# Du Val singularity of type G=A,D,E and “small” representations of G

We all know that a simple singularity $W_G(x_1,x_2,x_3)=0$ of type G=A,D,E has the following nice deformation involving the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$. Namely, there is a polynomial function $W_G(x_1,x_2,x_3;h)$ on $\mathbb{C}^3 \times \mathfrak{h}$ invariant under the Weyl group such that

1. $W_G(x_1,x_2,x_3;0)=W_G(x_1,x_2,x_3)$
2. the surface $W_G(x_1,x_2,x_3;h)=0$ inside $\mathbb{C}^3$ for a given $h\in\mathfrak{h}$ is singular iff there is a root $\rho$ of $\mathfrak{g}$ such that $\rho\cdot h=0$.

So far, it's really a classic result by now. My question concerns how we add the data of a representation of $G$ to the story.

Physics suggests that for a representation $R$ of $G$ which is "sufficiently small" in the sense that the quadratic Casimir of $R$ is smaller or equal to half of the quadratic Casimir of the adjoint representation, there is a polynomial function $X_R(x_1,x_2,x_3;h;m)$ on $\mathbb{C}^3\times \mathfrak{h}\times \mathbb{C}$ invariant under the Weyl group such that

The three-dimensional hypersurface $W_G(x_1,x_2,x_3;h)+ t X_R(x_1,x_2,x_3;h;m) =0$ inside $\mathbb{C}^4$ (with coordinates $x_i$ and $t$) is singular if there is a weight $w$ of $R$ such that $w\cdot h=m$.

And indeed, physicists did construct $X_R$ satisfying these properties one by one "by hand" in the late 90s, using various string dualities, see e.g. this and this.

One easy example is $G=A_{N-1}$ and $R$: the defining $N$ dimensional representation. Let's parameterize $\mathfrak{h}$ by $a_1,\ldots,a_N$ s.t. $\sum a_i=0$. Then

1. $W_G(x_1,x_2,x_3;a_i)=\prod_i (x_1-a_i) + x_2^2+x_3^2$
2. $X_R(x_1,x_2,x_3;a_i;m)=(x_1-m).$

And indeed, when $m=a_i$, $W_G + t X_R \sim (t-c) (x_1-a_i) + x_2^2 + x_3^2$ for some $c$, and becomes singular.

Now my question is: is the existence of such an $X_R$ a known fact in singularity theory? If so, which books or articles should I have a look at?

--- an update --- If you'd like to know the explicit forms of $X_R$ for various $R$, please see Appendix A of arXiv:1108.2315.

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It doesn't sound at all familiar to me.... –  Ben Webster Jul 8 '11 at 21:48
I'd second Ben's remark. This question is intriguing but illustrates once again the distance between mathematical physics and classical pure mathematics. By the way, I guess G denotes a simple complex Lie (or algebraic) group of the given type? And a tag rt.representation-theory would be useful, along with maybe lie-groups or algebraic-groups. –  Jim Humphreys Oct 21 '11 at 18:24
The relation between $2$-dimensional Rational Double Points (i.e, Du Val singularities of surfaces) and the Weyl group of the corresponding Dynkin diagram is nowadays well-known in singularity theory and algebraic geometry.
Thanks, but that paper is about $W_G$, not about $X_R$, right? Sorry for my unclear phrasing of the question; I tried to clarify it by editing it. –  Yuji Tachikawa Jul 8 '11 at 15:33