论文标题
弱Weyl换向关系的表示
Representations of the weak Weyl commutation relation
论文作者
论文摘要
让$ g $成为一个本地紧凑的Abelian Group，其中Pontraygin Dual $ \ wideHat {g} $。假设$ p $是包含身份元素$ 0 $的$ g $的封闭子群。我们假设$ p $具有密集的内饰，$ p $会产生$ g $。令$ u：= \ {u_χ\} _ {χ\ in \ wideHat {g}} $是一个非常连续的一级人员组，让$ v：= \ {v_ {v_ {a} \} \} _ {a {a \ in P} $是一个强烈的ISOMETOMERTIES的连续连续的半group。如果\ [u_χv_{a} =χ（a）v_ {a}u_χ\ \ \]，我们将$（u，v）$称为弱的Weyl对。 我们根据上述换向关系的表示理论（阶乘和不可约表示）的假设是$ \ {v_ {a} v_ {a} v_ {a}^{*}：a in p \} $是通勤的项目家庭。这不仅概括了[4]和[5]的结果，我们的证明还带出了结果背后的莫里塔对等。对于$ p = [0，\ infty）\ times [0，\ infty）$，我们证明，如果我们放弃对范围投影的交换性假设，那么弱魏尔换向关系的表示理论就会变得非常复杂。
Let $G$ be a locally compact abelian group with Pontraygin dual $\widehat{G}$. Suppose $P$ is a closed subsemigroup of $G$ containing the identity element $0$. We assume that $P$ has dense interior and $P$ generates $G$. Let $U:=\{U_χ\}_{χ\in \widehat{G}}$ be a strongly continuous group of unitaries and let $V:=\{V_{a}\}_{a \in P}$ be a strongly continuous semigroup of isometries. We call $(U,V)$ a weak Weyl pair if \[ U_χV_{a}=χ(a)V_{a}U_χ\] for every $χ\in \widehat{G}$ and for every $a \in P$. We work out the representation theory (the factorial and the irreducible representations) of the above commutation relation under the assumption that $\{V_{a}V_{a}^{*}:a \in P\}$ is a commuting family of projections. Not only does this generalise the results of [4] and [5], our proof brings out the Morita equivalence that lies behind the results. For $P=[0,\infty)\times [0,\infty)$, we demonstrate that if we drop the commutativity assumption on the range projections, then the representation theory of the weak Weyl commutation relation becomes very complicated.
