'Nudging' overset characters downwards

\documentclass{minimal}
\usepackage{amsmath}
\usepackage{lipsum}
\usepackage{tikz}

% Adjust the -0.75ex to taste
\newcommand{\tightoverset}[2]{%
  \mathop{#2}\limits^{\vbox to -.5ex{\kern-0.75ex\hbox{$#1$}\vss}}}

% Just for fun, a tikz solution
\newcommand{\tikzoverset}[2]{%
  \tikz[baseline=(X.base),inner sep=0pt,outer sep=0pt]{%
    \node[inner sep=0pt,outer sep=0pt] (X) {$#2$}; 
    \node[yshift=1pt] at (X.north) {$#1$};
}}

\begin{document}
\lipsum[1]
$\overset{\rightharpoonup}{\Gamma_{\!G}}$
$\tightoverset{\rightharpoonup}{\Gamma_{\!G}}$
$\tikzoverset{\rightharpoonup}{\Gamma_{\!G}}$
\lipsum[2]

\end{document}

(This will also give you a larger harpoon, which is closer to your Mathematica example.)

You can reduce the height using \smash and insert a fake height to move the accent up/down:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}

$\overset{\rightharpoonup}{\Gamma_{\!G}}\ 
   \overset{\rightharpoonup}{\vphantom{a}\smash{\Gamma_{\!G}}}$

\end{document}

\vphantom{a} is just to insert an object of height a (no ascenders). Alternatively, you could use \rule{0pt}{.6ex}, say.

You may want to consider using the \overrightharp command that's provided by the harpoon package. The symbol's overall length adapts automatically to the width of its argument.

Note that unlike most math-accent symbols, the \overrightharp macro operates in text mode. If it's going to be used regularly in math contexts, it's probably a good idea to create a wrapper macro, say \orh, that operates correctly while in math mode. The following MWE shows how this might be done:

\documentclass{standalone}
\usepackage{harpoon,amsmath}
\newcommand*{\orh}[1]{\text{\overrightharp{\ensuremath{#1}}}}
\begin{document}
\overrightharp{$\Gamma_{\!G}$} and $\orh{\Gamma_{\!G}}$
\end{document}

Separately, if the term \Gamma_{\!G} is going to be used frequently in your document, you may want to create a dedicated macro, say \def\GG{\Gamma_{\!G}}. This will save you you a lot of typing (and avoid lots of opportunities for creating typos...) in the document itself.