Hecke $L$-functions and Fourier coefficients of covering Eisenstein series

Gao, Fan

doi:10.4171/dm/819

Cited by 3 publications

(1 citation statement)

References 45 publications

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“…The following result regarding W γ t is shown in [24,38,13] for coverings of GL r . For a general covering group, the idea is the same; it is implicit in [37] and explicated in [17].…”

Section: Unramified Whittaker Functionmentioning

confidence: 99%

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

Gao¹

2021

J. Inst. Math. Jussieu

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For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

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“…The following result regarding W γ t is shown in [24,38,13] for coverings of GL r . For a general covering group, the idea is the same; it is implicit in [37] and explicated in [17].…”

Section: Unramified Whittaker Functionmentioning

confidence: 99%

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

Gao¹

2021

J. Inst. Math. Jussieu

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Local Coefficients and Gamma Factors for Principal Series of Covering Groups

Gao¹,

Shahidi²,

Szpruch³

2023

Memoirs of the AMS

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We consider an n n -fold Brylinski–Deligne cover of a reductive group over a p p -adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of G L 2 GL_2 to S L 2 SL_2 . In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.

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Rankin–Selberg Integrals and L-Functions for Covering Groups of General Linear Groups

Kaplan

2022

International Mathematics Research Notices

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Let ${\operatorname {GL}}_c^{(m)}$ be the covering group of ${\operatorname {GL}}_c$, obtained by restriction from the $m$-fold central extension of Matsumoto of the symplectic group. We introduce a new family of Rankin–Selberg integrals for representations of ${\operatorname {GL}}_c^{(m)}\times {\operatorname {GL}}_k^{(m)}$. The construction is based on certain assumptions, which we prove here for $k=1$. Using the integrals, we define local $\gamma $-, $L$-, and $\epsilon $-factors. Globally, our construction is strong in the sense that the integrals are truly Eulerian. This enables us to define the completed $L$-function for cuspidal representations and prove its standard functional equation.

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Hecke $L$-functions and Fourier coefficients of covering Eisenstein series

Cited by 3 publications

References 45 publications

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

Local Coefficients and Gamma Factors for Principal Series of Covering Groups

Rankin–Selberg Integrals and L-Functions for Covering Groups of General Linear Groups

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