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Register a new userFinite multiplicities beyond spherical pairs
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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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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.
In this paper, we prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results from [KO13,KS16] on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. In order to deduce this application we prove relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $G$ defined over $mathbb{R}$, the space of $G(mathbb{R})$-equivariant distributions on the manifold of real points of any algebraic $G$-manifold $X$ is finite-dimensional if $G$ has finitely many orbits on $X$.
Let ${mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $mathbb C^n$.
In this paper, we prove several stability theorems for multiplicities of naturally defined representations of symmetric groups. The first such theorem states that if we consider the diagonal action of the symmetric group $S_{m+r}$ on $k$ sets of $m+r$ variables, then the dimension of the invariants of degree $m$ is the same as the dimension of the invariants of degree $m$ for $S_{m}$ acting on $k$ sets of $m$ variables. The second type of stability result is for Weyl modules. We prove that the dimension of the $S_{n+r}$ invariants for a Weyl module, ${}_{m+r}F^{lambda}$ (the Schur-Weyl dual of the $S_{|lambda|}$ module $V^{lambda}$) with $leftvert lambda rightvert leq m$ is of the same dimension as the space of $S_{m}$ invariants for ${}_{m}F^{lambda}$. Multigrad
Using the twisted denominator identity, we derive a closed form root multiplicity formula for all symmetrizable Borcherds-Bozec algebras and discuss its applications including the case of Monster Borcherds-Bozec algebra. In the second half of the paper, we provide the Schofield constuction of symmetric Borcherds-Bozec algebras.
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