13 dezembro 2017, 11:00
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Gustavo Granja
Decomposition of a finite dimensional representations into weight spaces for the Cartan subalgebra: V=\osum_\lambda V_\lambda with \lambda \in \mathfrak h^*. L_\alpha \cdot V_\lambda \subset V_{\lambda+\alpha} and \lambda(h_\alpha) are integers for each \alpha \in \Phi (or equivalently because the h_\alpha_i are a base for the dual root system) for each \alpha_i \in \Delta). Lambda=\{ \lambda \colon \lambda(h_\alpha) \in \Z\} is called the weight lattice. If {\lambda_i} denote the fundamental dominant weights (i.e the dual basis to h_\alpha_i for a base {\alpha_i}) then \Lambda is the lattice spanned by the \lambda_i and the matrix which expresses the inclusion of the root lattice in the weight lattice is the Cartan matrix. Example of sl(3;C).
B=\mathfrak h \osum \osum_{\alpha \in \Phi^+} L_\alpha is solvable with [B,B]=\osum_{\alpha \in \Phi^+} L_\alpha (nilpotent) so by Lie's theorem each representation contains an eigenvector v for B with [B,B] acting trivially. These are called highest weight vectors. By the classification of sl(2)-modules, \lambda_i(v)>=0. The set . If V is irreducible then V=U(L)v for any highest weight vector. The set of weights for which this happens are called the dominant weights \Lambda^+=\Z_{\geq 0} \langle \lambda_1, \ldots, \lambda_l\rangle.
Theorem: There is a 1-1 correspondence between \Lambda^+ and the set of isomorphism classes of irreducible finite dimensional representations of L, sending V to V(\lambda) with (unique) highest weight \lambda.
Idea of construction: Induce from a one dimensional representation of the Borel subalgebra B with weight \lambda on the Cartan subalgebra: V = U(L) \otimes_{U(B)} \langle v \rangle. By the PBW theorem this breaks up as a sum of weight spaces lower than \lambda. One can show this has a unique irreducible quotient with highest weight v and this is the irreducible representation V(\lambda). This universal construction shows V(\lambda) is unique.
The set \Pi(\lambda) of weights appearing in V(\lambda) is preserved by the action of the Weyl group and for each \alpha the \alpha-string through \lambda has length \langle \lambda,\alpha \rangle. From this one can actually get the following characterization of \Pi(\lambda): it is the set of weights \mu such that \sigma \mu is less than or equal to \lambda for all \sigma in the Weyl group (for the partial order determined by \Delta).
Example: Some representations of sl(3)
G compact Lie group. For any representation W there is a canonical isomorphism \oplus_{V irred} \Hom^G(V,W) \otimes V \to W given by evaluation. Can do the usual algebraic constructions with representations - dual tensor, and so on. Over \C there is also the conjugate representation. An invariant Hermitean inner product on V gives an isomorphism between \overline V and V^*. Definition of character and representative function. Example: For G=S^1 the representative functions are the trigonometric polynomials.
Theorem: Representative functions coming from non-isomorphic irreducible representations are orthogonal in L^2(G)
Elementary properties of characters: (1) If V and W are isomorphic, \chi_V=\chi_W (2) \chi_V(h)=\chi_V(ghg^{-1}) (3) \chi_{V \oplus W} = \chi_V + \chi_W (4) chi_{V\otimes W) = \chi_V \chi_W (5) \chi_{V^*}(g)=\chi_{\overline V}(g)=\overline{\chi_V(g)} = \chi_V(g^{-1}) (6) \chi_V(e)=\dim V
Proposition: (7) \int_G \chi_V(g) = \dim V^G
(8) \langle \chi_V, \chi_W \rangle = \dim \Hom^G(V,W). Hence if V and W are irreducible this is 1 if V and W are isomorphic and 0 otherwise.
Corollary: V is isomorphic to W iff \chi_V=\chi_W; If V=\osum V_j^{n_j} with V_j irreducible (and distinct) then \|\chi_V\|^2 = \sum n_j^2. In particular V is irreducible if and only if its character is unital in L^2(G)
Proposition: Let G and H be compact Lie groups. Irreducible representations of G\times H are exactly the tensor products of irreducibles of G with irreducibles of H (with the action (g,h) v\otimes w = (gv) \otimes (gw)).
Example: Representations of compact abelian groups.
