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1. Dimensions of automorphic representations, L-functions and liftings

   https://doi.org/10.1007/978-3-030-68506-5_3  

   Friedberg, Solomon; Ginzburg, David (January 2021, Relative Trace Formulas)

   Müller, Werner; Shin, Sug Woo; Templier, Nicolas (Ed.)

   The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers. 

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2. A reciprocal branching problem for automorphic representations and global Vogan packets

   https://doi.org/10.1515/crelle-2019-0016  

   Jiang, Dihua; Liu, Baiying; Xu, Bin (August 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G ,determine the occurrence of an irreducible representation σ of H in the restriction of π to H . The reciprocal branching problem of this classical branching problemis to ask: For an irreducible representation σ of H , find an irreducible representation π of G such that σ occurs in the restrictionof π to H . For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture.In this paper, we investigate the reciprocal branching problem for automorphic representationsof special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied toother classical groups as well. 

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3. An exceptional Siegel–Weil formula and poles of the Spin L-function of

   https://doi.org/10.1112/S0010437X20007186  

   Gan, Wee Teck; Savin, Gordan (June 2020, Compositio Mathematica)

   null (Ed.)

   We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $$\text{PGSp}_{6}$$ discovered by Pollack, we prove that a cuspidal representation of $$\text{PGSp}_{6}$$ is a (weak) functorial lift from the exceptional group $$G_{2}$$ if its (partial) Spin L-function has a pole at $s=1$ . 

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4. The cohomology of semi-infinite Deligne–Lusztig varieties

   https://doi.org/10.1515/crelle-2019-0039  

   Chan, Charlotte (January 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ⁢ ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ⁢ ( X h ) ⁢ [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p -adic groups in general. 

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5. On the local converse theorem for split 𝑆𝑂_{2𝑙}

   https://doi.org/10.1090/ert/687  

   Hazeltine, Alexander; Liu, Baiying (March 2025, Representation Theory)

   In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic $p\ne <\#comment/>2$. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply a new idea by considering a certain sum of partial Bessel functions to overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups. 

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