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In this paper, we introduce geometric multiplicities, which are positive varieties with potential fibered over the Cartan subgroup $H$ of a reductive group $G$. They form a monoidal category and we construct a monoidal functor from this category to the representations of the Langlands dual group $G^vee$ of $G$. Using this, we explicitly compute various multiplicities in $G^vee$-modules in many ways. In particular, we recover the formulas for tensor product multiplicities of Berenstein- Zelevinsky and generalize them in several directions. In the case when our geometric multiplicity $X$ is a monoid, i.e., the corresponding $G^vee$ module is an algebra, we expect that in many cases, the spectrum of this algebra is affine $G^vee$-variety $X^vee$, and thus the correspondence $Xmapsto X^vee$ has a flavor of both the Langlands duality and mirror symmetry.
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In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.
In this paper we discuss the highest weight $frak k_r$-finite representations of the pair $(frak g_r,frak k_r)$ consisting of $frak g_r$, a real form of a complex basic Lie superalgebra of classical type $frak g$ (${frak g} eq A(n,n)$), and the maximal compact subalgebra $frak k_r$ of $frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$.
Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections.
The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30-page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example - that of HilbpC2, nq, hoping to spark the curiosity of colleagues in those numerous fields of study where one should expect applications.
Let $G$ be a real reductive algebraic group, and let $Hsubset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is tame if this space is spherical. In particular, the multiplicities of the space $mathcal{S}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper we formulate and analyze a generalization of sphericity that implies finite multiplicities in $mathcal{S}(G/H)$ for small enough irreducible representations of $G$.
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