Stability property of multiplicities of group representations

Abstract

Let M be a compact complex manifold acted on by a compact Lie group G. Let L → M be a G-equivariant holomorphic line bundle that is assumed to be ample. Note that the G-action on L → M extends to the complex reductive group G C [9].In this context, we are interested in the family of G-modules Γ(M, L ⊗n ) formed by the holomorphic sections, and more particularly to the sequence H(n) := dim Γ(M, L ⊗n ) G , n ≥ 1. For any holomorphic G-complex vector bundle E → M , we consider also the sequenceOur main result, that we will detail in the next Section, can be summarized as follows : if the sequence H(n) is bounded, then the sequence H E (n) is bounded for any holomorphic G-complex vector bundle E and we can compute its value for large n. Stability resultSince the line bundle L is ample, there exists an Hermitian metric h on L such that the curvature Ω := i(∇ h ) 2 of its Chern connection ∇ h is a Kähler class : Ω is a symplectic form on M that is compatible with the complex structure. By an averaging process we can assume that the Gaction leaves the metric and connection invariant. Hence we have a moment map Φ : M → g * defined by Kostant's relationsHere L(X) is the Lie derivative on the sections of L, and X M (m) := d ds e −sX · m| s=0 is the vector field generated by X ∈ g.The [Q, R] = 0 Theorem of Meinrenken [16] and Meinrenken-Sjamaar [17] says that the moment map Φ gives a geometric interpretation of the sequence H(n). An important object here is the reduced space M 0 := Φ −1 (0)/G which is homeomorphic to the Mumford GIT quotient M/ /G C [11].

A geometric approach to the stabilisation of certain sequences of Kronecker coefficients

“…S. Sam and A. Snowden proved shortly after, in [19], that it is indeed verified. We also learned during the redaction of this article about a prepublication by P.-E. Paradan [14], who demonstrated this kind of result in a more general context which in particular contains the case of Kronecker coefficients (as well as the plethysm case). In the first part of this article, we give another new proof of this result: Theorem 1.2.…”

“…In Sections 4 and 5, using our method, we prove that weak stability also implies stability for some other coefficients arising in Representation Theory: at first for plethysm coefficients (the main result was already in [19] and [14]), and then for the multiplicities in the tensor product of two irreducible representations of the hyperoctahedral group, which is the Weyl group of type B n .…”

Witten Non Abelian Localization for Equivariant K-Theory, and the [𝑄,𝑅]=0 Theorem

“…Consequences of this asymptotic result [37] includes many stability results on asymptotic behavior of branching coefficients. We introduce a notion of Φ-positivity for equivariant vector bundles and Clifford bundles which extends a similar notion introduced by Tian-Zhang in the Hamiltonian context [47].…”

“…Theorem 9.32 plays an essential role when one studies the asymptotic behaviour of branching law coefficients (see [37]).…”

“…We also prove some generalizations, in the context of almost complex manifolds. In particular, we prove an asymptotic [Q, R] = 0 theorem for Clifford bundles (Theorem 9.32) which plays an essential role when one studies the asymptotic behavior of branching law coefficients (see [37]) and the notion of stability.…”