# Polynomial invariants for simple algebraic groups

Let $$G$$ be a simple complex algebraic group. Let $$V$$ be a finite-dimensional algebraic representation of $$G$$. Thus, we can write $$V=V_1\oplus \cdots \oplus V_n$$ where $$V_i$$'s are irreducible representations. Let $$I:=\mathbb{C}[V]^G$$ denote the space of invariant polynomials. We know that $$I$$ is a finitely generated $$\mathbb{C}$$-algebra. Let $$V/\!/G:=\mathrm{Spec}(I)$$. Then $$V/\!/G$$ is an affine variety.

Question: What do we know about $$V/\!/G$$? For instance, for which pairs $$(G,V)$$ do we know the dimension of $$V/\!/G$$? For which pairs $$(G,V)$$ is $$V/\!/G$$ isomorphic to an affine space?

The most familiar case is when $$V$$ is the adjoint representation, in which case, $$V/\!/G$$ is an affine space of dimension equal to the rank of $$G$$. Vinberg's invariant theory for finite gradings of Lie algebras provides a generalisation. I'm looking for more examples, or a general theory if there is one.

Of particular interest to me is $$G=\mathrm{Sp}_{2n}$$ and $$V=L(\omega_1)\oplus L(\omega_1)\oplus L(\omega_2)$$ where $$\omega_i$$ is the $$i$$th fundamental weight.

• There is a well-known classification of representations with free module of coinvariants, due to Vinberg, Popov and Kac. This condition is a priori stronger than freeness of the subalgebra of invariants, but is equivalent to it for "equidimensional" (ED) actions. (A linear representation $V$ of $G$ is ED if all fibers of the quotient map $V\to V/\!/G$ have the same dimension.) An overview with references appears in the volume on geometric invariant theory (Algebraic Geometry 4) of the encyclopaedia of mathematics (green VINITI Russian edition or yellow Springer English translation). – Victor Protsak Dec 25 '18 at 0:50

## 1 Answer

Generally, one has $$\dim V//G=\dim V-\dim G$$ but there are exceptions. For simple $$G$$ the exceptions have been classified in

Èlašvili, A. G. Canonical form and stationary subalgebras of points in general position for simple linear Lie groups (MSN English article). Funkcional. Anal. i Priložen. 6 (1972), no. 1, 51–62

Popov, A. M. Stationary subgroups in general position for certain actions of simple Lie groups (MSN English article). Funkcional. Anal. i Priložen. 10 (1976), no. 3, 88–90.

Generally, the quotient $$V//G$$ is singular. For simple $$G$$, the representations with $$V//G$$ being smooth (i.e. isomorphic to an affine space) were classified in

Schwarz, Gerald W. Representations of simple Lie groups with regular rings of invariants (MSN article). Invent. Math. 49 (1978), no. 2, 167–191

Concerning your example $$G=Sp(2n)$$ and $$V=L(\omega_1)⊕L(\omega_1)⊕L(\omega_2)$$: Here $$V//G$$ is smooth of dimension $$2n-1=\dim V-\dim G$$ (see the paper of Schwarz above). The ring of invariants is freely generated by polynomials of multidegrees $$(0,0,d)$$, $$d=2,\ldots, n$$ and $$(1,1,d)$$, $$d=0,\ldots,n-1$$. The quotient map $$V\to V//G$$ is equidimensional (by another paper of Schwarz).