Representations of the normalizers of maximal tori of simple Lie groups (Q974338)

The zero weight space of a representation \(V\) of a complex semisimple Lie group \(G\) naturally affords a representation of the Weyl group \(W\) of \(G\), because \(W\) is by definition the quotient group \(N/T\), where \(T\) is a maximal torus of \(G\) and \(N= N_G(T)\) is the normalizer of \(T\) in \(G\). This representation of \(W\) is called the zero weight representation for \(V\), which gives an interesting relationship between the representations of \(G\) and those of \(W\). In particular, the \(W\)-module structure of the zero weight space has a deep-rooted connection to various problems in the representation theory of Lie groups such as the analysis of the plethysm and generalized exponents, etc. A natural extension of this problem is to study the restriction of the whole representation of \(G\) to \(N\). This is what the authors pursue in this paper. We have a similar problem if we replace \(G\) by its compact real form, which is really an equivalent problem. The authors study the branching rule for the restriction from a complex simple Lie group \(G\) to the normalizer of a maximal torus of \(G\). The paper is organized as follows. In \S 2, ``Representation of finite extensions of groups'', the authors recall Clifford's theory on the representations of an extension group \(E\) of a group \(H\) by a finite group \(F\) over the field \(\mathbb C\) of the complex numbers, which is a purely algebraic version of the method of little groups or Mackey machine. For an irreducible representation \(\chi\) of \(H\), define \(E_\chi\) to be the subgroup leaving the equivalence class of \(\chi\) invariant and take an irreducible representation \(\tau\) of \(E_\chi\) such that \(\tau|_H\) is a multiple of \(\chi\). Then the representation of \(E\) induced from \(\tau\) is irreducible. Every irreducible representation of \(E\) is given in this way. If \(\chi\) can be extended to an ordinary representation (by which we mean a linear representation) of \(E_\chi\), then these two projective representations can be replaced by ordinary representations. If \(E\) is a semidirect product of \(H\) and \(F\), then this condition holds for all \(\chi\) and therefore all irreducible representations of \(E\) can be obtained from irreducible representations of \(H\) and subgroups of \(F\). In \S 3, ``Relation to semidirect products'', the authors study the case where \(H\) is an abelian group and give a sufficient condition for all irreducible representations \(\chi\) to be extended to ordinary representations of \(E_\chi\). Under this condition, the equivalence classes of irreducible representations of \(E\) are parametrized by the conjugacy classes, under the action of \(E\), of the pairs \((\chi,\varphi)\) where \(\chi\) is an irreducible character of \(H\) and \(\varphi\) is an irreducible character of the factor group \(E_\chi/H\). The characters of \(E\) are given in \S 4, ``Formula for irreducible characters of finite extensions of abelian groups''. In \S 5, ``Case of normalizers of maximal tori'', the case is studied where \(H\) is a maximal torus \(T\) of a connected complex semisimple Lie group \(G\) and \(E\) is its normalizer \(N\) in \(G\), which is an extension of \(T\) by the Weyl group \(W\). Although \(N\) is not a semidirect product of \(T\) and \(W\) in general, the equivalence classes of irreducible holomorphic representations of \(N\) can be parametrized by the conjugacy classes of \((\chi,\varphi)\), where \(\chi\) is a holomorphic character of \(T\) and \(\varphi\) is an irreducible representation of the parabolic subgroup \(N_\chi/T\) of \(W\). In \S 6, ``Branching from \(G\) to \(N\): reduction to zero weight representations'', the authors discuss the structure of the \(N\)-module \(V\!\downarrow_N^G\) obtained from a \(G\)-module \(V\) by restriction from \(G\) to \(N\), by applying the results of the preceding sections for a complex simple Lie group \(G\) and the normalizer \(N\) of a maximal torus \(T\) of \(G\). The problem is reduced to the determination of the structure of ``zero weight representations'' for \(V\!\downarrow_{L_p'}^G\), where \(L_p'\) varies over the derived groups of the Levi parts of parabolic subgroups of \(G\). The same result is also formulated starting with the compact real forms of \(G\). In the last four sections the authors study in detail the case where \(G =SL(n,\mathbb C)\) by using Young diagrams and Schur functions. The multiplicity of an irreducible representation of \(N\) in the restriction of an irreducible representation of \(G\) to \(N\) is calculated in two ways. In \S 7, ``The case of \(SL(n,\mathbb C)\)'', and \S 8, ``Explicit formula for branching from \(SL(n,\mathbb C)\) to \(N\) (I)'', the authors apply the results of \S 6 and determine the zero weight representations for \(V\!\downarrow_{L_p'}^G\). The multiplicities are written in terms of Littlewood-Richardson's coefficients, characters of parabolic subgroups of \(W\) and generalized \(q\)-binomial coefficients. In \S 9, ``Explicit formula for branching from \(SL(n,\mathbb C)\) to \(N\) (II)'', the authors adopt the compact group formulation and consider the unitary group \(U(n)\) and the normalizer \(N'\) of a maximal torus of \(U(n)\). Since an element of \(N'\) is a product of a permutation matrix and a diagonal matrix, the restriction of an irreducible character of \(U(n)\) to a connected component of \(N'\) can be regarded as a function on \(T\). This enables one to calculate the multiplicities combinatorially in terms of the Schur functions and Weyl groups. In the last section, ``On the irreducibility of the \(N\)-span of a weight space for \(GL(n,\mathbb C)\)'', a series is given of examples of irreducible modules \(V\) for \(GL(n,\mathbb C)\) and their weights \(\mu\) such that \(\bigoplus_{\nu\in W_\mu}V_\nu\) is irreducible as an \(N\)-module, where \(V_\nu\) denotes the \(\nu\)-weight space of \(V\) and \(W_\mu = N_\mu/T\). For the case where \(\mu = 0\), every irreducible representation of the symmetric group \(\mathfrak S_n\) is obtained as the zero weight representation of a suitable irreducible representation of \(SL(n,\mathbb C)\).