Bott towers, complete integrability, and the extended character of representations (Q1345224)

A Bott tower is an iterated family of \({\mathbf C} {\mathbf P}^1\) bundles over a point with distinguished sections of the fiber maps; if \(M_0\) (the zeroth stage is a point), \(M_1 = {\mathbf C} {\mathbf P}^1\), then \(M_2\) is defined as the projectivization of \(\mathbf{1} \oplus {\mathbf L}_1\), where \({\mathbf L}_1\) is a holomorphic line bundle over \(M_1\), \(M_3\) is the projectivization of \(\mathbf{1} \oplus {\mathbf L}_2\), where \({\mathbf L}_2\) is a holomorphic line bundle over \(M_2\), and so on. The distinguished sections are defined by the zero sections of \textbf{1} and of \({\mathbf L}_j\). The \(n\)th step in this construction can also be obtained from any collection of \(n(n-1)/2\) integers \(\{c_{ij}\}_{1 \leq i < j \leq n}\) by \(M_n = ({\mathbf C}_2 \setminus \mathbf{0})^n / ({\mathbf C}^*)^n\), where the \(i\)-th factor \(a_i\) of \(({\mathbf C}^*)^n\) acts on the right by \((z_1, w_1, \dots, z_n, w_n) \cdot a_i = (z_1,w_1, \dots, z_i a_i, w_i a_i, \dots, z_j, a_i^{c_{ij}} w_j, \dots)\). All Bott towers arise in this way and the authors prove that the maps \(\mathbb{Z}^{n(n-1)/2} \to\) \{Isomorphism classes of \(n\)-step Bott towers\} and, for a given \(M_n\) in the image of this map, \(\mathbb{Z}^n \to\) \{Isomorphism classes of holomorphic line bundles over \(M_n\}\) are bijections. This notion is related to representation theory in a natural way using ``twisted cubes''. A twisted cube \(C\) in an affine space \(V\) with lattice \(l_V\) is a subset of \(V\), called the support of \(C\) and a density function \(\rho : V \to \mathbb{R}\) which takes the values 1 and \(-1\) on the support. These data determine a signed measure \(m_C\) in \(V\). The authors prove: If \(M_n\) is a Bott tower, \(T\) acts on it by a complete torus action. \(\omega\) is a closed \(T\)-invariant 2-form, and \(\Phi\): \(M_n \to {\mathfrak t}^*\) is a moment map, then the corresponding Duistermaat-Heckman measure \(\Phi_* \omega^n/n!\) coincides with the measure \(m_C\) for a twisted cube in \({\mathfrak t}^*\). If \(T = T^{n + 1}\) acts completely on a line bundle \(\mathbf L\) over a Bott tower \(M_n\), this action be restricted to the Bott tower; then the multiplicity function for the (virtual) character of the irreducible representation with highest weight \(\lambda \in i{\mathfrak t}^*\) is given by the density function of the twisted cube determined by the integers \((c_{ij})\) that determine the Bott tower and \(n\) real numbers \((l_j)\) defining the image of the 2-form \(\omega\) under the moment map. Demazure described a complex structure on a Bott-Samelson manifold \(M\) which relates it to a flag variety. Bott found a non-holomorphic action on \(M\) of a torus of half the dimension of the space. The authors describe a second complex structure on \(M\) for which the action is holomorphic and which makes the Bott-Samelson manifold into a Bott tower. These two complex structures are connected by a one-parameter family of complex structures. When this is done, the character \(\chi\) of an irreducible representation of a connected compact Lie group \(K\) where \(M = K/T\) as the restriction of a virtual character \(\widetilde {\chi}\) which comes from a Bott tower. This restriction depends only on the choice of a reduced expression for the longest element of the Weyl group. Then the authors are able to express the multiplicities in a representation as the number of points inside certain polygonal regions and also obtain a formula for \(\widetilde {\chi}\) which implies Demazure's character formulas. These results all hold if the flag varieties are replaced by Schubert varieties.