Twisted geometric Satake equivalence

Finkelberg, Michael; Lysenko, Sergey

doi:10.1017/s1474748010000034

Cited by 32 publications

(60 citation statements)

References 8 publications

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“…To the covering group G (n) , one may associate a complex dual group which may be distinct from the dual of G (see [36], [17], and [40]). Let B Q : Y × Y → Z be the symmetric bilinear form associated to the quadratic form Q.…”

Section: Brylinski-deligne Extensionsmentioning

confidence: 99%

A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie

2019

Trans. Amer. Math. Soc.

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We study a new lifting of automorphic representations using the theta representation Θ on the 4-fold cover of the symplectic group, Sp 2r (A). This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to Θ.We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to Sp 2r (A). Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of Sp 2r (A).Proof. See [29, Section7.1]. 7.2. Local Root Exchange. Suppose now that F is a non-archimedean local field, and let G be the F -rational points of a split algebraic group over F or finite cover thereof. As in the global setting, we consider unipotent subgroups C, X, Y ⊂ G, and a nontrivial characterAs before, set B = Y C, D = XC, and A = BX = DY . We also assume that these subgroups satisfy the local analogue of the six properties preceding Lemma 7.1 (For a precise statement and proof, see [13, Section 6.1]).Lemma 7.2. Suppose that π is a smooth representation of A, and extend the character ψ C trivially to ψ B on B and ψ D on D. Then we have an isomorphism of C-modules J B,ψ B (π) ∼ = J D,ψ D (π).Moreover, J C,ψ C (π) = 0 ⇐⇒ J B,ψ B (π) = 0 ⇐⇒ J D,ψ D (π) = 0.

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Section: Brylinski-deligne Extensionsmentioning

confidence: 99%

A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie

2019

Trans. Amer. Math. Soc.

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“…A morphism from a collection { 1 L µ n } to another collection { 2 L µ n } is a collection of maps 1 L µ n → 2 L µ n in Perv ζ (X µ n ) compatible with the isomorphisms (21). Let j poles :Ẋ n ֒→ X n be the complement to all the diagonals.…”

Section: The Fs Categorymentioning

confidence: 99%

“…We fix the following data as in ( [32], Section 2.3). Write Gr G = G(F )/G(O) for the affine grassmannian of G. For j ∈ J let L j denote the (Z/2Zgraded purely of parity zero) line bundle on Gr G with fibre det(g j (O) : g j (O) g ) at gG(O) (the definition of this relative determinant is found in [21]). Let E a j be the punctured total space of the pull-back of L j to G(F ).…”

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confidence: 99%

Twisted Whittaker models for metaplectic groups

Lysenko

2017

Geom. Funct. Anal.

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Let G be a reductive group (over an algebraically closed field) equipped with the metaplectic data. In this paper we study the corresponding twisted Whittaker category for G. We construct and study a functor from the latter category to the corresponding category of factorizable sheaves. It plays the role of the restriction functor from the category of representations of the big quantum group to those of the graded small quantum group. We also prove an analog in our setting of the Lusztig-Steinberg tensor product theorem for quantum groups describing the semi-simple part of the Whittaker category as a module over the Hecke algebra.Comment: 77 pages, final version. An erratum to the published version is also adde

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“…B.y 1 ; y 2 / D Q.y 1 C y 2 / .Q.y 1 / C Q.y 2 //: Recently, a number of authors [7,24,26] have found a "dual group" to a metaplectic group, or at least the root datum thereof. The combinatorics of this construction are below.…”

Section: Cartan Datamentioning

confidence: 99%

Split metaplectic groups and their L-groups

Weissman

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We adapt the conjectural local Langlands parameterization to split metaplectic groups over local fields. When Q G is a central extension of a split connected reductive group over a local field (arising from the framework of Brylinski and Deligne), we construct a dual group Q G _ and an L-group L Q G _ as group schemes over Z. Such a construction leads to a definition of Weil-Deligne parameters (Langlands parameters) with values in this L-group, and to a conjectural parameterization of the irreducible genuine representations of Q G. This conjectural parameterization is compatible with what is known about metaplectic tori, Iwahori-Hecke algebra isomorphisms between metaplectic and linear groups, and classical theta correspondences between Mp 2n and special orthogonal groups.

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Twisted geometric Satake equivalence

Cited by 32 publications

References 8 publications

A generalized Theta lifting, CAP representations, and Arthur parameters

A generalized Theta lifting, CAP representations, and Arthur parameters

Twisted Whittaker models for metaplectic groups

Split metaplectic groups and their L-groups

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