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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.

I think that the primary sources are Brieskorn's paper Singular elements of semi-simple algebraic groups and at the references given there.

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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
@Yuji: As Francesco points out, the group-theoretic realization of simple singularities has been worked out by Brieskorn, and later by his student Peter Slodowy in more detailed ways. The setting here is mainly the internal structure and adjoint action of a simple algebraic group in characteristic 0. But the literature may or may not help directly to answer your question. – Jim Humphreys Oct 21 '11 at 18:29

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