Specify alignment per column in align equation

Your output is what you'd expect from aligned: you have four columns, the 1st and 3rd right aligned, the 2nd and 4th left aligned, and some space between the first and the second "lr pair". What you actually want is best achieved with a cases environment; @{} is used where you don't want space between the columns.

Note that the vertical space between the lines is smaller than with your aligned code. If you want the same vertical spacing as with aligned, use \\[\jot] instead of \\ in the code below.

\documentclass{article}
\usepackage{mathtools}  % extends amsmath and fixes some of its quirks
\usepackage{amssymb}
\begin{document}
\begin{align}\label{eq:mergefunction_with_weighting}
  \curlyvee(v, \omega) := l_1 \triangleright l_2 = 
  \left\{
    \begin{array}{@{}r@{}l@{}ll}
       &l_1(v)               &                          &\text{wenn $v \in G_1 \setminus G_2$}\\
\omega(&l_1(v)) + (1-\omega)(&\hat{R}l_2(v) +\hat{t})   &\text{wenn $v \in G_1 \cap G_2$}\\
       &                     &\hat{R}l_2(v) +\hat{t}    &\text{wenn $v \in G_2 \setminus G_1$}.
    \end{array}
  \right.
\end{align}
\end{document}

You can also use \begin{cases}...\end{cases} instead of \left\{...\right., but then in this particular example the equation number doesn't fit anymore. Moreover, note that there's no need to use align as the outer environment since that one has no tab stops; I'd use equation instead.

You can use the cases environment and \phantom to achieve the desired alignments

\[
\curlyvee(v, \omega) := 
l_1 \triangleright l_2 = 
\begin{cases}
\phantom{\omega(}l_1(v) &   \text{wenn $v \in G_1 \setminus G_2$}\\
\omega(l_1(v)) + (1-\omega)(\hat{R}l_2(v) +\hat{t})   &\text{wenn $v \in G_1 \cap G_2$}\\
\phantom{\omega(l_1(v)) + (1-\omega)(}\hat{R}l_2(v) +\hat{t} &\text{wenn $v \in G_2 \setminus G_1$}.
\end{cases}
\]

A second option is to use a tabbing environment inside a minipage

\[
\curlyvee(v, \omega) :=  l_1 \triangleright l_2 = 
\begin{cases}
  \begin{minipage}{0cm}
    \begin{tabbing}
      $\omega($\=$l_1(v)) + (1-\omega)($\=$\hat{R}l_2(v)+\hat{t})$ \quad\= \kill
      \> $l_1(v)$ \>\>   wenn $v \in G_1 \setminus G_2$\\
      $\omega(l_1(v)) + (1-\omega)(\hat{R}l_2(v) +\hat{t})$ \>\>\> wenn $v \in G_1 \cap G_2$\\ 
      \>\> $\hat{R}l_2(v) +\hat{t}$ \> wenn $v \in G_2 \setminus G_1$.
    \end{tabbing}
  \end{minipage}
\end{cases}
\]

I think you want to use the alignedat environment instead of aligned. This is similar to the aligned environment in that it provides pairs of r/l equations, but without the inter equation space that the aligned environment adds. So the only changes required to your code as posted above are:

1. Replace the \begin{aligned} with \begin{alignedat}{4}, and the corresponding end{}.

2. Change the second and third alignment point from & to &&. This doubling changes the alignment from right aligned to left aligned which is what you want.

3. Change text{wenn...} to text{ wenn...}, or use \quad\text{wenn...} instead so that you have some space there.

Notes:

• But as the other solutions mention, the outed environment really is a cases (or dcases) environment and should not be an align as you do not have multiple equations that are aligned. You have one equation.

Code:

\documentclass{article}
\usepackage{mathtools}  % extends amsmath and fixes some of its quirks
\usepackage{amssymb}
\begin{document}


\begin{align}\label{eq:mergefunction_with_weighting}
\curlyvee(v, \omega) := 
l_1 \triangleright l_2 = 
\left\{
 \begin{alignedat}{4}
       &l_1(v)               &&                          &&\quad\text{wenn $v \in G_1 \setminus G_2$}\\
\omega(&l_1(v)) + (1-\omega)(&&\hat{R}l_2(v) +\hat{t})   &&\quad\text{wenn $v \in G_1 \cap G_2$}\\
       &                     &&\hat{R}l_2(v) +\hat{t}    &&\quad\text{wenn $v \in G_2 \setminus G_1$}.
 \end{alignedat}
\right.
\end{align}


\end{document}