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I am wondering if there is a good motivation for geometric representation theory from within the questions of classical representation theory.

In other words, I'd be curious to see something using geometry that is "meatier" than, say, just using geometric techniques to construct the exceptional isomorphisms between low-dimensional Lie groups --- but something that can still be stated in the framework of classical representation theory (unlike, say, the Borel-Weil theorem and friends).

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Does the geoemetric proof of Gabriel's theorem on the representation type of quivers give a good motivation? It is an immensely natural argument and gets you right there into GRT. –  Mariano Suárez-Alvarez Apr 25 '13 at 4:58
Kazhdan--Lusztig conjectures? en.wikipedia.org/wiki/Kazhdan%E2%80%93Lusztig_polynomial –  Simon Wadsley Apr 25 '13 at 7:55
As a soft question, this might be designated community-wiki? Clearly there's more than one answer. For instance, Springer theory comes to mind relative to the traditional representation theory of Weyl groups, including symmetric groups. –  Jim Humphreys Apr 25 '13 at 14:12
I'm a little worried that by "geometric representation theory" you mean things like Borel-Weil, which is very much not what people who call themselves "geometric representation theorists" mean by GRT -- they mean Springer theory, quiver varieties, geometric Satake, etc. –  Allen Knutson May 4 '13 at 13:52

4 Answers 4

I'm struggling to see what the actual question is but here is an example of a non-trivial use of geometry to prove a simple statement in representation theory; moreover, it is the only known way to obtain the result (apologies if this is not what you're after): the $n!$ conjecture states that the dimension of a certain bigraded $S_{n}$-module is $n!$ (in fact, something stronger is true, the bigraded module is the left regular representation). This statement is equivalent to a certain morphism being Gorenstein and Cohen-Macaulay, namely the morphism $\rho: X_{n}\to H_{n}$, where $H_n$ is the Hilbert scheme of n points in $\mathbb{C}^2$ and $X_n$ is the isospectral Hilbert scheme. Mark Haiman gave a proof of the geometric statement in 2000 (math.AG/0010246).

The bigraded module (call it $D_{\mu}$) considered here is the span of partial derivatives (with respect to $x$'s and $y$'s) of a bihomogeneous polynomial $\Delta_{\mu}(x_1,\ldots,x_n;y_1,\ldots,y_n)$, where $\mu$ is a partition of $n$. Furthermore, the 'bigraded multiplicity' of the simple $S_n$-module $V^{\lambda}$ in $D_{\mu}$ gives the coefficients of the Macdonald-Kostka polynomials, thereby proving their positivity (Macdonald's conjecture).

Again, I'm not sure this is an example of 'geometric representation theory' as most people see it, but it's a nice example of using geometry to solve a 'classical' representation theoretic problem. (I suppose my struggle to see the question is more to do with understanding what comes under 'geometric representation theory')

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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).

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Read GIT as GRT :( –  plusepsilon.de Apr 25 '13 at 15:05
I edited that for you (but don't know why you didn't fix it). –  Allen Knutson May 4 '13 at 13:49

Consider triples $(\lambda,\mu,\nu)$ of dominant weights of $G$ such that the irrep $V_\nu$ occurs in $V_\lambda \otimes V_\mu$. Then this space of triples is closed under addition.

Proof. An intertwiner can be identified with a $G$-invariant section of the $(\lambda^*,\mu,\nu)$ equivariant line bundle over $(G/B)^3$. Tensoring two such sections together, we get a third, which is again nonzero because $(G/B)^3$ is reduced and irreducible.

(Moreover, this monoid is finitely generated, also not hard to prove with this approach.)

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Are there simple non-geometric proofs ? –  Alexander Chervov Jul 9 '13 at 14:48

a nice geometric problem to introduce discrete subgroups of semisimple Lie groups and some representation theory is presented in: Armand Borel, Values of indefinite quadratic forms at integral points and flows on spaces of lattices. Bull. Amer. Math. Soc. (N.S.) 32 (1995)

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