arXiv subjects

Joël Cohen

Publications and source records attributed to Joël Cohen.

2 recordsLinked to original sources

arXiv2015

A spectral expression for a certain orbital integral

Let $F$ be a $p$-adic field, $G = \text{GL}_{2n}(F)$ and $θ_0$ be the exterior automorphism of $G$ that fixes a pinning of a Borel pair. Consider the set $\widetilde{G} = G θ_0$ on which $G$ acts by conjugacy and the orbital integral $J_{\widetilde{G}}(θ_0, f)$ at $θ_0$. We prove a Plancherel-Harish-Chandra type formula for this orbital integral, namely as an integral over the irreducible tempered auto-dual representations of $G$ that we call "symplectic" (meaning their Langlands parameter factors through $\text{Sp}_{2n}(\mathbb{C})$). This solves a problem raised by G. Chenevier and L. Clozel. Our method uses the endoscopic transfer to $\text{SO}_{2n+1}$. Along the way, we also prove that the Plancherel measure is constant on $L$-packets.

math.RT↗

arXiv2014

A matrix Paley-Wiener theorem for non-connected $p$-adic reductive groups

Let $F$ be a local non archimedian field of characteristic $0$, and $G$ a non-connected reductive group over $F$. We denote $G^0$ the connected component of the identity and assume the quotient $G/G^0$ is abelian. For $f$ a locally constant compactly supported function on $G$ and $π$ a complex smooth representation of $G$, we define the Fourier transform of $f$ evaluated at $π$ to be $π(f) = \int_{G} f(g) π(g) \, dg$, which is an endomorphism of the underlying vector space of $π$. We give a description of the image of this Fourier transform map : given, for every $π$ in a certain family of induced representations of $G$, an endomorphism $φ(π)$ of the underlying vector space, we provide necessary and sufficient conditions under which there exists a function $f$ (necessarily unique) such that $π(f) = φ(π)$ for all $π$ in the family.

math.RT↗