## A problem of Kollár and Larsen on finite linear groups and crepant resolutions.(English)Zbl 1257.14013

Let $$V=\mathbb C^d$$. The following notion of age of elements of $$\text{GL}(V)$$ originates from the work of M. Reid: If an element $$g\in \text{GL}(V)$$ is conjugate to $$\text{diag}(e^{2\pi i r_1},\ldots, e^{2\pi i r_d})$$, where $$0\leqslant r_j<1$$, then, by definition, $$\text{age}(g):=\sum_{j=1}^{d}r_j$$. In this paper, the authors solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age $$\leqslant 1$$. They prove the following theorems:
Theorem 1.4. Let $$d\geqslant 11$$ and let $$G<\text{GL}(V)$$ be a finite irreducible subgroup. Assume that, up to scalars, $$G$$ is generated by its elements with $$\text{age}\leqslant 1$$. Then $$G$$ contains a complex bireflection of order $$2$$ or $$3$$, and one on the following statements holds.
(i) $$Z(G)\times {\mathsf A}_{d+1}\leqslant G\leqslant (Z(G)\times {\mathsf A}_{d+1})\cdot 2$$, with $${\mathsf A}_{d+1}$$ acting on $$V$$ as on its deleted natural permutation module.
(ii) $$G$$ preserves a decomposition $$V=V_1\oplus\cdots \oplus V_d$$, with $$\dim V_i=1$$ and $$G$$ inducing either $${\mathsf S}_d$$ or $${\mathsf A}_d$$ while permuting the $$d$$ subspaces $$V_1,\ldots, V_d$$.
(iii) $$2\,|\,d$$, and $$G=D: {\mathsf S}_{d/2}< \text{GL}_2(\mathbb C)\wr {\mathsf S}_{d/2}$$, a split extension of $$D<\text{GL}_2(\mathbb C)^{d/2}$$ by $${\mathsf S}_{d/2}$$. Furthermore, if $$g\in G\setminus D$$ has $$\text{age}(g)\leqslant 1$$, then $$g$$ is a bireflection (and $$\text{age}(g)=1$$).
Theorem 1.5. Let $$d\geqslant 9$$ and let $$G<\text{GL}(V)$$ be a finite irreducible subgroup. Assume that, up to scalars, $$G$$ is generated by its elements with $$\text{age}\leqslant 1$$, and that $$G$$ contains a scalar multiple of a noncentral element $$g$$ with $$\text{age}(g)<1$$. Then one of the following statements holds.
(i) One of the conclusions (i), (ii) of Theorem 1.4 holds, and $$G$$ contains a scalar multiple of a complex reflection.
(ii) The conclusion (iii) of Theorem 1.4 holds, and, modulo scalars, $$G$$ cannot be generated by its elements of $$\text{age}<1$$.
More generally, the authors bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence, they derive some properties of symmetric spaces $$\text{GU}_d(\mathbb C)/G$$ having shortest closed geodesics of bounded length, and of quotients $${\mathbb C}^d/G$$ having a crepant resolution.

### MSC:

 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 20C15 Ordinary representations and characters

### Keywords:

finite linear group; complex reflection group; quotient

GAP
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### References:

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