## Kloosterman sheaves for reductive groups.(English)Zbl 1272.14012

Fix a prime $$p$$, a positive integer $$n$$, a finite field extension $$\mathbb F_q/\mathbb F_p$$ of the field with $$p$$ elements, and an element $$a\in\mathbb F_q^\times$$. The Kloosterman sums $\text{Kl}_n(a; q) = (-1)^{n-1} \sum_{x_i\in\mathbb F_q, \prod x_i=a} \exp\left(\frac{2\pi i}p \text{Tr}_{\mathbb F_q/\mathbb F_p}(x_1+\cdots +x_n) \right).$ are of great importance in number theory and have been intensely studied since their introduction by H. Kloosterman in 1926 [Proc. Lond. Math. Soc. (2) 25, 143–173 (1926; JFM 52.0170.02)]. For instance, they occur in the Fourier coefficients of modular forms. A powerful method to study them, and to prove results such as the Weil bound $$| \text{Kl}_n(a; q) | \le nq^{(n-1)/2}$$, is a sheaf-theoretic interpretation (“sheaf-function dictionary”) of Kloosterman sums. This permits to use the tool box of algebraic geometry, and also adds considerable insight from a conceptual point of view. The key is Deligne’s definition of the Kloosterman sheaf $$\text{Kl}_n = R\pi_! \sigma^* \text{AS}_\psi[n-1]$$, where $$\mathbb G_a$$ and $$\mathbb G_m$$ are the additive and multiplicative group over $$\mathbb F_p$$, $$\sigma: \mathbb G_m^n\rightarrow\mathbb G_a$$ is the sum, $$\pi: \mathbb G_m^n\rightarrow \mathbb G_m$$ the product, $$\text{AS}_\psi$$ is the Artin-Schreier local system attached to a nontrivial character $$\psi: \mathbb F_p\rightarrow \mathbb Q_{\ell}(\mu_p)^\times$$ ($$\ell$$ a prime $$\neq p$$). After fixing an embedding $$\iota: \mathbb Q_{\ell}(\mu_p) \rightarrow\mathbb C$$ such that $$\iota(\psi(x)) = \exp(2\pi i/p)$$ for $$x\in\mathbb F_p$$, the Grothendieck-Lefschatz trace formula shows that the Kloosterman sum above can be expressed as the trace of Frobenius $$\text{Frob}_a$$ on the stalk at a geometric point over $$a$$.
The Kloosterman sheaves were studied by Deligne, Katz and others. For example, Katz determined the Zariski closure of the image of the monodromy representation $$\pi_1(\mathbb G_m, \overline{\eta}) \rightarrow \mathrm{GL}_n(\mathbb Q_\ell(\mu_p))$$, where $$\overline{\eta}$$ is a geometric point of $$\mathbb G_m$$. Depending on $$n$$ and $$p$$, the algebraic group arising in this way can be $$\mathrm{SL}_n$$, $$\mathrm{Sp}_n$$, $$\mathrm{SO}_n$$ or, quite surprisingly, of Dynkin type $$G_2$$. The $$G_2$$ case occurs for $$n=7$$, $$p=2$$.
The goal of the paper at hand is to study the following question, raised by Katz: Do all semisimple groups appear as geometric monodromy groups of local systems on $$\mathbb G_m$$? Using the sheaf-function dictionary, one can translate this question to the question whether there exist exponential sums whose equidistribution laws are governed by arbitrary simple groups, in particular exceptional groups.
In the paper under review, a uniform construction of such local systems is given: For a split reductive group $$G^\vee$$ a $$G^\vee$$-local system $$\text{Kl}_{G^\vee}$$ on $$\mathbb G_m$$ is constructed. This sheaf has similar local ramification properties as $$\text{Kl}_n$$, and for $$G^\vee=GL_n$$ the construction yields the sheaf $$\text{Kl}_n$$ as defined by Deligne. The authors determine the Zariski closure of the global geometric monodromy, prove purity of the sheaf and deduce equidistribution laws. They also give a conjecture about the unicity of such local systems.
Roughly speaking, the construction is based on work of B. H. Gross and M. Reeder [Duke Math. J. 154, No. 3, 431–508 (2010; Zbl 1207.11111)] who construct an automorphic representation which should correspond to the (yet to be defined) Kloosterman sheaf. Writing down explicitly a Hecke eigenfunction for this automorphic representation, the authors are able to find a Hecke eigensheaf in terms of which the desired Kloosterman sheaf can be defined.
For more details, we refer to the introduction of the paper.

### MSC:

 14D24 Geometric Langlands program (algebro-geometric aspects) 11L05 Gauss and Kloosterman sums; generalizations 11T23 Exponential sums 22E57 Geometric Langlands program: representation-theoretic aspects

### Citations:

Zbl 1207.11111; JFM 52.0170.02
Full Text:

### References:

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