I interpret GRT as explained here: http://ncatlab.org/nlab/show/geometric+representation+theory

I give you two examples from number theory, in particular from the Langlands program and explain, how geometry might be useful.

The main concern of the Langlands program are automorphic representations of a given reductive group $G$ over a global field $F$, how they can be transferred to different groups (**functoriality**) and how they are related to Galois representations/motives (**correspondence**). One of the key tool in this study is the Arthur trace formula: It relates

**automorphic representations of $G/F$** $\qquad\leftrightarrow\qquad$ **conjugacy classes in $G(F)$.**

**Functoriality**: Automorphic forms per se are in general really hard to attack, specifically those related transcendental Maass cusps forms. Merely to show their existence, Selberg exploited the above comparison (Selberg trace formula). Also, if you want to proof that certain automorphic representation can be *mapped* to an other reductive group $G'$, you can try to *compare* the conjugacy classes of $G(F)$ and $G'(F)$. The Jacquet-Langlands correspondence is proven along these lines. Other famous conjectures, which are known to follow from such *maps* are the Selberg eigenvalue conjecture or more generally the Ramanujan-Petersson conjecture.

**Correspondence**: If you can realize Galois representations geometrically as operators on certain Homology classes, and compare the Lefschetz trace formula with the Arthur trace formula, you can prove an equality of the Artin L-function of the Galois representation with the L-function os some automorphic representation. A famous example is the Shimura-Taniyama conjecture and the proofs of the Langlands correspondence for global function fields by Drinfeld/Lafforgue. I don't know much abut Deligne's proof of the modularity of modular forms of weight $k\geq 2$ and generalizations by Harris-Taylor, etc., but I guess it is the same principle. These give you also special cases of the Ramanujan-Petersson conjectures, simply because the Artin L-function of a Galois representation necessarily satisfies it.

At least morally, one should be interested in concrete models/geometric realizations of irreducible representations. Harish-Chandra has classified all discrete series of semi-simple connected real Lie groups via computing their traces way before Atiyah-Schmid found a geometric realization for all of them via K-theory. The known classification of admissible reps of reductive groups like $GL(n)$ over non-archimedean fields are geometric to my knowledge, i.e., given as induced reps. Also Harish-Chandra himself expressed the Plancherel formula of real reductive Lie groups in terms of geometric data like conjugacy classes, very similar to the ideas in the Arthur trace formula, yielding a tool for a similar local analysis as given in the Langlands program (local for local field).