Character of Verma Module I'm currently learning about the representation theory of semisimple Lie algebras from the lecture notes supplied by my professor. However, as they are partially incomplete, I am consulting the lecture notes by Wolfgang Soergel in case of questions.
My question regards the character of a Verma module, so I'll give the definitions I'm working with.
Let $\mathfrak{g}$  be a complex finite-dimensional semisimple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra in $\mathfrak{g}$. Let $R^+$ denote the set of positive roots. For some $\lambda \in R^+$ consider the Verma module $\Delta(\lambda)$ with highest weight $\lambda$.
We introduce a partial order on $\mathfrak{h}^*$ (dual space) by $\lambda \leq \mu$ iff $\mu - \lambda \in \mathbb{N}R^+$. This allows us to define the partial completion of the group algebra associated to the additive group of $\mathfrak{h}^*$:
$\widehat{\mathbb{Z}[\mathfrak{h}]} := \lbrace (c_\lambda) \in \prod\limits_{\lambda \in \mathfrak{h}^*} \mathbb{Z}e^\lambda \, \vert \, \exists \mu_1, \dots, \mu_n \in \mathfrak{h}^*: c_\lambda \neq 0 \Rightarrow \exists i \in {1, \dots, n}: \lambda \leq \mu_i \rbrace$,
where $e^\lambda$ denotes the basis vector associated to $\lambda$.
Let $V$ be a weight module over $\mathfrak{g}$ such that the weights are suitably bounded from above by some $\mu_1, \dots, \mu_n \in \mathfrak{h}^*$ and denote by $V_\lambda$ its weight space with regards to the weight $\lambda \in \mathfrak{h}^*$. We define the character of $V$ to be
$$ \operatorname{ch} V := \sum\limits_{\lambda \in \mathfrak{h}^*} \dim V_\lambda \cdot e^\lambda \in \widehat{\mathbb{Z}[\mathfrak{h}^*]}$$
My difficulty arises when trying to compute the character of $\Delta(\lambda)$. One has $\dim \Delta(\lambda)_\mu = P(\lambda - \mu)$, where $P$ denotes the Kostant partition function (it assigns to a $\nu \in \mathfrak{h}^*$ the number of ways one can write $\nu$ as a sum of positive roots). My first steps in the calculation of the character would be to use this identity as follows:
$$ \operatorname{ch} \Delta(\lambda) = \sum\limits_{\mu \in \mathfrak{h}^*} \dim \Delta(\lambda)_\mu \cdot e^\mu
= \sum\limits_{\mu \in \mathfrak{h}^*} P(\lambda - \mu) \cdot e^\mu$$
However, both my professor and Soergel start off like this:
$$ \operatorname{ch} \Delta(\lambda) = \sum\limits_{\mu \in \mathfrak{h}^*} P(\lambda - \mu) \cdot e^{\lambda - \mu}$$
I can't explain this shift in indices. It is relevant as later in the calculation the following identity is used:
$$\prod\limits_{\alpha \in R^+} (1 + e^{-\alpha} + e^{-2\alpha} + \dots) = \sum\limits_{\mu \in \mathfrak{h}^*}P(-\mu) \cdot e^{-\mu}$$
which obviously requires the same index in the argument of $P$ and in the exponent of $e$.
I haven't been able to find this result in Humphreys or Hilgert-Neeb, so I would highly appreciate if anyone could clarify my misunderstanding here or point me to a place in the literature where this argument is properly spelled out.
If this isn't the right place to ask this kind of question, I apologize. This is my first question on MSE and I'm not super acquainted with the structures yet.
 A: First, since your answer does not contain the precise definition of Verma module I will fix the convention here, with axioms that incorporate the minimal assumptions needed and seem as natural as possible (at least, to me; there is some disagreement in the literature around whether or not to build a certain $\rho$-shift into the definition). I am not sure precisely where your confusion is, but comparing with the calculation below will surely clear it up.
Given a Lie algebra $\mathfrak{g}$ over $\mathbf{C}$ equipped with a direct sum decomposition
$$\mathfrak{g}=\mathfrak{n}^- \oplus \mathfrak{b},$$ where $\mathfrak{n}^-$ and $\mathfrak{b}$ are subalgebras, and a character $\lambda: \mathfrak{b} \to \mathbf{C}$, thought of as a one-dimensional representation of $\mathfrak{b}$ on $\mathbf{C}$, we form the induced module
$$\Delta(\lambda)=\mathrm{Ind}_{U(\mathfrak{b})}^{U(\mathfrak{g})}(\mathbf{C}).$$ If $f_{\beta_i}$, where $i=1,2,\dots,N$ (notation chosen as suggestively as possible), is a $\mathbf{C}$-basis of $\mathfrak{n}^-$ and we fix a basis element $v \in \mathbf{C}$ of the one-dimensional $\mathfrak{b}$-module we are inducing, then the Poincaré-Birkhoff-Witt theorem for Lie algebras implies that the set of monomials
$$\{f_{\beta_1}^{e_1} \cdots f_{\beta_N}^{e_N} v \ | \ e_i \in \mathbf{Z}_{\geq 0} \}$$ are a $\mathbf{C}$-basis of $\Delta(\lambda)$.
If moreover $\mathfrak{t} \subseteq \mathfrak{b}$ is an abelian subalgebra such that
$$[x,f_{\beta_i}]=\beta_i(x) f_{\beta_i} \quad \hbox{for all $x \in \mathfrak{t}$ and $1 \leq i \leq N$,}$$ then the definition of $\Delta(\lambda)$ shows that the $\mathfrak{t}$-action on $\Delta(\lambda)$ is determined by
$$x \cdot f_{\beta_1}^{e_1} \cdots f_{\beta_N}^{e_N} v=(\lambda(x)+\beta(x)) f_{\beta_1}^{e_1} \cdots f_{\beta_N}^{e_N} v \quad \hbox{for all $x \in \mathfrak{t}$,}$$ where
$$\beta=\sum_{i=1}^N e_i \beta_i.$$
In order for the $\mathfrak{t}$-weight spaces to be finite dimensionsal, we impose the further requirement that there exists a partial ordering $<$ on $\mathfrak{t}^*$, compatible with $+$ and such that $\beta_i < 0$ for all $i=1,2,\dots,N$. Now the expression
$$\mathrm{ch}(\Delta(\lambda))=\sum_{\gamma \in \mathfrak{t}^*} \mathrm{dim}(\Delta(\lambda)_\gamma) e^\gamma,$$ where
$$\Delta(\lambda)_\gamma=\{u \in \Delta(\lambda) \ | \ x \cdot u=\gamma(x) u \},$$ makes sense and is equal to
$$\mathrm{ch}(\Delta(\lambda))=e^\lambda \prod_{i=1}^N (1-e^{\beta_i})^{-1},$$ by using the geometric series expansion of each factor together with the above implementation of the PBW theorem.
Of course, all this specializes to the case you are interested in by taking $\beta_1,\dots,\beta_N$ to be the negative roots (ordered in some arbitrary fashion), and moreover applies, mutatis mutandis, e.g. to the case of parabolic Verma modules for Kac-Moody Lie algebras.
