$\boldsymbol{e}$-Central Elements

Abstract

In order to practise a more detailed analysis of monomial representations, we suppose in this chapter that $G = \text{exp}\ \mathfrak{g}$ is a connected and simply connected nilpotent Lie group with Lie algebra $\mathfrak{g}$. Let us introduce e-central elements due to Corwin and Greenleaf [17]. Let

$$\displaystyle{ \{0\} = \mathfrak{g}_{0} \subset \mathfrak{g}_{1} \subset \cdots \subset \mathfrak{g}_{n-1} \subset \mathfrak{g}_{n} = \mathfrak{g},\ \mbox{ dim}(\mathfrak{g}_{k}) = k\ (0 \leq k \leq n) }$$

(9.1.1)

be a composition series of ideals of $\mathfrak{g}$. Let {X _j }_{1 ≤ j ≤ n} be a Malcev basis of $\mathfrak{g}$ according to this composition series, i.e. $X_{j} \in \mathfrak{g}_{j}\setminus \mathfrak{g}_{j-1}\ (1 \leq j \leq n)$ and $\{X_{j}^{{\ast}}\}_{1\leq j\leq n}$ its dual basis in $\mathfrak{g}^{{\ast}}$. We denote the coordinates of $\ell\in \mathfrak{g}^{{\ast}}$ by $(\ell_{1},\ldots,\ell_{n}),\ell_{j} =\ell (X_{j})$. Then $\mathfrak{g}_{j}^{\perp } = \langle X_{j+1}^{{\ast}},\ldots,X_{n}^{{\ast}}\rangle _{\mathbb{R}} \subset \mathfrak{g}^{{\ast}},\mathfrak{g}_{j}^{{\ast}}\mathop{\cong}\mathfrak{g}^{{\ast}}/\mathfrak{g}_{j}^{\perp }$ and the projection $p_{j}: \mathfrak{g}^{{\ast}}\rightarrow \mathfrak{g}_{j}^{{\ast}}$ intertwines the actions of G on $\mathfrak{g}^{{\ast}}$ and $\mathfrak{g}_{j}^{{\ast}}$. For $\ell\in \mathfrak{g}^{{\ast}}$, we define $e_{j}(\ell) = \mbox{ dim}\big(G\cdot p_{j}(\ell)\big),e(\ell) = (e_{1}(\ell),\ldots,e_{n}(\ell))$ and set $\mathcal{E} =\{ e(\ell);\ell\in \mathfrak{g}^{{\ast}}\}$. We also recognize e _j (ℓ) = dim(G _j ⋅ ℓ) with $G_{j} = \text{exp}(\mathfrak{g}_{j})$. In fact,

$$\displaystyle\begin{array}{rcl} & & \mbox{ dim}\big(G\cdot p_{j}(\ell)\big) = \mbox{ dim}\big(\mathfrak{g}/\mathfrak{g}_{j}^{\ell}\big) = \mbox{ dim}(\mathfrak{g}/\mathfrak{g}(\ell)) -\mbox{ dim}\big(\mathfrak{g}_{ j}^{\ell}/\mathfrak{g}(\ell)\big) {}\\ & & = \mbox{ dim}(\mathfrak{g}/\mathfrak{g}(\ell)) -\big (\mbox{ dim}(\mathfrak{g}/\mathfrak{g}(\ell)) -\mbox{ dim}\big(\mathfrak{g}_{j}/\mathfrak{g}_{j}(\ell)\big)\big) = \mbox{ dim}(G_{j}\cdot \ell). {}\\ \end{array}$$

For $e \in \mathcal{E}$ we define the G-invariant layer $U_{e} =\{\ell\in \mathfrak{g}^{{\ast}};e(\ell) = e\}$. With e ₀ = 0 we define the set of jump indices $S(e) =\{ 1 \leq j \leq n;e_{j} = e_{j-1} + 1\}$ and that of non-jump indices $T(e) =\{ 1 \leq j \leq n;e_{j} = e_{j-1}\}$. $\mathcal{U}(\mathfrak{g})$ being the enveloping algebra of $\mathfrak{g}_{\mathbb{C}}$, $A \in \mathcal{U}(\mathfrak{g})$ is called an e-central element if, with $\pi _{\ell} =\hat{\rho } _{G}(\ell)$, π _ℓ (A) = d π _ℓ (A) is a scalar operator for any ℓ ∈ U _e .