论文标题
在海森堡小组上的共同不变差分运算符和最小的表示
Conformally invariant differential operators on Heisenberg groups and minimal representations
论文作者
论文摘要
对于Heisenberg抛物线亚组$ P $的简单真实的Lie Group $ G $，我们研究了相应的退化主要系列表示。对于一定的感应参数，由Barchini，Kable和Zierau构建的二阶差分算子的形式不变系统的内核是一个子代表，事实证明是最小的表示。为了研究这种子代理，我们将海森堡集团傅立叶变换在非紧凑型图片中，并表明它在$ l^2 $ functions的空间中对最小表示产生了新的实现。 Lie代数操作由订单$ \ leq3 $的差分运算符给出，我们找到了构成最低$ k $ type的功能的明确公式。 这些$ l^2 $ -Models以前以$ \ perperatorname {so}（n，n）$，$ e_ {6（6）} $，$ e_ {7（7）} $和$ e_ {8（8）} $ a kazhdan and Savin的组合，用于$ g_ {2（2）$ by Geld， $ \ widetilde {\ peripatorname {sl}}}（3，\ mathbb {r}）$ by torasso，使用不同的方法。我们的新方法为这些情况提供了统一的系统处理，并为$ e_ {6（2）} $，$ e_ {7（-5）} $和$ e_ {8（-24）} $构建了新的$ l^2 $ models $ $ \ widetilde {\ operatoratorname {so}}}（p，q）$，带有$ p \ geq q = 3 $或$ p，q \ geq4 $和$ p+q $偶数。 作为我们建筑的副产品，我们找到了一个明确的公式，用于非平凡的Weyl组元素的组动作，该公式与抛物线子组的简单动作一起生成$ g $。
For a simple real Lie group $G$ with Heisenberg parabolic subgroup $P$, we study the corresponding degenerate principal series representations. For a certain induction parameter the kernel of the conformally invariant system of second order differential operators constructed by Barchini, Kable and Zierau is a subrepresentation which turns out to be the minimal representation. To study this subrepresentation, we take the Heisenberg group Fourier transform in the non-compact picture and show that it yields a new realization of the minimal representation on a space of $L^2$-functions. The Lie algebra action is given by differential operators of order $\leq3$ and we find explicit formulas for the functions constituting the lowest $K$-type. These $L^2$-models were previously known for the groups $\operatorname{SO}(n,n)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$ by Kazhdan and Savin, for the group $G_{2(2)}$ by Gelfand, and for the group $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ by Torasso, using different methods. Our new approach provides a uniform and systematic treatment of these cases and also constructs new $L^2$-models for $E_{6(2)}$, $E_{7(-5)}$ and $E_{8(-24)}$ for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups $\widetilde{\operatorname{SO}}(p,q)$ with either $p\geq q=3$ or $p,q\geq4$ and $p+q$ even. As a byproduct of our construction, we find an explicit formula for the group action of a non-trivial Weyl group element that, together with the simple action of a parabolic subgroup, generates $G$.
