α∨∈ R∨.10. Let h1, . . . , hrbe linear forms in variables x1, . . . , xnwith integer coefficients. Let Fqdenote the finite field with q = peelements. Prove that except in a finite number of “bad”characteristics p, the number of vectors v ∈ Fnqsuch that hi(v) 6= 0 for all i is given for allq by a polynomial χ(q) in q with integer coefficients, and that (−1)nχ(−1) is equal to thenumber of connected