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Register a new userRankin-Selberg convolutions for $mathrm{GL}(n)times mathrm{GL}(n)$ and $mathrm{GL}(n)times mathrm{GL}(n-1)$ for principal series representations
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Let $mathsf k$ be a local field. Let $I_ u$ and $I_{ u}$ be smooth principal series representations of $mathrm{GL}_n(mathsf k)$ and $mathrm{GL}_{n-1}(mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $mathrm{GL}_n(mathsf k)times mathrm{GL}_n(mathsf k)$.
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Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. In this paper we study the {em Whittaker coinvariants functor}, which is an exact functor from category $mathcal O$ for $mathfrak{gl}_{m|n}(mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergels functor $mathbb V$ in the setting of category $mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $mathcal O$ up to Morita/derived equivalence.
We prove that the local Rankin--Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin--Selberg subgroups, up to certain constants given by the local gamma factors.
Following the paradigm of cite{MR3117742}, we are going to explore the stable transfer factors for $mathrm{Sym}^{n}$ lifting from $mathrm{GL}_{2}$ to $mathrm{GL}_{n+1}$ over any local fields $F$ of characteristic zero with residue characteristic not equal to $2$. When $F=mathbb{C}$ we construct an explicit stable transfer factor for any $n$. When $n$ is odd, we provide a reduction formula, reducing the question to the construction of the stable transfer factors when the $L$-morphism is the diagonal embedding from $mathrm{GL}_{2}(mathbb{C})$ to finitely many copies of $mathrm{GL}_{2}(mathbb{C})$ under mild assumptions on the residue characteristic of $F$. With the assumptions on the residue characteristic, the reduction formula works uniformly over any local fields of characteristic zero, except that for $p$-adic situation we need to exclude the twisted Steinberg representations.
In this note we consider representations of the group GL(n,F), where F is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field F. We study reducibility and composition series of such representations. Similar results were obtained already in [HL99,Al12,Zel80], but we give a short uniform proof in the general case, using the tools from [AGS15a]. We also indicate some applications to cosine transforms in integral geometry.
In this paper we establish a new case of Langlands functoriality. More precisely, we prove that the tensor product of the compatible system of Galois representations attached to a level-1 classical modular form and the compatible system attached to an n-dimensional RACP automorphic representation of GL_n of the adeles of Q is automorphic, for any positive integer n, under some natural hypotheses (namely regularity and irreducibility).
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