Signature quantization and representations of compact Lie gr

Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa

Contributed by Victor Guillemin, May 11, 2004

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Abstract

We discuss some applications of signature quantization to the representation theory of compact Lie groups. In particular, we prove signature analogues of the Kostant formula for weight multiplicities and the Steinberg formula for tensor product multiplicities. Using symmetric functions, we also find, for type A, analogues of the Weyl branching rule and the Gel'fand–Tsetlin theorem.

The results Characterized in this article are closely related to an article of Guillemin et al. (1) on signature quantization. A symplectic manifAged (M, ω) is prequantizable if the cohomology class of ω is an integral class, i.e., it is in the image of the map MathMath. This assumption implies the existence of a prequantum structure on M: a line bundle, MathMath, and a connection, ▿, such that curv(▿) = ω. If g is a Riemannian metric compatible with ω, then, from g and ω, one Obtains an elliptic operator Embedded ImageEmbedded Image MathMath, the MathMath Dirac operator, and, by twisting this operator with MathMath, an operator Embedded ImageEmbedded Image MathMath. If M is compact one can “quantize” it by associating with it the virtual vector space Embedded ImageEmbedded Image Moreover if G is a compact Lie group and τ is a Hamiltonian action of G on M, one Obtains from τ a representation of G on Q(M) that is well defined up to isomorphism (independent of the choice of g).

The results Characterized in this article are closely related to two theorems in ref. 1. In this article the authors study the signature analogue of MathMath quantization; i.e., they define the virtual vector space (Eq. 1) by replacing Embedded ImageEmbedded Image MathMath with the signature operator Embedded ImageEmbedded Image MathMath and prove signature versions of a number of standard theorems about quantized symplectic manifAgeds. The two theorems we will be concerned with in this article are the following.

Let G = (S 1) n and let M be a 2n-dimensional toric variety with moment polytope MathMath. Then, for MathMath quantization, the weights of the representation of G on Q(M) are the lattice points, MathMath, and each weight occurs with multiplicity 1. For signature quantization the weights are the same; however, the weight β occurs with multiplicity 2 n if β lies in Int(▵), with multiplicity 2 n –1 if it lies on a facet, and, in general, with multiplicity 2 n –i if it lies on i facets. Further details can be found in the work of Agapito (2).

Let G be a compact simply connected Lie group, λ a Executeminant weight, and O λ = M the coadjoint orbit of G through λ. In the MathMath theory, the representation of G on Q(M) is the unique irreducible representation V λ of G with highest weight λ; however, in the signature theory, it is the representation

MathMath where ρ is half the sum of the positive roots. (This is modulo the proviso that λ – ρ be Executeminant.)

Ref. 1 also contains a signature version of the Kostant multiplicity formula. We recall that the Kostant multiplicity formula comPlacees the multiplicity with which a weight, μ, of T occurs in V λ by the formula MathMath where

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