# Representations with a reduced null cone

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## Abstract

Let *G* be a complex reductive group and *V* a *G*-module. Let \(\pi: V \rightarrow V/\!\!/G\) be the quotient morphism defined by the invariants and set \(\mathcal{N}(V ):=\pi ^{-1}(\pi (0))\). We consider the following question. Is the null cone \(\mathcal{N}(V )\) reduced, i.e., is the ideal of \(\mathcal{N}(V )\) generated by *G*-invariant polynomials? We have complete results when *G* is \(\mathop{\mathrm{SL}}\nolimits _{2}\), \(\mathop{\mathrm{SL}}\nolimits _{3}\) or a simple group of adjoint type, and also when *G* is semisimple of adjoint type and the *G*-module *V* is irreducible.

## Keywords

Null cone Null fiber Quotient morphism Semisimple groups Representations## Mathematics Subject Classification

20G20 22E46## References

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