align inside \left ... \right

There's a solution with Sébastien Gouezel's \MTkillspecial command and \DeclarePairedDelimiter command from mathtools: I define a \brkparens command, which adds an implicit pair of \left … \right in its starred version. For fine tuning, the non-starred version accepts an optional argument (\big, \Big, &c.):

\documentclass[twocolumn]{article}
\usepackage{mathtools}
   \newcommand\MTkillspecial[1]{% helper macro
    \bgroup
    \catcode`\&=9
    \let\\\relax%
    \scantokens{#1}%
    \egroup
    }
    \DeclarePairedDelimiter\brkparens()
    \reDeclarePairedDelimiterInnerWrapper\brkparens{star}{
    \mathopen{#1\vphantom{\MTkillspecial{#2}}\kern-\nulldelimiterspace\right.}
    #2
    \mathclose{\left.\kern-\nulldelimiterspace\vphantom{\MTkillspecial{#2}}#3}}

\begin{document}

\def\leftside{\upsilon_{N, cw}(\omega)}
\def\prefix{- \frac{1}{\Delta} \Gamma}
\def\firstline{i\omega - \xi_{cc}\bar N + \frac{\Delta\Gamma}{2}}
\def\lastline{{+}\:\frac{|\kappa|^2}{2\Delta\Gamma} \exp(-i\omega\tau)}
\begin{align}
    \begin{split}
        \leftside =& \prefix
            \begin{aligned}[t]
                &\brkparens*{\firstline\\
                &\lastline}
            \end{aligned}
    \end{split}
\end{align}

\end{document} 

In the definition of \lastline at line 7, you have one unbalanced \left., you should either remove this or remove the other one repeated at line 13. Also, you may get rid of the split environment since the \\ inside aligned will automatically split the equation over two lines.

\documentclass[twocolumn]{article}
\usepackage{amsmath}
\begin{document}
\def\leftside{\upsilon_{N, cw}(\omega)}
\def\prefix{- \frac{1}{\Delta} \Gamma}
\def\firstline{i\omega - \xi_{cc}\bar N + \frac{\Delta\Gamma}{2}}
\def\lastline{{+}\:\frac{|\kappa|^2}{2\Delta\Gamma} \exp(-i\omega\tau)}

\begin{equation}
  \begin{aligned}
    \leftside = \prefix & \left( \firstline \right. \\
                        & \left. \lastline  \right)
  \end{aligned}
\end{equation}

\end{document}

I'm curious, if the following simple elementary solution enabled by use of package mathtools fit to your expectation:

\documentclass[twocolumn]{article}
\usepackage{mathtools}

\usepackage{lipsum}

\begin{document}
\lipsum[1]
\begin{equation}
  \begin{multlined}
\upsilon_{N, cw}(\omega) 
    = - \frac{1}{\Delta} \Gamma 
        \left(i\omega - \xi_{cc}\bar N + \frac{\Delta\Gamma}{2}\right.       \\
        \left. + \frac{|\kappa|^2}{2\Delta\Gamma} \exp(-i\omega\tau)  \right)
  \end{multlined}
\end{equation}
\lipsum[2]
\end{document}