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Register a new userBessel Models for General Admissible Induced Representations: The Compact Stabilizer Case
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A holomorphic continuation of Jacquet type integrals for parabolic subgroups with abelian nilradical is studied. Complete results are given for generic characters with compact stabilizer and arbitrary representations induced from admissible representations. A description of all of the pertinent examples is given. These results give a complete description of the Bessel models corresponding to compact stabilizer.
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Recently a remarkable map between 4-dimensional superconformal field theories and vertex algebras has been constructed cite{BLLPRV15}. This has lead to new insights in the theory of characters of vertex algebras. In particular it was observed that in some cases these characters decompose in nice products cite{XYY16}, cite{Y16}. The purpose of this note is to explain the latter phenomena. Namely, we point out that it is immediate by our character formula cite{KW88}, cite{KW89} that in the case of a textit{boundary level} the characters of admissible representations of affine Kac-Moody algebras and the corresponding $W$-algebras decompose in products in terms of the Jacobi form $ vartheta_{11}(tau, z)$.
In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called {em PSH-algebras}. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bi-algebra that contains a large PSH-algebra that extends Zelevinskys algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.
In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems.
The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using.
Let $K/F$ be a quadratic extension of $p$-adic fields, $sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $pi^{vee}$ the smooth contragredient of $pi$, and by $pi^{sigma}$ the representation $picirc sigma$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $pi^vee otimes pi^sigma$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields.
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