Related papers: Periods and harmonic analysis on spherical varieti…

We present a conceptual and uniform interpretation of the methods of integral representations of L-functions (period integrals, Rankin-Selberg integrals). This leads to: (i) a way to classify of such integrals, based on the classification…

Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k)…

We study the right regular representation on the space $L^2(N_0\setminus G;\psi)$ where $G$ is a quasi-split $p$-adic group and $\psi$ a non-degenerate unitary character of the unipotent subgroup $N_0$ of a minimal parabolic subgroup of…

In this paper we study the Plancherel formula for a new class of homogeneous spaces for real reductive Lie groups; these spaces are fibered over non-Riemannian symmetric spaces, and they exhibit a phenomenon of uniform infinite…

In a recent preprint, Sakellaridis and Venkatesh considered the spectral decomposition of the space $L^2(X)$, where $X = H\G$ is a spherical variety and $G$ is a real or $p$-adic group, and stated a conjecture describing this decomposition…

We consider the variation of canonical local periods on spherical varieties proposed by Sakellaridis-Venkatesh in families. We formulate conjectures for the rationality and meromorphic property of canonical local periods and establish these…

In this paper, we compute the local relative character for 10 strongly tempered spherical varieties in the unramified case. We also study the local multiplicity for these models. By proving a multiplicity formula, we show that the summation…

Yiannis Sakellaridis and Akshay Venkathesh have determined, when the group $G$ is split and the field $\F$ is of characteristic zero, the Plancherel formula for any spherical space $X$ for $G$ modulo the knowledge of the discrete spectrum.…

In this paper, we calculate the periods of Deligne-Lusztig characters $R_{T,\chi}^G$ associated to $X^F$, where $X=G/H$ for a connected spherical subgroup $H$ of $G$. As an application, we deduce a formula that partially generalizes…

In this paper, we will determine the factorization of global period attached to the spherical variety $X:=\mathrm{SO}(n-1)\backslash \mathrm{SO}(n)$, which is a special case of the Sakellaridis-Venkatesh conjecture. The main idea is to…

We consider two $S$-dual hyperspherical varieties of the group $G_2 \times \text{SL}(2)$: an equivariant slice for $G_2$, and the symplectic representation of $G_2 \times \text{SL}_2$ in the odd part of the basic classical Lie superalgebra…

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.…

Let $G$ be a real linear reductive group and let $H$ be a unimodular, locally algebraic subgroup. Let $\operatorname{supp} L^2(G/H)$ be the set of irreducible unitary representations of $G$ contributing to the decomposition of $L^2(G/H)$,…

Given a real semisimple connected Lie group $G$ and a discrete subgroup $\Gamma < G$ we prove a precise connection between growth rates of the group $\Gamma$, polyhedral bounds on the joint spectrum of the ring of invariant differential…

We establish the analog for real homogeneous spherical varieties of the Scattering Theorem of Sakellaridis and Venkatesh (Periods and harmonic analysis on spherical varieties, Asterisque 396, (2017), Theorem 7.3.1) for p-adic wavefront…

Let $X=H\backslash G$ be a spherical variety over a $p$-adic field. Assume $G$ is split. Let $\widehat{G}$ be the Langlands dual group of $G$. There is a complex group $\widehat{G}_X$ whose root datum is the little Weyl group of $X$. It was…

Let $X$ be a spherical variety for a connected reductive group $G$. Work of Gaitsgory-Nadler strongly suggests that the Langlands dual group $G^\vee$ of $G$ has a subgroup whose Weyl group is the little Weyl group of $X$.…

This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the…

Given a quasi-split reductive group $G$ and a symmetric variety $X$, we introduce a notion of endoscopic varieties for $(G,X)$, and establish the foundational properties of these varieties such as matching of stable semi-simple orbits. To…

Consider the space B of complex $p\times q$ matrces with norm <1. There exists a standard one-parameter family $S_a$ of unitary representations of the pseudounitary group U(p,q) in the space of holomorphic functions on B (i.e. scalar…