Geometric Langlands duality and representations of algebraic groups over commutative rings.

*(English)*Zbl 1138.22013
Ann. Math. (2) 166, No. 1, 95-143 (2007); erratum ibid. 188, No. 3, 1017-1018 (2018).

From the authors’ introduction: In this paper we give a geometric version of the Satake isomorphism [I. Satake, Publ. Math., Inst. Hautes Étud. Sci. 18, 229–293 (1963; Zbl 0122.28501)]. As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and co-roots appear in a symmetric manner, and so the connected reductive algebraic groups come in pairs. If \(G\) is a reductive group, we write \(\check G\) for its companion and call it the dual group. The notion of dual group itself does not appear in Satake’s paper, but it was introduced by Langlands, together with its various elaborations [R. P. Langlands, Lect. Notes Math. 170, 18–61 (1970; Zbl 0225.14022)] and is a cornerstone of the Langlands program. It also appeared later in physics. In this paper we discuss the basic relationship between \(G\) and \(\check G\).

Let \(G\) be a reductive algebraic group over the complex numbers. We write \(G_{\mathfrak O}\) for the group scheme \(G(\mathbb C[[z]])\) and \(\mathfrak{Gr}\) for the affine Grassmannian of \(G(\mathbb C((z)))/G(\mathbb C[[z]])\); the affine Grassmannian is an ind-scheme, i.e. a direct limit of schemes. Let \(k\) be a Noetherian, commutative unital ring of finite global dimension. Let us write \(P_{G_{\mathfrak O}}(\mathfrak{Gr},k)\) for the category of \(G_{\mathfrak O}\)-equivariant perverse sheaves with \(k\)-coefficients. Furthermore, let \(\text{Rep}_{\check G_k}\) stand for the category of \(k\)-representations of \(\check G_k\); here \(\check G_k\) denotes the canonical smooth split reductive group scheme over \(k\) whose root datum is dual to that of \(G\). The goal of this paper is to prove that the categories \(P_{G_{\mathfrak O}}(\mathfrak{Gr},k)\) and \(\text{Rep}_{\check G_k}\) are equivalent as tensor categories’.

Let \(G\) be a reductive algebraic group over the complex numbers. We write \(G_{\mathfrak O}\) for the group scheme \(G(\mathbb C[[z]])\) and \(\mathfrak{Gr}\) for the affine Grassmannian of \(G(\mathbb C((z)))/G(\mathbb C[[z]])\); the affine Grassmannian is an ind-scheme, i.e. a direct limit of schemes. Let \(k\) be a Noetherian, commutative unital ring of finite global dimension. Let us write \(P_{G_{\mathfrak O}}(\mathfrak{Gr},k)\) for the category of \(G_{\mathfrak O}\)-equivariant perverse sheaves with \(k\)-coefficients. Furthermore, let \(\text{Rep}_{\check G_k}\) stand for the category of \(k\)-representations of \(\check G_k\); here \(\check G_k\) denotes the canonical smooth split reductive group scheme over \(k\) whose root datum is dual to that of \(G\). The goal of this paper is to prove that the categories \(P_{G_{\mathfrak O}}(\mathfrak{Gr},k)\) and \(\text{Rep}_{\check G_k}\) are equivalent as tensor categories’.

Reviewer: Anatoly N. Kochubei (Kyïv)