[Paper Review] Perverse sheaves on a Loop group and Langlands' duality

Victor Ginzburg|ArXiv.org|Nov 13, 1995

Advanced Algebra and GeometryMathematics23 references232 citations

TL;DR

This paper establishes a geometric Langlands duality equivalence between the category of $G(ar{F}[[z]])$-equivariant perverse sheaves on the affine Grassmannian $\mathrm{Gr} = G(\overline{F}((z)))/G(\overline{F}[[z]])$ and the category of finite-dimensional rational representations of the Langlands dual group $G^\lor$. Using the function-sheaf correspondence and equivariant cohomology, it constructs a tensor equivalence that induces an isomorphism on Grothendieck groups, realizing the Satake isomorphism geometrically via perverse sheaves on an infinite-dimensional variety.

ABSTRACT

An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. The construction is applied to the study of the topology of the affine Grassmannian of G and to establishing a Langlands type correspondence for "automorphic" sheaves on the moduli space of G-bundles.

Motivation & Objective

• To provide a geometric interpretation of the Satake isomorphism using perverse sheaves on the affine Grassmannian.
• To establish a tensor equivalence between the category of $G(\overline{F}[[z]])$-equivariant perverse sheaves on $\mathrm{Gr}$ and the category of finite-dimensional rational representations of the Langlands dual group $G^\lor$.
• To realize the Hecke algebra of $G(\mathbb{Z}_p)\backslash G(\mathbb{Q}_p)/G(\mathbb{Z}_p)$ as the Grothendieck group of perverse sheaves on $\mathrm{Gr}$, via the function-sheaf correspondence.
• To extend the classical Satake isomorphism to a geometric setting by replacing $p$-adic fields with complex formal power series, enabling the use of algebraic geometry and perverse sheaf theory.

Proposed method

• Introduce the affine Grassmannian $\mathrm{Gr} = G(\overline{F}((z)))/G(\overline{F}[[z]])$ as a complex algebraic variety with a $G(\overline{F}[[z]])$-action.
• Define the category $P(\mathrm{Gr})$ of semisimple $G(\overline{F}[[z]])$-equivariant perverse sheaves on $\mathrm{Gr}$, equipped with a convolution tensor product structure.
• Apply the function-sheaf correspondence to assign to each perverse sheaf $\mathcal{F} \in P(\mathrm{Gr})$ a function $\chi_{\mathcal{F}}$ on $G(\mathbb{F}((z)))/G(\mathbb{F}[[z]])$, inducing an isomorphism $\mathbb{C} \otimes_{\mathbb{Z}} K(P(\mathrm{Gr})) \simeq \mathbb{C}[G(\mathbb{F}[[z]])\backslash G(\mathbb{F}((z)))/G(\mathbb{F}[[z]])]$.
• Use the Satake isomorphism and its reinterpretation via the Langlands dual group $G^\lor$, identifying $\mathbb{C}[X^*(T^\lor)]^W \simeq \mathbb{C}[G^\lor]^{G^\lor}$, the algebra of polynomial class functions on $G^\lor$.
• Establish a canonical isomorphism between the hypercohomology of a perverse sheaf $\mathcal{P}(V)$ and the underlying vector space of the corresponding representation $V \in \mathrm{Rep}_{G^\lor}$.
• Apply equivariant cohomology techniques, including the localization theorem and the K"{u}nneth formula, to analyze the cohomology of equivariant complexes and relate them to ordinary cohomology via specialization at regular elements of the Lie algebra.

Experimental results

Research questions

• RQ1 How can the Satake isomorphism between the Hecke algebra of $G(\mathbb{Z}_p)\backslash G(\mathbb{Q}_p)/G(\mathbb{Z}_p)$ and the $W$-invariant part of the group algebra of the coweight lattice be realized geometrically via perverse sheaves on the affine Grassmannian?
• RQ2 What is the precise categorical equivalence that underlies geometric Langlands duality in the context of loop groups and their perverse sheaves?
• RQ3 How does the function-sheaf correspondence on the infinite-dimensional variety $\mathrm{Gr}$ recover the class functions on the Langlands dual group $G^\lor$?
• RQ4 Can the representation theory of $G^\lor$ be reconstructed from the equivariant cohomology of perverse sheaves on $\mathrm{Gr}$, and if so, how?
• RQ5 What role does the equivariant derived category and the localization theorem play in relating the cohomology of perverse sheaves to the representation theory of $G^\lor$?

Key findings

• There exists a tensor equivalence of categories $P(\mathrm{Gr}) \simeq \mathrm{Rep}_{G^\lor}$, which induces the isomorphism $\mathbb{C} \otimes_{\mathbb{Z}} K(P(\mathrm{Gr})) \simeq \mathbb{C} \otimes_{\mathbb{Z}} K(\mathrm{Rep}_{G^\lor})$ on Grothendieck groups.
• The underlying vector space of a representation $V \in \mathrm{Rep}_{G^\lor}$ is canonically isomorphic to the hypercohomology of the corresponding perverse sheaf $\mathcal{P}(V) \in P(\mathrm{Gr})$, providing a geometric realization of the representation space.
• The function-sheaf correspondence assigns to each perverse sheaf $\mathcal{F} \in P(\mathrm{Gr})$ a $G(\mathbb{F}[[z]])$-invariant function on $G(\mathbb{F}((z)))/G(\mathbb{F}[[z]])$, and this correspondence induces an isomorphism with the algebra of class functions on $G^\lor$.
• Equivariant cohomology $H^\bullet_T(M)$ for a $T$-equivariant perverse sheaf $M$ on a $T$-variety $Y$ is a free module over $H^\bullet(BT)$, and its specialization at a regular point $t \in \mathfrak{t}$ yields $\mathrm{gr}^W H_t(M) \cong H^\bullet(M)$, the ordinary cohomology of $M$.
• The localization theorem identifies the equivariant cohomology $H_t(Y,M)$ with the cohomology of the fixed point subvariety $Y^T$, via the pushforward $i_!$ from $Y^T$ to $Y$, under the condition that $t$ is regular.
• The K"{u}nneth formula holds for equivariant cohomology: $H^\bullet_T(M \boxtimes M') \cong H^\bullet_T(M) \mathbin{\otimes_{H^\bullet(BT)}} H^\bullet_T(M')$, enabling decomposition of cohomology for products of spaces.